diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbalo" "b/data_all_eng_slimpj/shuffled/split2/finalzzbalo" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbalo" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n It is known that quenched disorder can strongly affect the\n large-scale, long-time behaviour of nonequilibrium driven systems\n with interacting constituents. The interplay of disorder,\n interactions and drive opens up the possibility of new regimes of\n complex and interesting behaviour arising in these systems\n \\cite{Fisher}. In the theoretical effort to delineate and explore\n regimes of new behaviour, an important role is played by simple\n models which capture some features of more complex physical systems.\n In this paper, we study disordered driven diffusive systems by\n analysing stochastically evolving lattice gas models, with quenched\n disordered hopping rates \\cite {prl97}. \n\n Driven diffusive systems in the absence of disorder have been studied\n extensively and are reviewed in \\cite{DDS}. Also, systems with\n disorder and drive but no interactions between particles are well\n studied and understood \\cite{BouGeo}. But there have been only\n sporadic studies of disordered driven diffusive systems of\n interacting particles. It has been argued that strong enough random\n site dilution can substantially affect the transport properties of\n particles with hard-core interactions, and can make the system\n respond nonmonotonically to the driving field \\cite{RB,BR}. On the\n other hand, a low concentration of blocked sites was found\n numerically not to affect the critical behaviour of a driven lattice\n gas with additional attractive inter-particle interactions\n \\cite{Lauritsen}. Finally, a driven lattice gas with a quenched\n noise distribution was studied using field-theoretic techniques in\n \\cite{Janssen}, but the connection of this study with\n particle-conserving disordered lattice gas models is not clear.\n\n In this paper, we study disordered lattice gas models with a view\n towards identifying different sorts of generic behaviour that can\n arise on large scales as a consequence of disorder. The only\n interaction included is the hard-core constraint which limits the\n allowed occupancy of each site. Our results pertain mostly, but not\n exclusively, to one dimension. In the remainder of this\n Introduction, we discuss the different types of behaviour displayed\n by the lattice gas models under study.\n\n We find three distinct regimes in disordered driven diffusive systems\n in one dimension:\n\n In the {\\it Homogeneous} regime, the state of the system is\n characterized by a single density and a nonzero current. Quenched\n disorder induces variations of the density on the microscopic scale,\n of the order of a few lattice spacings. However, the system has a\n macroscopically homogeneous density. In the thermodynamic\n limit, the current approaches a finite value.\n\n In the {\\it Segregated-Density} regime, the state of the system is\n characterized by two distinct values of density, and a nonzero\n current. Besides microscopic-scale variations of the density, there\n are macroscopic regions with differing high and low\n densities. The state is thus characterized by phase separation of\n the density, and a spatially constant time-averaged current which\n remains finite in the thermodynamic limit. \n\n\n In the {\\it Vanishing-Current} regime, the state of the system is\n characterized by two distinct values of the density, and an\n essentially zero current. The hallmark of this regime is that the\n current decreases as the system size increases, and vanishes in\n the thermodynamic limit. This is a consequence of rare but\n rate-limiting backbends, or stretches of bonds which disfavour the\n forward flow of current. The density is inhomogeneous on a\n macroscopic scale.\n\n The density profiles in typical states in each of the three regimes\n are depicted in Figure 1, while Figure 2 shows the variation of the\n current with system size in the three cases.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=1.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{Representative steady-state density profiles \nfor the (a) Homogeneous (b) Segregated-Density and (c) Vanishing-Current\nregimes in the Disordered Asymmetric Exclusion Process (DASEP). }\n\\label{fig:regimes1}\n\\end{figure}\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=2.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\noindent\\caption{Variation of the steady-state current with the system size\nfor the three DASEP regimes of Fig. 1: (a) Homogeneous (circles), (b)\nSegregated-Density (triangles) and (c) Vanishing-Current (squares). In (a)\nand (b), the current approaches a finite value in the thermodynamic limit\nwhereas in (c) the current vanishes as a power of the system size. The\ndashed line corresponds to $J=0.125$ which is the limiting value of the\ncurrent in the regime (b) for the chosen values of the parameters.}\n\\label{fig:regimes2}\n\\end{figure}\n\n Examples of these behaviours are discussed in this paper for two\n types of lattice-gas models, namely the disordered drop-push process\n (DDPP) and the disordered asymmetric simple exclusion process\n (DASEP). The models are defined in detail in Sections \\ref{sec:DDPP}\n and \\ref{sec:DASEP} respectively, but for the purpose of discussion\n here, it suffices to note that the models are similar in that there\n is a maximum occupancy of each site in both, and are different in the\n dynamical moves --- attempted nearest neighbour jumps in\n the DASEP, and slightly longer-ranged leapfrogging moves in the DDPP.\n\n\n The absence of detailed balance, together with the breaking of\n translational invariance, in disordered off-equilibrium systems makes\n the characterization of even the stationary state difficult in\n general. It is shown that the steady state of the disordered\n drop-push process can be found explicitly -- the first such instance\n we are aware of, in a system with disorder, interactions and drive\n \\cite{Krug_Evans}. This determination -- which is based on the\n condition of pairwise balance \\cite{SRB} -- shows that a product\n measure form is valid in all dimensions. The form reflects the\n microscopic inhomogeneities coming from the underlying disorder, and\n results in a macroscopically homogeneous state.\n\n For the disordered asymmetric exclusion process, the steady state\n measure is not analytically characterizable, and we study the problem\n within a site-wise inhomogeneous mean-field theory and by numerical\n simulation. The result depends crucially on whether or not the\n system has backbends, which are stretches of the lattice where the\n local bias is against the particle flow. In the no-backbend case,\n when the average particle density is sufficiently away from 1\/2, the\n spatial profile of the density has microscopic shocks, but is uniform\n on macroscopic scales (Fig.~\\ref{fig:regimes1}a). However, in a\n finite region around half-filling, disorder induces phase separation\n into macroscopic regions of high and low\n density (Fig.~\\ref{fig:regimes1}b). We give approximate arguments to\n understand the origin and nature of this phase separation, and to\n obtain the form of the phase diagram in the current-density plane.\n This sort of behaviour has also been seen earlier in a model with a\n single weak bond \\cite{Janowsky}. We argue that disorder-induced\n phase separation is a generic feature of systems in which the current\n $J$ versus density $\\rho$ shows a maximum at some intermediate\n density, in the absence of disorder.\n\n In the version of the DASEP in which the easy direction of hopping is\n itself a quenched random variable, the model represents a system of\n hard-core particles in a random potential with an overall downward\n tilt, but with backbends of arbitrary length. Long backbends\n severely limit the maximum current that can flow through the system,\n and in fact the current decreases to zero as the system size\n increases (Fig.~\\ref{fig:regimes2}); the system is in the\n vanishing-current regime.\n\n Although our emphasis in this paper is on the analysis of lattice\n models, we comment briefly on certain constraints that are important\n in a continuum description. Such a description is expected to be\n valid for the large-scale, long-time behaviour, and is based on\n stochastic differential equations involving appropriate\n coarse-grained variables. It is argued that quenched randomness is\n manifest in random multiplicative coefficients in a gradient\n expansion. Conservation of particle number -- which implies spatial\n constancy of the current in the steady state -- imposes strong\n constraints on these terms.\n\n In one dimension, using a well known mapping \\cite{solids}, the\n particle models are equivalent to stochastic growth models of a 1-$d$\n interface moving in a 2-$d$ medium. The interface moves with a speed\n proportional to the current in the particle model. The\n disordered jump rates now become local growth rates which are\n disordered in a columnar fashion \n for the moving interface \\cite{columnar}. The three principal\n regimes of behaviour discussed above for the particle models\n translate into distinct regimes for interface motion, namely (i) a\n moving interface with normal roughness, (ii) a moving interface with\n large segments with different mean slopes, and (iii) an interface\n with different-slope segments, which is stationary in the\n thermodynamic limit.\n\n The paper is organized as follows. In Section \\ref{sec:DDPP} we\n define and discuss the steady state properties of the disordered\n drop-push process in arbitrary dimensionality. The disordered\n asymmetric exclusion process with only forward-easy-direction of\n hopping, but quenched random rates, is discussed in Section\n \\ref{sec:DASEP}; the case in which there are some\n backward-easy-direction bonds is discussed in Section\n \\ref{sec:Sinai}. In Section \\ref{sec:continuum} we discuss the\n constraints on a continuum description, while Section\n \\ref{sec:height} discusses the implications of our results for models\n interface growth in the presence of columnar disorder. Section\n \\ref{sec:conclusion} is the conclusion.\n\n\\section{Disordered Drop-Push Process : DDPP} \\label{sec:DDPP}\n\n The drop-push process was initially introduced in \\cite{BR,SRB} as a\n model of activated flow involving transport through a series of traps\n of equal depths. The dynamics consists of activated hops together with\n a cascade of overflows following each move. The disordered version of\n the model may be considered as a discrete model of activated fluid flow\n down an inclined rugged slope with lakes of varying depths; see\n Figs.~\\ref{fig:lakes},\\ref{fig:ddpp-1d}. This is similar to\n above-threshold behaviour of the model considered in\n \\cite{Narayan_Fisher}. In this section we show that the steady state\n and current can be found exactly in all dimensions for the DDPP and its\n generalizations.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=3.eps,width=5.5cm}\n\\end{center}\n\\narrowtext\n\\noindent\\caption{Schematic diagram of water flowing down a rugged\nhill-side. Water from a lake higher up cascades downhill, under the action\nof gravity, until it finds a partially filled lake. The unequal capacities\nof the lakes are the quenched variables in the system.}\n\\label{fig:lakes}\n\\end{figure}\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=4.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{A Disordered Drop-Push Process (DDPP) configuration and move in \n$d=1$. }\n\\label{fig:ddpp-1d}\n\\end{figure}\n\n \\subsection{The model} \\label{sub:defDDPP}\n\n The model in $d$-dimensions is defined on a hypercubic lattice with\n periodic boundary conditions along all the $d$ axes (with unit vectors\n \\{$\\e_\\nu |\\nu=1\\cdots d$\\}). At each site ${r}$ is a well which can hold\n at most $l_\\r$ particles (Figs.~\\ref{fig:ddpp-1d},\\ref{fig:ddpp-2d}) with\n $l_\\r$'s chosen independently from some probability distribution\n $P(l)$. The configuration $\\cal{C}$ of the system is specified by\n specifying the set of occupation numbers \\{$n_\\r$\\} with ($0\\len_\\r\\le\n l_\\r~, \\forall{r}$). Further, with each site ${r}$ is assigned a set\n \\{$\\epsilon(\\nr|\\lr); n_\\r=1,\\cdots,l_\\r$\\} of positive random numbers chosen from\n some given distribution \\cite{well_distbn}. The dynamics is\n stochastic. In a time interval $dt$, with a probability\n $p_{\\pm\\nu}\\epsilon(\\nr|\\lr) dt$, the topmost particle in the well ${r}$ hops out,\n and drops into well ${r}\\pm\\e_\\nu$, i.e. into the adjacent well in the\n $\\pm\\nu$th direction. Here $\\{p_{\\pm\\nu}; \\nu=1,..,d\\}$ are a set of\n site-independent positive numbers satisfying $\\sum_{\\nu=1}^d\n (p_\\nu+p_{-\\nu})=1$. Now, if well ${r}\\pm\\e_\\nu$ is already full, then the\n particle gets pushed further {\\it preserving the direction of the\n initial jump} to the next site and so on. The cascade of transfers\n terminates once a partially full well is encountered. Note that here\n the set of jump-rates \\{$\\epsilon(\\nr|\\lr)$\\} are site-dependent as well as\n functions of the occupation numbers. These rates, together with the\n well-depths \\{$l_\\r$\\}, constitute the quenched random variables in the\n model. The set of probabilities \\{$p_{\\pm\\nu}$\\} determines the\n direction of the global bias ${\\vec E} = \\sum_{\\nu=1}^d\n (p_\\nu-p_{-\\nu})\\e_\\nu$ and, as will be shown in Section\n \\ref{sub:imDDPP}, also the direction and magnitude of the steady state\n current in the model. However, they do not enter the expression for\n the normalized invariant measure.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=5.eps,width=7cm,angle=-90}\n\\end{center}\n\\narrowtext\n\\caption{The DDPP model in $d=2$. The model can be generalized to $d>2$\n(see text). The rates $\\epsilon(\\nr|\\lr)$ depend on the well depth $l_{r}$ as well as\nthe occupation number $n_{r}$.}\n\\label{fig:ddpp-2d}\n\\end{figure}\n\n Though all the results we will discuss holds for any arbitrary choice\n of the $\\epsilon$'s, in a physical system they should be determined\n from the details of the trapping mechanisms etc., e.g. they may be\n taken to be of the Kramers form $\\epsilon(\\nr|\\lr)\\propto exp[-g(l_\\r-n_\\r)]$ for\n situations where the jumps are activated \\cite{kehr87}.\n\n\n \\subsection{Invariant measure} \n \\label{sub:imDDPP} \n\n The time evolution of the probability\n $\\cal{P({\\cal C})}$ for the system to be in configuration ${\\cal C}$ is\n given by the master equation \\cite{vanKamp}\n\\begin{equation} {d\\over dt}{\\cal{P({\\cal C})}}\n =\\sum_{{\\calC^{\\prime\\prime}}}\nW({\\cdprime \\rightarrow \\calC}){\\cal P}({{\\cal C}''})-\\sum_{{\\cal{C}^{\\prime}}} W(\\calC \\rightarrow \\cprime){\\cal P}({\\cal C}).\n\\label{eq:master}\n\\end{equation}\n Here the $W$'s are the transition matrix elements identified with the\n rates $\\epsilon$'s defined in the model. e.g. if the transition\n $\\calC \\rightarrow \\cprime$ involves moving the topmost of the $n_{r}$ particles at ${r}$ to\n ${r^\\prime}$ along the $\\nu$th direction, then\n $W({\\calC \\rightarrow \\cprime})=p_\\nu\\epsilon_{r}(n_{r}|l_{r})$. The steady-state or the\n invariant measure of the dynamics is the set of time independent\n weights $\\{\\mu({\\cal C})\\}$ satisfying (\\ref{eq:master}) above.\n Hence the problem of finding the invariant measure reduces to that of\n finding a set of positive weights $\\{\\mu({\\cal C})\\}$ such that the\n total incoming flux into any configuration ${\\cal C}$ (the first sum\n in (\\ref{eq:master})) equals the total flux out of ${\\cal C}$ (the\n second sum in (\\ref{eq:master})). The uniqueness of the invariant\n measure is ensured by the connectedness property of the $W$-matrix,\n i.e. every configuration can be reached from any other by a sequence\n of transitions \\cite{vanKamp}.\n\n We claim that the (unnormalized) measure of configuration\n ${\\cal C}(\\{n_\\r\\})$ in the steady state has the product form\n\\begin{equation} \\mu({\\cal C})=\\prod_{{r}} u_{r}(n_\\r).\n\\label{eq:imDDPP}\n\\end{equation}\nHere $u_{r}$ are the single-site weights defined as\n\\begin{equation}\n u_{r}(n_\\r) = \\left\\{ \\begin{array}{ll}\n 1 &\\mbox{~~~if~$n_\\r=0$} \\\\\n \\tau_{r}(1)\\cdots \\tau_{r}(n_\\r) &\\mbox{~~~if~$01$}: To generalize the above results to\n $d>1$ we note that for DDPP in $d>1$, the invariant measure\n (\\ref{eq:imDDPP}) is the same if we single out a particular direction,\n say $\\nu$, and allow jumps only along that direction. Together with\n the direction preservation of individual jumps, this allows us to\n write the expression for the current in any dimension:\n\n\\begin{eqnarray}\n\\vec J_0 &=& \\left[\\sum_{\\nu=1}^d (p_\\nu-p_{-\\nu})\\e_\\nu\\right] \\epsilon_0 z \n = \\epsilon_0 z {\\vec E}\n\\label{eq:J0-dd}\n\\end{eqnarray}\n where ${\\vec E}\\equiv\\sum_{\\nu=1}^d (p_\\nu-p_{-\\nu})\\e_\\nu$ is the external\n drive. As in the $d=1$ case, the magnitude and the direction of\n the steady-state current does not depend upon the detailed\n arrangements of the wells.\n\n\\subsection{Static two-point correlation functions}\n\n Because of the product form of the measure, the connected part of\n the equal-time density-density correlation function\n\\begin{equation}\nG_{r}(\\Delta{r}) = \\langle n_{r} n_{{r}+\\Delta{r}}\\rangle-\\langle\nn_{r}\\rangle\\langle n_{{r}+\\Delta{r}}\\rangle\n\\label{eq:partcorr}\n\\end{equation}\n vanishes identically for $\\Delta{r} \\ne 0$. Consequently, the\n fluctuation of the number of particles in $r$ consecutive sites\n along a straight line can be computed exactly:\n\\begin{eqnarray}\n\\Gamma_i^2(r)& = & \\left\\langle\\left[\\sum_{j=i+1}^{i+r}\\left(n_j-\\langle\n n_j\\rangle\\right)\\right]^2 \\right\\rangle \\nonumber\\\\ \n & = & \\sum_{j=i+1}^{i+r}\n \\left( z{\\partial\\over \\partial z}\\right)^2 \\ln Z_j.\n\\label{eq:hhcor-DDPP}\n\\end{eqnarray}\n\\noindent The second step follows from the product form of the measure.\n\n For $d = 1$, a standard mapping discussed in Section VI introduces\n height variables defined by\n\\begin{equation}\nh_i=\\sum_{j\\le i} 2(\\langle n_j\\rangle-n_j).\n\\label{eq:heightdef}\n\\end{equation}\n\\noindent Evidently, $\\Gamma_i^2(r)$ is the equal-time height-height \n correlation $\\langle (h_{i+r}-h_i)^2\\rangle$. Averaging over the\n disorder distribution gives $\\overline {\\Gamma^2}(r)\\sim r$\n implying that the `roughness' exponent $\\alpha$ (defined by\n $\\overline {\\Gamma^2}(r) \\sim r^{2\\alpha}$) is $1\/2$.\n \n \\subsection{Two-rate DDPP model: Explicit results}\n \\label{subsec:2rate}\n \n Let us consider a drop-push model where the maximum occupancy of each site\nis restricted to one, i.e. $l_\\r=1, \\forall{r}$, but the hopping rates\n$\\epsilon_{r}(1)$ are\ndisordered, and chosen independently from the binary distribution\n\n\\begin{equation}\nProb(x=\\epsilon_a) = 1-f, ~~~~~~Prob(x=\\epsilon_b)=f.\n\\end{equation}\n\n\\noindent This model has the essential ingredients of disorder present in the\n original DDPP, yet it is simple enough that explicit, closed form\n relations between the mean density $\\rho$ and fugacity $z$, and hence\n the steady-state current $\\vec{J_0}$, can be written down.\n\n Let us denote by $Z_a$ and $Z_b$ the site generating functions\n for the $a$ and the $b$ sites respectively. Using\n (\\ref{eq:sitegen-ddpp}) and (\\ref{eq:imDDPP}), these are given by\n $Z_a=1+\\epsilon_0 z\/\\epsilon_a$ and $Z_b=1+\\epsilon_0 z\/\\epsilon_b$. Now, since the fractions of\n$a$ and $b$ sites are $1-f$ and $f$ respectively, (\\ref{eq:implicit})\nreduces to \n\\begin{equation}\n\\rho=(1-f){\\epsilon_0 z\\over \\epsilon_a+\\epsilon_0 z}+f{\\epsilon_0 z\\over \\epsilon_b+\\epsilon_0 z}.\n\\end{equation}\nThis can be easily\ninverted to obtain $z$ as a function of $\\rho$, e.g. for $f=1\/2$ and\n$\\epsilon_a=\\epsilon_0=\\epsilon_b\/q $ we obtain\n\n\\begin{equation}\nz(\\rho) = {\\sqrt{(1-q)^2(1\/2-\\rho)^2+q}-(1+q)(1\/2-\\rho)\\over 2(1-\\rho)}\n\\label{eq:fugacity}\n\\end{equation}\nSince $z(\\rho)$ is known, the steady-state current is trivially obtained\nfrom (\\ref{eq:J0-dd}).\n\nFinally, the correlation function $\\Gamma_i^2(r)$ of (\\ref{eq:hhcor-DDPP}),\nupon disorder averaging, may be written as\n\\begin{eqnarray}\n{\\overline{\\Gamma^2}(r)} & = & r\\left( z{\\partial\\over \\partial z}\\right)^2\n \\left[(1-f)\\ln Z_a + f \\ln Z_b\\right] \\nonumber \\\\\n & = &\n {1\\over 2}\\left[{z\\over(1+z)^2}+{qz\\over(q+z)^2}\\right] r\n\\end{eqnarray}\nwhere $z$ is given by (\\ref{eq:fugacity}) above.\n\n \\subsection{Generalized Disordered Drop-push Process: GDDP}\n \\label{subsec:GDDP}\n\n We may consider a generalized version of the drop-push process in\n which, in addition to the particle moves, independent hole moves are\n also allowed. For simplicity we restrict ourselves to the generalized\n version of the single occupancy DDPP introduced above. This generalized\n model may be regarded as the disordered lattice gas analogue of the\n Toom interface dynamics in the low-noise limit \\cite{Toom}; see Section\n \\ref{sec:height}. The techniques developed for the DDPP may be used to\n obtain the exact steady-state measure and other quantities such as\n current and static correlations provided a certain condition\n [(\\ref{eq:constr-GDDP}) below] is met.\n\n The model in $d$ dimensions is defined on a hypercubic lattice with\n periodic boundary conditions along all the $d$ axes (with unit vectors\n \\{$\\e_\\nu |\\nu=1\\cdots d$\\}), similar to the DDPP. Each site ${r}$ of the\n lattice can hold either a particle ($n_\\r=1$) or a hole ($n_\\r=0$). The\n configuration $\\cal{C}$ of the system is specified by specifying the\n occupation number of each well \\{$n_\\r$\\} with ($n_\\r\\in\\{0,1\\},\n \\forall{r}$). Further, to each site ${r}$ is assigned a pair of positive\n random numbers $(\\alpha_{r},\\beta_{r})$ chosen from some\n distribution. The dynamics is stochastic and is very similar to that\n for the DDPP dynamics: in a time interval $dt$, a particle at site ${r}$\n is exchanged with the closest hole in the $\\pm\\nu$th direction with a\n probability $p_{\\pm\\nu}\\alpha_\\r dt$ (Fig.~\\ref{fig:gddp-model}). For\n identical particles this move is equivalent to a cascade of particle\n moves terminating at the first vacant site as in the drop-push\n dynamics. Likewise, in interval $dt$, a hole at site ${r}$ is exchanged\n with the closest particle along the $\\nu$th direction with probability\n $q_{\\nu}\\beta_\\r dt$. This can be looked upon of as a cascade of hole-moves\n analogous to the cascade of particle moves. Here, as in the DDPP, the\n $p_\\nu$'s and $q_\\nu$'s are all non-negative and satisfy\n $\\sum_{\\nu=1}^d (a_\\nu+a_{-\\nu})=1; a=p,q$. Further we chose $\\alpha$'s\n and $\\beta$'s such that\n\\begin{equation}\n\\alpha_\\r\\beta_\\r= K,\n\\label{eq:constr-GDDP}\n\\end{equation} \n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=7.eps,width=5cm,angle=-90}\n\\end{center}\n\\narrowtext\n\\caption{Generalized Disordered Drop-push Process (GDDP) configuration and\nmoves in $d=2$.}\n\\label{fig:gddp-model}\n\\end{figure}\n \\noindent where $K$ is a constant independent of ${r}$. As we will see\n below, this particular choice of the jump-rates allows the exact\n determination of the invariant measure of the model. Physically this\n choice is quite reasonable since it implies that the sites which act as\n traps for particles (low $\\alpha_\\r$) are more transparent to holes (high\n $\\beta_\\r$) and vice versa. As pointed out earlier the non-disordered\n version of this model, i.e. \\{$\\alpha_\\r=\\alpha, \\beta_\\r=\\beta, \\forall {r}$\\} in\n 1-$d$, is the lattice gas equivalent of the low-noise driven Toom\n interface dynamics \\cite{Toom}.\n\n The master equation (\\ref{eq:master}) governing the time evolution of\n the system now includes terms corresponding to hole moves as well.\n Since each micro-step involves either only particle moves or hole\n moves, we use the principle of pairwise balance for the particle moves\n and hole moves separately. We work in the grand canonical picture in\n the thermodynamic limit.\n\n If only particle moves were allowed the invariant measure would be\ngiven by\n\\begin{equation}\n\\mu_{pcle}({{\\cal C}})=\\prod_{r} u_{r}(n_r)\n\\label{eq:muC-p}\n\\end{equation}\n The single site weights $u_{r}$ are given by\n\\begin{equation}\n u_{r}(n_\\r) = \\left\\{ \\begin{array}{ll}\n 1 & \\mbox{~~~if~$n_\\r=0$} \\\\ \n \\epsilon_0\/\\alpha_\\r & \\mbox{~~~if~$n_\\r=1$}\n \\end{array} \\right. \n\\end{equation}\n\\noindent Introducing the fugacity $z$ and site generating functions \n $Z_{r}=1+z u_{r}(1)$ we can write the normalized single site\n probabilities as $P^{pcle}_{r}(0)=1\/Z_{r}$ and\n $P^{pcle}_{r}(1)=zu_{r}(1)\/Z_{r}$.\n \n Similarly, with only the hole moves, the invariant measure has the product\nform \n\\begin{equation}\n\\mu_{hole}({{\\cal C}})=\\prod_{r} v_{r}(n_r),\n\\label{eq:muC-h}\n\\end{equation}\nwhere $n_{r}=1$ ($0$) refers to the presence (absence) of a\n hole. The single site weights $v_{r}$ are given by\n\\begin{equation}\n v_{r}(n_\\r) = \\left\\{ \\begin{array}{ll}\n 1 & \\mbox{~~~if~$n_\\r=0$} \\\\ \n \\epsilon_0\/\\beta_\\r & \\mbox{~~~if~$n_\\r=1$}\n \\end{array} \\right. .\n\\end{equation}\n\\noindent Introducing the fugacity $y$ for holes and site generating \n functions $Y_{r}=1+y v_{r}(1)$ we can write the normalized single-site\n probabilities as: $P^{hole}_{r}(0)=1\/Y_{r}$ and\n $P^{hole}_{r}(1)=y v_{r}(1)\/Y_{r}$.\n\n Now, since each site is occupied either by a particle or a hole, we\n must have $P^{pcle}_{r}(0)=P^{hole}_{r}(1)$ and\n $P^{pcle}_{r}(1)=P^{hole}_{r}(0)$. Using the detailed forms of\n $P^{pcle}_{r}$'s and $P^{hole}_{r}$'s, we arrive at the condition\n (\\ref{eq:constr-GDDP}) with $K\\equiv\\epsilon_0^2 y z$. If this\n condition is satisfied then the invariant measure for GDDP is given by\n either (\\ref{eq:muC-p}) or (\\ref{eq:muC-h}), since both are\n equivalent.\n\n In a similar manner as for the DDPP the current due to the particle\n moves and hole moves may be computed. The total particle\n current due to both types of moves, in $d$ dimensions, is given by\n\\begin{equation}\n\\vec {J_0} = \\epsilon_0 z\\sum_{\\nu=1}^d (p_\\nu-p_{-\\nu})\\e_\\nu \n - \\epsilon_0 y\\sum_{\\nu=1}^d (q_\\nu-q_{-\\nu})\\e_\\nu\n\\end{equation}\n\n As for the DDPP, static density-density correlations in the steady\n state vanish identically on account of the product form of the\n steady-state measure. In $d=1$ the height-height correlation is given\n by (\\ref{eq:hhcor-DDPP}). \n \n\\section{Disordered Asymmetric Simple Exclusion Process: DASEP \\hfill} \n\\label{sec:DASEP}\n The asymmetric simple exclusion process (ASEP) is a prototype model\n for studying nonequilibrium phenomena in the context of lattice gases\n \\cite{ligget,spohn}. When discussing the effect of quenched disorder,\n it is important to distinguish between cases in which (a) the easy\n direction of hopping in each bond is the same but hopping rates are\n random, and (b) the easy direction is itself a random variable. The\n latter case is studied in Section \\ref{sec:Sinai}. In this section,\n we consider a 1-$d$ system with disorder of type (a) and show that\n quenched disorder can induce macroscopic phase separation. Using a\n variety of arguments we sketch the phase coexistence curve in the\n current ($J_0$) - mean density ($\\rho$) plane. This agrees\n qualitatively with the results obtained from the Monte Carlo (MC)\n simulations.\n\n \\subsection{The model} \n \\label{sub:defDASEP}\n \n In one dimension, we define the disordered asymmetric simple\n exclusion process on a ring of $L$ sites. Each site can hold either\n $1$ or $0$ particle. To each bond $(i,i+1)$ of the lattice is\n assigned a quenched random rate $\\alpha_{i,i+1}$ chosen independently\n from some probability distribution $Prob(\\alpha)$. The dynamics is\n stochastic: in a time interval $dt$ a particle at site $i$ attempts\n to hop, with probability $p\\alpha_{i,i+1} dt$, to site $i+1$. The\n move is completed if and only if site $i+1$ is unoccupied (see\n Fig.~\\ref{fig:dasep-1d}).\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=8.eps,width=7cm,angle=-90}\n\\end{center}\n\\narrowtext\n\\caption{The disordered fully asymmetric simple exclusion process \n(DASEP) and moves in $d=1$.}\n\\label{fig:dasep-1d}\n\\end{figure}\n For the model defined above no analytically exact characterization of\n the steady-state measure could be obtained. A simpler \n model in which there is only one defect bond has been studied in\n detail by Janowsky and Lebowitz \\cite{Janowsky}, but in this case too\n the exact steady-state measure is not known. We use Monte Carlo\n simulations and a mean-field approximation to demonstrate some\n striking effects of quenched disorder.\n\n \\subsection{Current-density plot and density profile in steady\n state\\hfill} \\label{sub:imDASEP}\n\n Figure \\ref{fig:dasep-jrho} shows the steady-state current $J_0$ vs\n mean density $\\rho$ plot, obtained from MC simulations, for\n a system of size $L=8000$ and the rates $\\alpha$ chosen from the\n binary distribution\n\\begin{equation}\n Prob(\\alpha=r)=f, ~~~~~Prob(\\alpha=1)=1-f.\n\\label{eq:binary}\n\\end{equation}\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=9.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{The current-density plot for the DASEP for a given realization of\ndisorder for a system of size $L=8000$. The hopping rates are chosen from\nthe distribution ({\\protect\\ref{eq:binary}}) with $r=f=1\/2$. The filled\ncircles are the MC results and the dashed line is the mean field curve. The\nsolid line represents the $J-\\rho$ curve for the Fully Segregated Model\n(FSM) for the same values of the parameters. }\n\\label{fig:dasep-jrho}\n\\end{figure}\n\\noindent Here $f$ is the fraction of weak bonds, and $r$ is a measure\n of the strength of the weak bonds. We used the values\n $r=1\/2,f=1\/2$ in our numerical work. For a specified mean density\n $\\rho$, a random initial distribution of $N_p=\\rho L$ particles\n on $L$ sites is chosen and the system is allowed to settle into a\n steady state by evolving it for a sufficiently large number of MC\n steps. Then the current across each bond is obtained by counting the\n total number of jumps across that bond, over a large number of MC\n steps. An average of all the bond currents thus computed is taken\n as $J_0$, as currents across all bonds are equal in the steady\n state. $J_0$ is a symmetric function of density around $\\rho=1\/2$\n as a result of a certain symmetry with respect to particle-hole\n interchange (see Appendix \\ref{apx:PHsym}). As may be expected, the\n current for the disordered system lies between the corresponding\n values of the two pure reference systems with $r=1$ and $r=1\/2$\n on all the bonds. The more striking qualitative difference\n between the disordered and pure systems is the appearance of a\n plateau (Regime B in Fig.~\\ref{fig:dasep-jrho}) for a range of\n densities $|\\rho-1\/2|\\le \\Delta$. In this regime, the current is\n independent of the mean density and equals the maximum allowed\n current in the system. The approximate size $\\Delta$ of regime\n B, which is a function of $r$ and $f$, is obtained in subsection\n \\ref{sub:FSM} below.\n\n We studied the steady state density profiles characterized by the set\n of site densities $\\{\\rho_i\\equiv\\langle n_i \\rangle\\}$ in both\n regimes A and B, using MC simulations. We found that in regime A the\n system is homogeneous on a macroscopic scale, while in regime B it\n shows macroscopic density segregation. Figure\n \\ref{fig:dasep-profiles} shows the steady state density profiles for\n three representative mean densities --- one from regime A and two from\n regime B. Evidently there is a large qualitative difference between\n the profiles in the two regimes. In regime A\n (Fig.~\\ref{fig:dasep-profiles}a), there are density variations\n (shocks) only over microscopic scales; coarse-graining over a few\n lattice spacings leads to a spatially uniform density. In contrast to\n this, in the profiles corresponding to regime B\n (Fig.~\\ref{fig:dasep-profiles}b), there are density inhomogeneities\n over length scales comparable to the system size $L$, in addition to\n the shocks on a microscopic scale. This segregation into high and low\n density phases, with large shocks separating them, is reminiscent of\n phase separation, and occurs over the full range of mean particle\n densities corresponding to regime B. A qualitatively similar\n phenomenon has been found in a system with one defect bond, studied in\n \\cite{Janowsky}.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=10.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{Density profiles for the DASEP for a system of size $L=8000$ for a\ngiven realization of disorder at three fillings (a) $\\rho=0.8$, (b)\n$\\rho=0.6$, and (c) $\\rho=0.5$. In d,e and f are shown the blow-ups of the\nregions enclosed in the dashed boxes in a,b and c respectively. Circles are\nMC profiles and the continuous lines are mean-field results.}\n\\label{fig:dasep-profiles}\n\\end{figure}\n\n The {\\it number} of different large-scale regions of high and low\n density shows fluctuations from one realization of disorder to\n another. As the size of the system is increased, we monitored the\n mean number of these regions, and found that it is nearly constant, or\n perhaps increases very mildly -- certainly, much less fast than\n linearly in the size of the system. This implies that the\n characteristic length scale of density segregation increases\n indefinitely in the thermodynamic limit.\n\n\n \\subsection{ Mean-field approximation \\hfill}\n \\label{sub:MFA}\n \n We now turn to a mean-field approximation, which assumes no\n correlations between site densities in the steady state. We will see\n that it captures most of the steady-state features found in the MC\n simulations above, including the steady-state density profile.\n \n The time-averaged steady-state current $J_{i,i+1}$ in the bond\n $(i,i+1)$ is given by $J_{i,i+1} = \\alpha_{i,i+1} \\langle\n n_i~(1-n_{i+1})\\rangle $. In view of the mean-field approximation\n $\\langle n_i n_j\\rangle =\\langle n_i\\rangle\\langle n_j\\rangle$ this\n reduces to\n\\begin{equation} J_{i,i+1} = \\alpha_{i,i+1}\\langle\n n_i\\rangle~\\langle 1-n_{i+1}\\rangle \\label{eq:MFA} \n\\end{equation}\n\n To compute the steady-state current $J_0$ as a function of the mean\n particle density $\\rho$ for a given realization of disorder, we use\n the two iteration schemes described below, which yield equivalent\n results.\n\n (i) {\\it Constant current iteration scheme.} For a given system\n of size $L$ and for a fixed value $J_0$ for the current, we\n iterate the following set of equations\n\\begin{equation}\n \\rho_{i+1}=1-J_0\/(\\alpha_{i,i+1}\\rho_i) ~~~~i=1,..,L\n\\end{equation} \n around the chain starting with, say, some value $\\rho_1$ (periodic\n boundary conditions imply $\\rho_{i+L}=\\rho_i$). If $J_0$ is less than a\n certain value $J^{MF}_{max}$, which is the maximum current supported\n by the system within the mean-field approximation, then the iteration\n scheme converges, i.e. we get all the site densities in the physically\n acceptable range $[0,1]$. The average of these site densities gives\n the mean density of the system corresponding to the stationary current\n $J_0$. There are in general two values of the mean particle density\n corresponding to an allowed value of $J_0$ and the\n iteration scheme converges to one or the other depending on the\n initial value of the density $\\rho_1$.\n \n However, in this scheme we find that the number of iterations required\n for convergence increases without bound, as the trial value $J_0$ gets\n closer to $J^{MF}_{max}$ from below. This divergence of the iteration\n scheme is presumably due to the existence of the plateau in the\n $J~vs~\\rho$ curve, i.e. there exist many values of $\\rho$ for $J_0$\n very close to $J^{MF}_{max}$. Hence to obtain the density profile for\n $\\rho$ in the density segregated regime we resort to the {\\it constant\n density} iteration scheme described below.\n\n (ii) {\\it Constant density iteration scheme.} In this scheme we begin\n by assigning site densities $\\{0\\le\\rho_i(0)\\le 1\\}$ to the lattice\n sites subjected to the constraint ${1\\over L}\\sum_i \\rho_i(0)\n =\\rho$. The site densities are updated in parallel according to:\n\\begin{equation}\n \\rho_i(t+1)=\\rho_i(t)+J_{i-1,i}(t)-J_{i,i+1}(t);~~~i=1,..,L \n\\end{equation}\n where $J_{i,i+1}(t)=\\alpha_{i,i+1}\\rho_i(t)[1-\\rho_{i+1}(t)]$ in view of\n (\\ref{eq:MFA}).\n\n We refer to this as the constant density scheme, since in each\n iteration the total density remains unchanged, i.e\n $\\sum_i\\rho_i(t+1)=\\sum_i\\rho_i(t)$. After a sufficient number of\n iterations, which depends upon the starting mean density $\\rho$,\n the set of site densities converge to a set of numbers\n \\{$\\rho_i$\\} and the current on each bond converges to the steady\n state current $J_0$.\n\n The steady-state density profiles and the $J_0~vs ~\\rho$\n plot ($0\\le\\rho\\le1$) for a given configuration of disorder obtained\n using these schemes is shown in Figs.~\\ref{fig:dasep-profiles} and\n \\ref{fig:dasep-jrho} respectively. It is evident that the mean-field\n approximation (\\ref{eq:MFA}) reproduces quite well not only the\n $J-\\rho$ relationship, but also the density profile, including the\n locations of shocks, though not the shapes of individual shocks.\n\n \\subsection{Qualitative explanation of phase separation}\n \\label{sub:MF-gross} \n\n Although the mean-field approximation of the previous subsection\n successfully reproduces many features in the steady state, it does not\n yield a simple understanding of the phase separation\n (Fig.~\\ref{fig:dasep-profiles}) or the plateau in the $J$--$\\rho$\n curve (Fig.~\\ref{fig:dasep-jrho}) in terms of the macroscopic\n parameters of the model. We conjecture that underlying the behaviour\n of the DASEP in different regimes is a Maximum Current principle: For\n a given mean density, the system settles into a steady-state \n which maximizes the stationary current. Such a maximum\n current principle has been used to describe phase separation in the\n asymmetric simple exclusion process with open boundary conditions in\n \\cite{Krug91}.\n \n To use the maximum-current principle to have a qualitative\n understanding of the phase separation in DASEP, let us assume that the\n density in each stretch of like bonds is uniform. This approximation\n is in fact exact in the Fully Segregated Model (FSM) discussed in the\n following subsection. Let us denote stretches of $\\alpha=1$ bonds by X\n and stretches of $\\alpha=r<1$ bonds by Y. The two parabolas in\n Fig.~\\ref{fig:phases-expl} are the steady-state $J~vs~\\rho$ curves for\n the two pure reference systems all X and all Y. In the disordered\n system, since the steady-state current has to be spatially constant,\n the possible densities are given by the four intersections of the line\n $J=J_0$ with the two parabolas. If the mean density is in the range\n $\\rho\\le 1\/2-\\Delta$ (or $\\rho\\ge 1\/2+\\Delta$), then the allowed\n densities for the X and Y stretches are $\\rho_1,\\rho_2$ (or\n $\\rho_4,\\rho_3$) respectively. The current is in fact determined by\n the density constraint\n $(1-f)\\rho_{1,4}(J_0)+f\\rho_{2,3}(J_0)=\\rho$. The variation of density\n between $\\rho_1$ and $\\rho_2$ (or $\\rho_3$ and $\\rho_4$) between X and\n Y stretches corresponds to the `sub-bands' seen in\n Fig.~(\\ref{fig:dasep-profiles}a). On a macroscopic scale, however, the\n system has uniform density. Now consider what happens when the mean\n density is brought closer to $1\/2$. From Fig.~\\ref{fig:phases-expl},\n it is evident that the current would tend to increase, and would\n eventually reach the maximum allowed value $J_{max}^Y$ (which equals\n $1\/4$ in the thermodynamic limit, as argued below). As the density is\n increased further, the current remains constant at $J_{max}^Y$, in\n accordance with the maximum current principle, and the excess density\n is taken care of by converting some of the X stretches of density\n $\\rho_1$ into ones with $\\rho_4$ (or vice versa if $\\rho > 1\/2$). This\n conversion takes place adjacent to the largest stretch of Y bonds,\n leading to two macroscopic regions of different mean densities -- one\n with densities $(\\rho_1, \\rho_2)$ for the X and Y stretches, and the\n other with $(\\rho_4, \\rho_3)$. The position of the principal shock\n separating these regions is at the location of the largest Y\n stretch. In the DASEP, the assumption of uniform density in each\n stretch is not really true, on account of the finite length of most of\n the stretches. However, the above argument provides a qualitative\n picture towards understanding the reason for phase separation in the\n DASEP.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=11.eps,width=6.5cm}\n\\end{center}\n\\narrowtext\n\\caption{Origin of phase separation in DASEP. The two parabolas\n$J=r\\rho(1-\\rho);~r=1,1\/2$, represent the\n$J-\\rho$ plots for the two reference non-disordered systems.}\n\\label{fig:phases-expl}.\n\\end{figure}\n In a certain respect, the reason for the phase separation in the DASEP\n is similar to that in the single defect bond model studied in\n \\cite{Janowsky} --- both have local segments in the system where the\n maximum allowed current is less than that allowed everywhere else. In\n the single defect bond model, phase separation takes place when the\n current carried by the rest of the system, with presumed uniform\n density, is larger than the maximum current allowed through the weak\n bond. In the DASEP, with an extensive number of weak bonds, the {\\it\n largest stretch} of weak bonds acts as the current-limiting\n segment. The length of this stretch increases as $\\ln L$ with system\n size, and in the thermodynamic limit the maximum allowed current in\n this stretch is ${1\\over 4}r$ -- equal to the maximal current in a\n pure system with only weak bonds.\n\n The essential point leading to phase separation is thus the maximum\n current principle, coupled with localized current-limiting regions in\n the system. In the DASEP, this limit is determined by long stretches\n of weak bonds, and similar considerations should apply in related\n models. Consequently, we would expect a density-segregated phase in\n disordered versions of models which, in the absence of disorder,\n display a maximum in the steady-state current as a function of\n density.\n \n \\subsection{The Fully Segregated Model} \\label{sub:FSM}\n\n It is useful to define a model for which many of the approximations\n made in the previous subsection are actually exact. To this end, we\n study a Fully Segregated Model (FSM), which is obtained from the\n binary random model above by arranging all like bonds together. Thus,\n in this model, one has {\\it two} large stretches of X and Y, of\n lengths $(1-f)L$ and $fL$ respectively. For the FSM in the\n thermodynamic limit, the assumption of uniform density within each\n stretch is justified, as correlations due to the junctions decay with\n increasing separation, and may be neglected in the bulk \\cite{FSM}.\n\n Steady-state MC density profiles for the FSM at three different\n fillings $\\rho\\le 1\/2$ are shown in Fig.~\\ref{fig:dp-seg} ---\n symmetry under particle-hole exchange implies analogous behaviour\n for $\\rho\\ge 1\/2$ (Appendix \\ref{apx:PHsym}). For low densities\n ($\\rho<\\rho_c^-$), the two stretches have uniform bulk\n densities $\\rho_x$ and $\\rho_y$ related to each other by the\n requirement of equality of the two bulk currents,\n\\begin{equation}\n\\rho_x(1-\\rho_x)=r\\rho_y(1-\\rho_y)=J_0,\n\\label{eq:J-xy}\n\\end{equation}\nand the density constraint \n\\begin{equation}\n(1-f)\\rho_x+f\\rho_y=\\rho. \n\\label{eq:contr-rho}\n\\end{equation}\nThese three equations determine $\\rho_x, \\rho_y$ and $J_0$ uniquely for a given\n$\\rho$. For $f=1\/2$ we obtain\n\\begin{eqnarray}\n\\rho_y &=& {4\\rho-1-r\\pm\\sqrt{(4\\rho-1-r)^2+8(1-r)\\rho(1-2\\rho)}\\over\n{2(1-r)}}\\nonumber \\\\ \n\\rho_x&=&2\\rho-\\rho_y, ~~J_0=\\rho_x(1-\\rho_x)\n\\end{eqnarray}\n This is analogous to the macroscopically homogeneous state of the\n fully random system.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=12.eps,width=7.5cm}\n\\end{center}\n\\narrowtext\n\\caption{Representative density profiles of the FSM with $r=f=1\/2$ at \nthree different fillings: (a) $\\rho=0.24$ (dotted) (b) $\\rho=\\rho_c^-\n=0.324$ (solid) and (c) $\\rho=0.4$ (dashed). The Y stretch ($r=1\/2$) is in\nthe range $i\\in[2000,6000]$. The inset shows the variation of the bulk\ndensities in the X and Y stretches as a function of the filling\n$\\rho$.}\n\\label{fig:dp-seg}\n\\end{figure}\n As the mean density is increased, the density of each stretch\n increases, until, at a critical density $\\rho=\\rho_c^-={1\\over\n 2}-{1\\over 4}\\sqrt{1-r}$ (the corresponding critical density on the\n higher side is $\\rho_c^+=1-\\rho_c^-$), $\\rho_y$ equals $1\/2$\n (Fig.~\\ref{fig:dp-seg}). At this density, the current equals the\n maximum possible current in Y, namely, $J_0=J_{max}=r\/4$. As $\\rho$\n is increased further, $\\rho_y$ and $J_0$ remain constant at $1\/2$ and\n $r\/4$ respectively. The density change is adjusted by creating a\n density inhomogeneity in stretch X. The two densities $\\rho_x^{high}$ and\n $\\rho_x^{low}$ are related by $\\rho_x^{high}+\\rho_x^{low}=1$ so that the\n currents in the two phases are equal,\n i.e. $\\rho_x^{high}(1-\\rho_x^{high})=\\rho_x^{low}(1-\\rho_x^{low})=r\/4$. This\n implies $\\rho_x^{high,low}=(1\\pm\\sqrt{1-r})\/2$ (see inset in\n Fig.~\\ref{fig:dp-seg}). The fraction of these phases can be determined\n from a lever rule and are given by\n $f^{high,low}=|\\rho-\\rho^\\mp_c|\/(\\rho_x^{high}-\\rho_x^{low})$. This locking\n of the density in the Y stretch at $\\rho_y=1\/2$ is a direct consequence\n of the maximum current principle introduced above: any change of $\\rho_y$\n from $1\/2$ would reduce the current in Y, and hence in the full\n system. All the arguments above can be applied for $\\rho>1\/2$ because\n of particle-hole symmetry. Thus for $\\rho^-_c<\\rho<\\rho^+_c$ the state\n of the segregated model is analogous to the phase separated regime B\n of the random model. The size of regime B in the FSM is given by\n $2\\Delta=\\rho_c^+-\\rho_c^-=\\sqrt{1-r}\/2$. It closely approximates the\n size of the B regime in the DASEP (Fig.~\\ref{fig:dasep-jrho}).\n\n\\subsection{Phase-coexistence curve}\n \\label{sub:pcc}\n\n For the FSM, as $r$ is varied we obtain\n different $J_0~vs~\\rho$ curves. The phase-coexistence curve in the\n current-density plane in the parametric form\n\\begin{equation}\nJ_c={r\\over 4}, ~~~~~\\rho_c={1\\over 2}\\pm{1\\over 4}\\sqrt{1-r}\n\\label{eq:ph-diag}\n\\end{equation}\n which is the parabola $J_c=1\/4-(1-2\\rho_c)^2$ in\n Fig.~\\ref{fig:phase-diag}.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=13.eps,width=7.6cm}\n\\end{center}\n\\narrowtext\n\\caption{The phase coexistence curve for the FSM and the DASEP for $f=1\/2$.\nThe solid parabola is the coexistence curve of the FSM. The circles and the\ntriangles are respectively the MC and mean-field phase-coexistence curves\nfor the DASEP. The dashed parabola $J=\\rho(1-\\rho)$ demarcates the allowed\nregion for the DASEP. }\n\\label{fig:phase-diag}\n\\end{figure}\n\n The difference between the phase boundaries of the DASEP and FSM\n Fig.~\\ref{fig:phase-diag} comes from the fact that the interspersed\n weak-bond stretches in the DASEP have finite lengths, and the density\n in these small stretches is not quite equal to $1\/2$. Close to the\n phase separation, we anticipate the mean density in a Y-stretch of\n length $l$ in the DASEP to be of the form $\\rho_Y(l)=1\/2\\pm\n A(r)\/l^\\alpha(r)$, with $\\alpha(r)>0$. Using the distribution of the\n $Y$ stretches, namely $P_Y(l)=2^{-l}$, we obtain the $r$ dependence of\n the critical density\n\\begin{equation}\n \\rho_c^\\pm = {1\\over 4} + \\rho_X \\pm A(r) \\sum_l P_Y(l) \\rho_Y(l) $$\n\\end{equation}\n \\noindent where $\\rho_X(r<<1)\\sim r$ is the density in the X stretches\n near the phase transition. Comparing with the phase diagram for the\n FSM in Fig.~\\ref{fig:phase-diag}, the correction term $A(r)$ seems to\n be positive for all $r$. Further, let us suppose that the current in\n the FSM is a lower bound to that in the DASEP, as suggested by\n Fig.~\\ref{fig:phase-diag}. Then the coexistence curve for the random\n system must be quadratic near the critical point ($J^0=1\/4,\n \\rho^0=1\/2$) -- being bounded by two quadratics, namely, the $J-\\rho$\n curve for the pure system $J=\\rho(1-\\rho)$, and the coexistence curve\n of the FSM $J=1\/4-(1-2\\rho)^2$.\n\n \\subsection{Correlations in the steady state}\n \n Figure \\ref{fig:dasep-corln} shows the Monte Carlo results for the\n site-averaged density-density and height-height correlation functions\n $\\overline{G}(\\Delta{r})~{\\rm and}~ \\overline{\\Gamma^2}(r)$ in both the\n homogeneous and density-segregated regimes of the DASEP.\n $\\overline{G}(\\Delta{r})$ is seen to decay rapidly over a few lattice\n spacings, accounting for the success of the mean field approximation.\n It is found that $\\overline{\\Gamma^2}(r)$ grows as $r$ --- implying a\n roughness exponent $\\alpha=1\/2$.\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=14.eps,width=7.6cm}\n\\end{center}\n\\narrowtext\n\\caption{ Height-height correlation function $\\overline{\\Gamma^2}(r)$\nfor DASEP: (a) in Homogeneous (circles), and (b) Segregated-Density\n(triangles) regimes. Inset shows the site averaged density-density\ncorrelation function $\\overline G(\\Delta{r})$ defined in\n({\\protect\\ref{eq:partcorr}}). The small negative values at large\n$|\\Delta{r}|$ arise due to the finite size of the system.}\n\\label{fig:dasep-corln}\n\\end{figure}\n\n\\section{DASEP with backbends}\n\\label{sec:Sinai}\n\n As discussed at the beginning of Section \\ref{sec:DASEP}, the\n introduction of randomness in the easy direction of individual bonds\n alters the properties of one-dimensional disordered exclusion process\n in a crucial way. We study this in this section.\n\n The model is defined as follows: Assign quenched arrows (pointing\n either right or left) independently to each bond of a periodic chain,\n with probability $f < {1\\over 2}$ for left arrows, and $1-f$ for right\n arrows. An arrow defines the easy direction of hopping on each bond:\n a particle-hole exchange across a bond occurs with rate $w (1+g)$ if\n the particle moves along the direction of the arrow, and $w (1-g)$ if\n it moves opposite to the arrow. Since $f < {1\\over 2}$, there is an\n overall tendency for particles to circulate rightward, and the\n question is whether there is a nonzero current even in the\n thermodynamic limit.\n\n The model represents a system of hard-core particles in a random\n potential with a downward tilt. A conglomeration of left pointing\n arrows constitutes a backbend, where the potential climbs up before\n going down again. Within mean-field theory it is possible to obtain\n an upper bound $J_\\l$ on the current that can be carried by mutually\n excluding particles through a backbend of length $\\l$ \\cite{RB}. To\n this end, consider biased diffusion of hard core particles in the\n segment [$0, l$] of a 1-d lattice, with the `optimal' boundary\n conditions $\\rho(0) = 1$ and $\\rho (l) = 0$; these boundary conditions\n force the largest possible current through the segment, opposite to\n the bias. The master equation that describes transport is invariant\n under interchange of particles and holes and simultaneous relabelling\n of sites in reverse order, i.e. $n_j \\rightarrow \\bar n_{l - j}$ The\n boundary conditions respect this symmetry, implying that the\n steady-state density $\\rho(j)$ at site $j$ satisfies $\\rho(j) = 1 -\n \\rho (l - j)$. Thus in the steady state the number of particles in\n the backbend is $l\/2$ irrespective of the strength of the bias $g$.\n The principal effect of increasing $g$ is to sharpen the region which\n marks the transition from the particle-rich half of the backbend to\n the hole-rich half. The steady-state profile approaches a step\n function centered at $j = l\/2$ as $ g \\rightarrow 1$.\n\n The current in the steady state is the number of particles crossing\n site $l$ in unit time. Results of a Monte Carlo study \\cite{RB} are\n consistent with the large-$l$ asymptotic behaviour \n\\begin{equation}\n J_l \\sim \\exp \\left( - {1 \\over 2}~l\/\\Lambda(g) \\right).\n\\label{eq:s1}\n\\end{equation}\n where $\\Lambda(g)$ is a bias-induced length given by $\\Lambda^{-1}(g)=\n \\ln \\{(1+g)\/(1-g)\\}$. This can be seen by writing the current within\n a mean field approximation as $J = W(1 + g) \\rho_j (1 - \\rho_{j+1}) -\n W(1 -g) \\rho_{j+1} (1 - \\rho_j)$, and finding the value of $J$ for\n which the boundary conditions $\\rho_0 = 1$, $\\rho_l = 0$ hold. For\n $l>>\\Lambda(g)>>1$,this leads to $J \\approx 2g~e^{-lg}$ \\cite{RB}, in\n agreement with (\\ref{eq:s1}) when $g$ is small.\n\n The origin of the factor $1\\over2$ in the exponent in (\\ref{eq:s1})\n has been discussed in \\cite{RB}, and we recount the argument in brief.\n The transport of a single particle through the backbend involves two\n (approximately) causally independent steps which occur in parallel:\n (i) the topmost particle (located at site $k \\approx l\/2$ in large\n fields) has to be activated a distance $l\/2$, which requires an\n activation time $\\tau_{1\/2} \\sim \\exp \\left({1\\over 2}~l \/\\Lambda(g)\n \\right)$ and, (ii) the consequent hole that remains in the\n steady-state distribution moves to the bottom and is filled up, by\n moving each of $l\/2$ particles up through a lattice spacing. The time\n required is $\\tau_{1\/2}$ again. The current $J$ is thus proportional\n to $\\tau^{-1}_{1\/2}$, and consequently follows (\\ref{eq:s1}).\n\n Since, for fixed $g$, the largest current that can flow through a\n long backbend ($l>>\\Lambda(g)$) is exponentially small in its\n length, the largest current through the 1-d lattice of length $L$\n is determined by the length $\\l^*(L)$ of the largest backbend\n encountered. Since the probability of occurrences of $\\l$\n consecutive left-pointing arrows on bonds is proportional to\n $f^\\l$, we may estimate $\\l^*$ from $Lf^{\\l^*} = C$ where $C$ is\n a constant of order unity. Substituting in (\\ref{eq:s1}), we\n find that the current falls with increasing lattice size as\n\\begin{equation}\nJ(L) \\sim L^{-{1\\over2} \\theta} \n\\label{eq:s2}\n\\end{equation}\n with $ \\theta^{-1} = \\Lambda(g)\\ln f$.\n Thus the current is expected to decay as a power law in $L$, with a\n bias-dependent power, and to vanish in the thermodynamic limit.\n Figure \\ref{fig:JL_sinai} shows the result of Monte Carlo simulations.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\leavevmode\n\\psfig{figure=15.eps,width=8cm}\n\\end{center}\n\\narrowtext\n\\caption{Size dependence of the steady-state current in the backbend\nmodel (DASEP with some reversed bonds) for two sets of parameters: (a)\n$g=0.33, f=0.25$ (triangles) and (b) $r=0.54, f=0.3$ (circles). Each point\nrepresents an average over $40$ realizations of disorder in (a) and $100$\nrealizations in (b). The straight lines have slopes of $-\\theta\/2$ with\n$\\theta=0.5$ and $1.0$ respectively as predicted by\n({\\protect\\ref{eq:s2}}).}\n\\label{fig:JL_sinai}\n\\end{figure}\n\n As with the milder sort of disorder discussed in Section III, the\n state is strongly inhomogeneous and shows macroscopic regions of high\n and low density. Figure \\ref{fig:regimes1}c shows the time-averaged\n density profile for a typical configuration of bonds. There is a large\n shock around the rate-limiting backbend, which separates the two\n regions.\n\n For fixed lattice size $L$ with an associated longest backbend\n $l^*(L)$, the current is a nonmonotonic function of $g$. This can\n be seen as follows. If $g$ is low enough that\n $\\Lambda(g)>>l^*(L)$, linear response theory would imply that\n current $J$ grows linearly with $g$. On the other hand, if $g$ is\n large enough that $\\Lambda(g)< \\! 200$ s, $P_\\mathrm{orb} \\! > \\! 200$ d;][]{reig1999,reig2011}. Although most BeXRBs have eccentric orbits, there are several systems (i.e. X Per, GS 0834--430, KS 1947+300) with low eccentricity ($e \\! < \\! 0.2$). \n\nTransient BeXRBS show two types of X-ray outbursts \\citep{stella1986}. Type I outbursts ($L_\\mathrm{x} \\! \\sim \\! 10^{36-37}$ erg s$^{-1}$) occur once in a while the pulsar passes from the periastron of the orbit, where accretion enhances. On the other hand, Type II outbursts are major events ($L_\\mathrm{x} \\! \\geq \\! 10^{37}$ erg s$^{-1}$) that occur during mass ejection episodes of the counterpart. Transient BeXRBs may have very low quiescence luminosities ($L_\\mathrm{x} \\! \\leq \\! 10^{33}$ erg s$^{-1}$). The luminosity increase during a Type II outburst can be 3--4 order of magnitudes, while it is only about one order of magnitude for a Type I outburst \\citep{reig2011}. When an X-ray outburst occurs in a BeXRB, the neutron star generally enters a spin-up episode due to enhanced accretion. \n\nAs the population of BeXRBs has grown in the past decades, the pulse periods ($P_\\mathrm{s}$) of pulsars in BeXRBs are still strongly correlated with the orbital periods ($P_\\mathrm{orb}$), as it is firstly demonstrated by \\cite{corbet1984}. Although there is a large scatter in data, a positive correlation is evident with $P_\\mathrm{s} \\! \\propto \\! P_\\mathrm{orb}^2$. The large BeXRB population in SMC also obeys this relation \\citep{knigge2011,yang2017}. Therefore in BeXRBs, pulsars with longer pulse periods reside in binary systems with wider orbits and consequently lower accretion rates.\n\nThe optical light curve of SXP 1062 (see Fig. \\ref{burst}) shows periodic variations devoted to an orbital period of $\\sim \\! 656$ days \\citep{schmidtke2012}. X-ray observations of the source reveal the occurrence of a Type I outburst ($L_\\mathrm{x} \\! \\! \\simeq \\! 1.3 \\times 10^{37}$ erg s$^{-1}$) that happen together with the optical enhancement. However, the long term spin-down of SXP 1062 is not interrupted by the outburst. Prior to the X-ray outburst a minimum luminosity of $2.4 \\times 10^{35}$ erg s$^{-1}$ is measured, hence the luminosity increases by a factor of $\\sim \\! 50$ during the outburst. The outburst of SXP 1062 is observed only during an observation with an exposure of 2.2 ks, however the actual duration of the outburst might be longer since neighbouring observations are 14 days apart. The luminosity drops to $3.6 \\times 10^{36}$ erg s$^{-1}$ therefore, the outburst finishes in the following observation. These characteristics classify SXP 1062 as a persistent BeXRB.\n\nWe are able to resolve the orbital motion of SXP 1062 from its X-ray emission observed for $ \\sim \\! 2$ years (see Fig. \\ref{orbit}). We determine the orbital epoch as 56351(10) MJD and the light travel time for projected semi-major axis as 1636(16) lt-s by considering a circular orbit with a period of 656(2) days. (see Table \\ref{soln}). We also report an upper limit of 0.2 to the eccentricity at 90 per cent confidence level, therefore SXP 1062 is claimed to be in a low eccentric orbit despite the fact that denser observational coverage is needed for a better assessment. The orbital and pulse periods of the system, position the source on a place in line with BeXRBs on the Corbet diagram, the uppermost right end of the existing correlation for BeXRBs. Moreover, the mass function of the system is calculated to be $f(M) \\!\\simeq\\! 10.9(3)$M$_{\\odot}$, which seems appropriate bearing in mind that the Be companion is suggested to have a mass of $ \\sim \\! 15$M$_{\\odot}$ \\citep{henault2012}. Consequently, the orbital inclination can be evaluated as $i \\!\\simeq\\! 73(2) \\degree$. If we allow variation of the donor mass, the effect on the inclination angle is plotted in Figure \\ref{inclination}. Using this relation, the minimum donor mass is determined to be 13.3(3)M$_{\\odot}$ for $i \\!=\\! 90 \\degree$.\n\n\n\\subsection{Magnetic Field Estimation from Secular Spin-down Trend before the Glitch}\n\nIf we consider that the source is accreting via a prograde accretion disc that is formed before the glitch, standard accretion disc theory \\citep{pringle1972,lamb1973,ghosh1979,wang1987,ghosh1994,torkelsson1998} can be used to estimate the surface magnetic field of the neutron star. For this scenario, the inner radius of the accretion disc at which the magnetosphere disrupts the Keplerian rotation depends on the accretion rate ($\\dot{M}$) and the magnetic dipole moment of the neutron star ($\\mu \\!=\\! BR^3$ where $B$ is the surface magnetic field and $R$ is the radius of the neutron star) as\n\n\\begin{equation}\nr_\\mathrm{o} = K \\, \\mu^{4\/7} \\, (GM)^{-1\/7} \\, \\dot{M}^{-2\/7} \\, ,\n\\label{eqn:r0}\n\\end{equation}\n\n\\noindent where $K$ is a dimensionless parameter of about 0.91 and $M$ is the mass of the neutron star \\citep{pringle1972,lamb1973}. The torque can then be estimated as\n\n\\begin{equation}\n2\\pi \\, I \\, \\dot{\\nu }= n(\\omega_\\mathrm{s}) \\, \\dot{M} \\, l_\\mathrm{K} \\, ,\n\\label{eqn:torque}\n\\end{equation} \n\n\\noindent where $I$ is the moment of inertia of the neutron star, $\\dot{\\nu}$ is the spin rate of the neutron star, $n(\\omega_\\mathrm{s})$ is the dimensionless torque which is a factor parametrising the material torque and magnetic torque contributions to the total torque, and $l_\\mathrm{K }= (GMr_\\mathrm{o})^{1\/2}$ is the angular momentum per mass added by the Keplerian disc at $r_\\mathrm{o}$. The dimensionless torque can be approximated as\n\n\\begin{equation}\nn(\\omega_\\mathrm{s}) \\approx 1.4 \\, (1-\\omega_\\mathrm{s}\/\\omega_\\mathrm{c}) \\, \/ \\, (1-\\omega_\\mathrm{s}) \\, ,\n\\label{eqn:dimtorque}\n\\end{equation}\n\n\\noindent where $\\omega_\\mathrm{s}$, being equal to the ratio of the neutron star's rotational frequency to the Keplerian frequency at the inner radius of the accretion disc, is known as the fastness parameter and can be expressed as\n\n\\begin{equation}\n\\omega_\\mathrm{s} = 2\\pi \\, K^{3\/2} \\, P^{-1} \\, (GM)^{-5\/7} \\, \\mu^{6\/7} \\, \\dot{M}^{-3\/7} \\, ,\n\\label{eqn:fastness}\n\\end{equation}\n\n\\noindent where $P$ is the pulse period of the neutron star. In Eqn. \\ref{eqn:dimtorque}, $\\omega_\\mathrm{c}$ is the critical fastness parameter which has been estimated to be $\\sim \\!0.35$ \\citep{ghosh1979,wang1987,ghosh1994,torkelsson1998}. For $\\omega_\\mathrm{s} \\!=\\! \\omega_\\mathrm{c}$, the total torque on the neutron star becomes zero (i.e. $n(\\omega_\\mathrm{s}) \\!=\\! 0$) due to the negative torque contribution coming from the magnetic torque exerted outside the co-rotation radius at which the neutron star's rotational frequency equals to the Keplerian frequency.\n\nFor $\\omega_\\mathrm{s} \\!>\\! \\omega_\\mathrm{c}$, spin-down contribution coming from the outer disc outside the co-rotation radius is greater in magnitude than the total spin-up contributions coming from the material torque at the inner radius and the magnetic torque inside the co-rotation radius. This leads to a net spin-down of the neutron star (i.e. $n(\\omega_\\mathrm{s}) \\!<\\! 0$). On the contrary, for $\\omega_\\mathrm{s} \\!<\\! \\omega_\\mathrm{c}$, spin-up contribution coming from the material and magnetic torques is greater in magnitude than the spin-down contribution coming from the magnetic torques from the outer disc (i.e. $n(\\omega_\\mathrm{s}) \\!>\\! 0$). \n\nFrom a quadratic fit to the arrival times prior to the glitch, SXP 1062 is found to show a secular spin-down with a rate of $-\\,4.29(7) \\times 10^{-14}$ Hz s$^{-1}$ when a maximum luminosity of $L_\\mathrm{x} \\!\\sim \\! 3.3 \\times 10^{36}$ erg s$^{-1}$ is observed. Considering this luminosity value to be nearly the total accretion luminosity (i.e. $L \\!=\\! GM \\dot{M} \/ R$) and assuming a typical neutron star with $I \\!=\\! 10^{45}$ g cm$^2$, $M \\!=\\! 1.4$M$_{\\odot}$ and $R \\!=\\! 10^6$ cm; Eqn.s \\ref{eqn:r0} - \\ref{eqn:fastness} are solved numerically to obtain $\\mu$ of about $1.5 \\times 10^{32}$ G cm$^{3}$ leading to a magnetic field estimate of about $1.5 \\times 10^{14}$ G with $n(\\omega_\\mathrm{s}) \\!\\approx \\! -\\,0.0123$ and $r_\\mathrm{o} \\!=\\! 8.78 \\times 10^9$ cm. \n\nSXP 1062 can be considered to be a member of a class of accretion powered pulsars in high-mass X-ray binaries with very slow pulsations and persistent spin-down states \\citep{reig2012,fu2012}. Long spin periods together with the spin-down behaviour of these pulsars are argued as an indication of their magnetar-like magnetic fields. Thus, this class is sometimes classified as ``accreting magnetars''. Alternatively by using a theoretical model based on quasi-spherical subsonic accretion, long spin periods of these systems have also been considered not to be necessarily related to magnetar fields \\citep{shakura2013}. \n\n\\cite{fu2012} previously made use of three different theoretical approaches to obtain an estimate of the magnetic field of SXP 1062: Firstly, they estimated the magnetic field strength by considering the time scale for the ejector phase being comparable to the estimated age of the pulsar. Secondly, they estimated the magnetic field strength assuming the short-term spin-down rate of $-2.6\\times 10^{-12}$ Hz s$^{-1}$ as being near the maximum spin-down rate in disk or spherical accretion \\citep{lynden1974,lipunov1982,bisnovatyi1991}. Their final approach was to make use of the spin-down mechanism proposed by \\cite{illarionov1990}. All these three approaches lead to a surface magnetic field of SXP 1062 as $\\gtrsim 10^{14}$ Gauss.\n\nOur timing analysis shows that the source has a long-term secular steady spin-down trend with a rate of $-4.29(1)\\times 10^{-14}$ Hz s$^{-1}$ which could be as a result of a steady disc accretion. Thus, using standard accretion theory, our magnetic field estimate for SXP 1062 follows consideration of accretion via prograde accretion disc with a small negative dimensionless torque. According to this theoretical framework, observed spin-down rate and luminosity of the source leads to a magnetar-like surface magnetic field estimation which is consistent with the previous estimations by \\cite{fu2012}. \n\n\\subsection{The Glitch}\n\n\\begin{figure}\n \\center{\\includegraphics[width=4.1cm,angle=270]{freqs.eps}} \n \\caption{Pulse frequency evolution of SXP 1062. Frequencies are calculated from the slopes of linear fits to the TOAs shown in Fig. \\ref{glitch}. The time intervals of linear fits are represented as x-axis error bars. The frequency jump on MJD 56834.5 is identified as a spin-up glitch event with $\\Delta \\nu = 1.28(5) \\times 10^{-6}$ Hz. The source continues to spin-down after the glitch with a change of frequency derivative $\\Delta \\dot \\nu = 1.5(9) \\times 10^{-14}$ Hz s$^{-1}$.}\n \\label{freqs}\n\\end{figure}\n\nA glitch in the pulse frequency is observed 25 days after the X-ray outburst of SXP 1062. The source has not shown any spin-up trend during the outburst which may be due to a very short duration of the outburst. Actually, the outburst is displayed only in one of the observations, which has an exposure of about 2.2 ks. As seen from Figure \\ref{glitch} and Table \\ref{soln}, the glitch occurred on MJD 56834.5 with a change of pulse frequency $\\Delta \\nu = 1.28(5) \\times 10^{-6}$ Hz and a change of pulse frequency derivative $\\Delta \\dot \\nu = - \\,1.5(9) \\times 10^{-14}$ Hz s$^{-1}$. In Figure \\ref{freqs}, we also show the pulse frequency evolution which is constructed by measuring the slopes of the TOAs (see Fig. \\ref{glitch}) for time intervals of approximately 30--70 days. Since the occurrence of the glitch does not coincide with the time of the X-ray outburst, it should be associated with the internal structure of the neutron star. SXP 1062 continues to spin-down with a constant rate after the glitch event.\n\nA glitch is a sudden fractional change in frequency which is mostly pursued by a change of spin-down rate of a previously rather stable rotating pulsar. Almost 10 per cent of pulsars are observed to glitch and pulsars of all ages seem to have glitches \\citep{haskell2015} with fractional change of frequency ($\\Delta \\nu\/\\nu$) ranging from $10^{-11}$ to $10^{-5}$ and fractional change of frequency derivative ($\\Delta\\dot{\\nu}\/\\dot{\\nu}$) varying between $10^{-4}$ and $10^{-1}$ \\citep{espinoza2011,yu2013,dib2014}. The core of a neutron star contains a significant amount of neutron superfluid \\citep{lamb1978a,lamb1978b,sauls1989,lamb1991,datta1993,lattimer2007} therefore, the moment of inertia of the star resides mainly in the core. Moreover, the inner part of the crust lattice also contains an amount of neutron superfluid which carries $10^{-2}$ of the star's moment of inertia. Coupling time scales between crustal neutron superfluid and the rest of the crust is typically very long extending from months to years \\citep{alpar1981,alpar1993,akbal2015}. For radio pulsars which spin-down due to electromagnetic dipole radiation, it is possible to resolve moment of inertia of the crustal superfluid during the post glitch \\citep{espinoza2011,yu2013}. Like canonical pulsars, magnetars also exhibit glitches however, there are some distinguishing characteristics between these two groups. While almost all pulsar are radiatively quiet i.e. they are not accompanied by any burst or pulse profile changes after the glitch \\citep{espinoza2011,yu2013} \\citep[see][for exceptions]{archibald2016,manchester2011,livingstone2010}, magnetar glitches can either be radiatively loud i.e. they can be accompanied by flares, bursts and\/or pulse profile changes; or radiatively quiet \\citep{dib2014}. \n\nIn magnetars, glitches are resolved with high fractional frequency changes at the order of $\\Delta \\nu \/ \\nu \\! \\sim \\! + \\, 10^{-5}$ and $- \\, 10^{-4}$ \\citep{kaspi2017}. Largest spin-down glitches observed are, the glitch of 1E 2259+586 with $\\Delta \\nu \/ \\nu \\! \\sim \\! 10^{-6}$ \\citep{archibald2013} and the glitch of SGR 1900+14 with $\\Delta \\nu \/ \\nu \\! \\sim \\! 10^{-4}$ within 80 days after a large outburst \\citep{woods1999,thompson2000}. There are a few net spin-down glitches \\citep{icdem2012,sasmaz2013,archibald2017} together with a large number of spin-up glitches \\citep{dib2014}. Large spin-down glitches can be explained by particle outflow from magnetic multipoles during an outburst, while this process induces vortex inflow from the crust. The density of vortex lines are proportional to the superfluid velocity therefore the angular momentum taken from the crust \\citep{thompson2000,duncan2013}. The spin-up glitches can be caused by sudden fractures of the crust and consequently vortex outflow in the crust superfluid \\citep{thompson2000}. For both cases of spin-down and spin-up glitches, vortex unpinning from the crust occurs and then the vortices creep and re-pin to the crustal nuclei, therefore the post glitch relaxation should be observed in both cases \\citep{gugercinoglu2014}.\n\nDue to the presence of dominant external torque noise, it is not easy to detect these types of glitches for accretion powered pulsars in X-ray binaries \\citep{baykal1997}. However, for KS 1947+300 \\cite{galloway2004} have discovered a spin-up glitch. KS 1947+300 was spinning up during this glitch, therefore the influence of the external torques is not clear yet; whether the glitch event is associated with internal or external torques. Recently, \\cite{ducci2015} have suggested that both glitches and anti-glitches are possible for accretion powered X-ray pulsars, furthermore glitches of binary pulsars should have longer rise and recovery time scales compared to isolated pulsars since they have pulse periods longer than those of isolated ones. \n\nSXP 1062 is found to be spinning down secularly until MJD 56834, that is 25 days subsequent to the X-ray outburst. Then, the source showed a spin-up glitch with a fractional frequency change of $\\Delta \\nu \/ \\nu \\! \\sim \\! 1.37(6) \\times 10^{-3}$ and a fractional change of frequency derivative $\\Delta \\dot \\nu \/ \\dot \\nu \\! \\sim \\! 0.3(2)$.\n\nDuring the secular spin-down of SXP 1062, the spin-down rate is measured to be $- \\, 4.29(7) \\times 10^{-14}$ Hz s$^{-1}$. If we consider that the observed glitch is due to a torque reversal (i.e. consider it as a frequency jump due to accretion torque) with a similar magnitude of spin-up rate, upper limit for ${{\\Delta \\nu} \/ {\\nu}}$ can be estimated to be about $\\sim \\! 1.5 \\times 10^{-5}$ for a maximum of $\\Delta t \\! \\sim \\! 4$ days (the time interval between two neighboring observations around the frequency jump) which is two orders of magnitude smaller than the observed ${{\\Delta \\nu} \/ {\\nu}}$ of the glitch. So, it is unlikely that the glitch is as a result of the accretion torques. Furthermore, the ratio of the core superfluid moment of inertia to the crust moment of inertia ($I_{\\mathrm{s}} \/ I_{\\mathrm{c}}$) should be at the order of $10^2$ \\citep{baykal1991,baykal1997}. Therefore the glitch event in SXP 1062 should be associated with the internal structure of the neutron star.\n\nRecently, \\cite{ducci2015} discussed observability of glitches in accretion powered pulsars by using the ``snowplow'' model of \\cite{pizzochero2011}. In the two component neutron star model \\citep{baym1969}, a neutron star consists of two components: the normal component where charged particles (protons and electrons) co-rotate with the neutron star's magnetic field with moment of inertia $I_{\\mathrm{c}}$ and the neutron superfluid with moment of inertia $I_{\\mathrm{s}}$. The rotating superfluid (both in the core and inner crust) is considered to be an array of vortices which are pinned to the crustal lattice of ions. When the neutron star slows down, a rotational lag is developed between the vortices and the normal component. Eventually, vortices are unpinned and suddenly move out after a certain critical value of rotational lag, leading to a glitch. The time required to build a glitch is inversely proportional to the spin-down rate therefore, pulsars with higher spin-down rates are expected to glitch more often. Moreover, the coupling time scales between the crust and core are proportional to the pulse period as $\\tau = 10-100 \\, P_{\\mathrm{s}}$ \\citep{alpar1984b,alpar1988,sidery2009}. Since SXP 1062 has a long pulse period along with a strong spin-down rate, it is a good candidate for observing such glitches. In accretion powered pulsars, the time scales for both glitch rise and decay are suggested to be long therefore, a glitch would appear as a single jump in frequency leaving the spin down-rate almost unchanged \\citep{ducci2015}. The jump in pulse frequency can be estimated via \\citep{ducci2015}\n\n\\begin{equation}\n \\Delta\\nu \\simeq 2\\times 10^{-5} \\, \\frac{Q_{0.95} \\, R_{6}^{2} \\, f_{15}}{M_{1.4} \\, [ \\, 1-Q_{0.95} \\, (1-Y_{0.05}) \\, ]} \\qquad \\mathrm{Hz} \\, \\mathrm{s^{-1}} \\, ,\n\\end{equation}\n\n\\noindent where $Q$ ($= I_{\\mathrm{s}} \/ (I_{\\mathrm{c}} + I_{\\mathrm{s}})$) is the fraction of superfluid in the neutron star ($Q_{0.95}$ in units of 0.95), $Y$ is the fraction of vortices coupled to normal crust ($Y_{0.05}$ in units of 0.05) and $f$ is the pinning force ($f_{15}$ in units of $10^{15}$ dyn cm$^{-1}$). The parameter $Y$ represents short time dynamics and approaches to 1 for long time scales (steady state). Assuming a neutron star with a mass of 1.4M$_{\\odot}$, a radius of 10 km, $f_{15} \\!\\simeq\\! 1$ dyn cm$^{-1}$ and by using the $\\Delta\\nu$ value observed for SXP 1062; we find that for a superfluid fraction around 95 per cent the fraction of coupled vortices is around 78 per cent. \n\nBoth glitch rise and decay times for SXP 1062 should be at the order of a day or less ($\\tau = 10-100 \\, P_{\\mathrm{s}} \\simeq 10^{4}-10^{5}$ s) however, the sampling of TOAs around the glitch is about 3--4 days. Therefore; we observe neither the rise nor the decay of the glitch, since the glitch rise and decay should have already finished within the observational gaps. Thus for SXP 1062, the observed step-like change in pulse frequency and its magnitude can be qualitatively explained by the model of \\cite{ducci2015}.\n\nSXP 1062 has a strong and steady spin-down rate among accretion powered X-ray pulsars. Moreover, SXP 1062 is associated with a young supernova remnant with an age of 10--40 kyr \\citep{henault2012,haberl2012}, therefore it is a young pulsar spinning down very fast in the remnant. The detection rate of glitches are observed to be higher for younger pulsars \\citep{espinoza2011} and long intervals of steady spin rates are expected to increase glitch possibility \\citep{ducci2015}. Therefore, these unique properties of SXP 1062 allows the vortices to creep and pin to the crustal nuclei \\citep{alpar1984a,alpar1984b}. Sudden unpinning of vortices may cause a large glitch event, which is observed in this case with $\\Delta \\nu \/ \\nu \\! \\sim \\! 10^{-3}$ being the largest value of fractional frequency jump reported as far. The fractional size of the glitch suggests that $I_{\\mathrm{s}}\/I_{\\mathrm{c}}$ is around $10^2$ which corresponds to soft equation of state \\citep{datta1993,delsate2016}. It is possible to observe a glitch in this source again. In addition, the long pulse period of SXP 1062 makes it possible to reveal glitch rise and crust core coupling time if future observations are sampled closely \\citep{newton2015}. Future monitoring of this source with \\textit{LOFT} and \\textit{NICER} can reveal more information about the interior of the neutron star. \n\n\\section*{Acknowledgements}\n\nWe acknowledge support from T\\\"{U}B\\.{I}TAK, the Scientific and Technological Research Council of Turkey through the research project MFAG 114F345. We thank M. Ali Alpar for helpful comments. We also thank the anonymous referee for the valuable comments that helped to improve the manuscript.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe rank of a finitely generated group is the minimum cardinality of a generating set. There are very few families of groups for which one knows how to compute the rank (see \\cite{kw} and references therein), and there exists no algorithm computing the rank of a word-hyperbolic group \\cite{sh}. \n\nBy Grushko's theorem, rank is additive under free product. It does not behave as nicely under direct product, even when one of the factors is $\\Z$: the solvable Baumslag-Solitar group $BS(1,2)=\\langle a,t\\mid tat\\m=a^2\\rangle$ and the product $BS(1,2)\\times \\Z$ both have rank 2.\n\nIn this paper we consider semi-direct products $G=A\\rtimes_\\varphi\\Z$ (also known as mapping tori), with the generator of the cyclic group $\\Z$ acting on $A$ by some automorphism $\\varphi\\in Aut(A)$. This was motivated by the remark that, when $A$ is a free group $F_d$ and $\\varphi$ has finite order in $Out(F_d)$, then $G$ is a generalized Baumslag-Solitar group and its rank may be computed \\cite{le}. But we do not know how to compute the rank when $\\varphi$ has infinite order. Abelianizing does not help much, so we ask:\n\n\\begin{qst*} Given $\\varphi\\in GL(d,\\Z)$, can one compute the rank of $G=\\Z^d\\rtimes_\\varphi\\Z$?\n\\end{qst*}\n\nWe can prove:\n\n\\begin{thm} \\label{decid} Given $\\varphi\\in GL(d,\\Z)$, one can decide whether $G=\\Z^d\\rtimes_\\varphi\\Z$ has rank 2 or not. \n\\end{thm}\n\n\nIt turns out that the rank of $G$ is 1 plus the minimum number $k$ such that $\\Z^d$ may be generated by $k$ orbits of $\\varphi$ (i.e.\\ there exist $g_1,\\dots,g_k\\in\\Z^d$ such that the elements $\\varphi^n(g_i)$, for $n\\in\\Z$ and $i=1,\\dots, k$, generate $\\Z^d$). \nIn particular, $G$\nhas rank 2 if and only if $\\Z^d$ may be generated by a single $\\varphi$-orbit. This happens precisely when\n$\\varphi$ is conjugate to the companion matrix having the same characteristic polynomial. This may be decided since the conjugacy problem is solvable in $GL(d,\\Z)$ \\cite{grunewald}. \n\nTheorem \\ref{decid} extends to the case when $\\varphi$ is an automorphism of an arbitrary finitely generated nilpotent group $A$.\n\nWhen $G$ has rank 2, one can classify generating pairs up to Nielsen equivalence. In particular: \n\n\\begin{thm}Suppose that $G=\\Z^d\\rtimes_\\varphi\\Z$ has rank 2.\n There are infinitely many Nielsen classes of generating pairs if and only if the cyclic subgroup of $GL(d,\\Z)$ generated by $\\varphi$ has infinite index in its centralizer. \n\\end{thm}\n\n\nOur next result is motivated by the following theorem due to J.\\ Souto:\n\\begin{thm}[\\cite{souto}] Let $A$ be the fundamental group of a closed orientable surface of genus $g\\ge 2$. Let $\\varphi$ be an automorphism of $A$ representing a pseudo-Anosov mapping class. Then there exists $n_0$ such that the rank of $G_n=A\\rtimes_{\\varphi^n}\\Z$ is $2g+1$ for all $n\\ge n_0$.\n\\end{thm}\n\nWe prove:\n\\begin{thm} \\label{powe} Given $\\varphi$ of infinite order in $ GL(d,\\Z)$, with $d\\ge2$, there exists $n_0$ such that the rank of $G_n=\\Z^d\\rtimes_{\\varphi^n}\\Z$ is $\\ge3$ for all $n\\ge n_0$.\n\\end{thm}\n\nThe theorem becomes false if the hypothesis that $\\varphi$ has infinite order is dropped, or if 3 is replaced by 4. We do not know hypotheses that would guarantee that the rank is $d+1$ for $n$ large. \n\n\nAn equivalent formulation of Theorem \\ref{powe} is:\n\\begin{thm} \\label{pow2} Given a matrix $M$ of infinite order in $ GL(d,\\Z)$, with $d\\ge2$, there exists $n_0$ such that $M^n$ is not conjugate to a companion matrix if $n\\ge n_0$.\n\\end{thm}\n\nOur proof is based on the Skolem-Mahler-Lech theorem on linear recurrent sequences \\cite{rs}. There are alternative approaches based on equations in $S$-units and Baker's theory on linear forms in logarithms. They are due to Amoroso-Zannier \\cite {AZ} and yield uniformity: \\emph{one may take $n_0=[C d^6(\\log d)^6]$ where $C$ is a universal constant (independent of $M$).}\n\n\nWe conclude with a few open questions. \n\nOur analysis on $\\Z^d$ uses the Cayley-Hamilton theorem. This is not available in a non-abelian free group $F_d$. Given $\\varphi\\in Aut(F_d)$, can one decide whether $F_d$ may be generated by a single $\\varphi$-orbit? More basically: given $\\varphi\\in Aut(F_d)$ and $g\\in F_d$, can one decide whether the $\\varphi$-orbit of $g$ generates $F_d$?\n\nWhat about ascending HNN extensions? For instance, let $\\varphi$ be an injective endomorphism of $\\Z^d$ (a matrix with integral entries and non-zero determinant). Let $G=\\Z^d*_\\varphi=\\langle \\Z^d,t\\mid tgt\\m=\\varphi(g)\\rangle$. Can one decide whether $G$ has rank 2?\n\\medskip\n\n{\\it \\small\nAcknowledgements. We wish to thank J.-L.\\ Colliot-Th\\'el\\`ene, F.\\ Grunewald, P.\\ de la Harpe, G.\\ Henniart, and number theorists in Caen, in particular F.\\ Amoroso, J.-P.\\ Bezivin, D.\\ Simon, for helpful conversations related to this work. The second author would also like to thank LMNO of Universit\\'e de Caen for their hospitality during the\npreparation of the present work.}\n\n \n \\section{Generalities}\n \nLet $A$ be a finitely generated group. The letters $a,b,v$ will always denote elements of $A$. We denote by $i_a$ the inner automorphism $v\\mapsto ava\\m$. \n\n\nGiven $\\varphi\n\\in Aut(A)$, we let $G$ \nbe the mapping torus $G=A\\rtimes_\\varphi \\Z= \\langle A,t\\mid tat\\m=\\varphi(a)\\rangle$. There is an exact sequence $1\\to A\\to G\\to \\Z\\to 1$. Up to isomorphism, $G$ only depends on the image of $\\varphi$ in $Out(A)$. Any $g\\in G$ has unique forms $at^n, t^na'$ with $n\\in\\Z$. \n\nIf $N$ is a characteristic subgroup of $A$, we denote by $\\bar \\varphi$ the automorphism induced on $A\/N$. There is an exact sequence $1\\to N\\to A\\rtimes_\\varphi\\Z\\to A\/N\\rtimes_{\\bar\\varphi}\\Z\\to 1$.\n\n \n The rank $rk(G)$ is the minimum cardinality of a generating set. We let $vrk(G)$ be the minimum number of elements needed to generate a finite index subgroup: $vrk(G)=\\inf _H rk(H)$ with the infimum taken over all subgroups of finite index.\n \n Two generating sets are Nielsen equivalent if one can pass from one to the other by Nielsen operations: permuting the generators, replacing $g_i$ by $g_i\\m$ or $g_ig_j$. For instance, any generating set of $\\Z$ is Nielsen equivalent to $\\{0,\\dots,0,1\\}$ by the Euclidean algorithm.\n \n The $\\varphi$-orbit of $a\\in A$ is $\\{\\varphi^n(a)\\mid n\\in\\Z\\}$. We denote by\n$\\Or(\\varphi)$ the minimum number of $\\varphi$-orbits needed to generate $A$. Clearly $\\Or(\\varphi)\\le rk(A)$.\n We also denote by \n $\\vor(\\varphi)$ the minimum number of $\\varphi$-orbits needed to generate a finite index subgroup of $A$, so $\\vor(\\varphi)\\le vrk(A)$.\n\n\n \\begin{lem} Given $a,a_1,\\dots,a_k\\in A$, the intersection $A'= \\langle a_1,\\dots,a_k, at\\rangle\\cap A$ is generated by the $(i_a\\circ \\varphi)$-orbits of $a_1,\\dots,a_k$. \n \n\n The $(i_a\\circ \\varphi)$-orbits of $a_1,\\dots,a_k$ generate $A$ if and only if $a_1,\\dots,a_k, at$ generate $G$.\n \\end{lem}\n \n \\begin{proof} One has $(i_a\\circ \\varphi)^n(v)=(at)^nv(at)^{-n}$ for $v\\in A$ and $n\\in\\Z$. This shows that the $(i_a\\circ \\varphi)$-orbit of $a_i$ is contained in $A'$. \n Conversely, if $v\\in A'$, write it in terms of $a_1,\\dots,a_k, at$. The exponent sum of $t$ is 0, so $v$ is a product of elements of the form $(at)^na_i(at)^{-n}$.\n \n If $A'=A$, then $\\langle a_1,\\dots,a_k, at \\rangle$ contains $A$ and $at$, so equals $G$. \n \\end{proof}\n \n \\begin{cor} $rk(G)=1+\\min_{a\\in A}\\Or(i_a\\circ \\varphi)$.\n\\end{cor}\n\n\\begin{proof} $\\le$ is clear. For the converse, use that any finite generating set of $G$ is Nielsen equivalent to a set $\\{a_1,\\dots,a_k, at\\}$ (Euclid's algorithm).\n\\end{proof}\n\n \\begin{cor} $vrk(G)=1+\\min_{a\\in A, n\\ne0}\\vor(i_a\\circ \\varphi^n)$.\n\\end{cor}\n\n\\begin{proof} If $n\\ne0$ and the $(i_a\\circ \\varphi^n)$-orbits of $a_1,\\dots,a_k$ generate a finite index subgroup of $A$, the subgroup of $G$ generated by $a_1,\\dots,a_k, at^n$ has finite index because it maps onto $n\\Z$ and it meets $A$ in a subgroup of finite index.\n\nAny finite subset of $G$ generating a finite index subgroup is Nielsen equivalent to $\\{a_1,\\dots,a_k, at^n\\}$ with $n\\ne0$, and the $(i_a\\circ\\varphi^n)$-orbits of $a_1,\\dots,a_k$ generate a finite index subgroup of $A$. \\end{proof}\n\n \n \\begin{cor} Suppose that \n $A$ is abelian.\n \\begin{enumerate}\n \n\\item \n $rk(G)=1+ \\Or( \\varphi)$ and $vrk(G)=1+ \\vor( \\varphi )$. \n\\item $G$ has rank $\\le2$ if and only if $A$ \nis generated by a single $\\varphi$-orbit. A pair $(a_1,at)$ generates $G$ if and only if the $\\varphi$-orbit of $a_1$ generates $A$. \n\\item $vrk(G)$ is computable.\n \\end{enumerate}\n\\end{cor}\n\n\\begin{proof} $i_a$ is the identity and $\\vor( \\varphi )\\le \\vor( \\varphi^n)$, so 1 follows from previous results. 2 is clear. \n\nFor 3, first suppose $A=\\Z^d$. View $\\varphi$ as an automorphism of the vector space $\\Q^d$. Then $\\vor( \\varphi )$ is the minimum number of $\\varphi$-orbits needed to generate $\\Q^d$. This is computable (it is the number of blocks in the rational canonical form of $\\varphi$). If $A$ has a torsion subgroup $T$, then $A\/T\\simeq \\Z^d$ for some $d$. Let $\\bar \\varphi$ be the automorphism induced on $\\Z^d$. Then $\\vor( \\varphi )=\\vor(\\bar \\varphi )$ is computable.\n\\end{proof}\n\n\\section{Computability}\n\nSuppose $A=\\Z^d$ with $d\\ge1$. We view $\\varphi\\in Aut(A)$ as an automorphism of $\\Z^d$ or as a matrix in $ GL(d,\\Z)$. Its companion matrix $M_\\phi$ is the unique matrix of the form $$ \\left(\\begin{array}{ccccc}\n0&&&&*\\\\\n1&0&&&*\\\\\n&\\ddots&\\ddots&&*\\\\\n&&1&0&*\\\\\n&&&1&*\n \\end{array}\\right)$$ having the same characteristic polynomial as $\\varphi$ (the empty triangles are filled with 0's, and $*$ denotes an arbitrary integer).\n \n \n \n \\begin{lem} \\label{comp} \n Let $\\varphi\\in GL(d,\\Z)$, with $d\\ge1$. \n \\begin{enumerate}\n \\item\n The following are equivalent:\n \\begin{enumerate}\n\\item $G=\\Z^d\\rtimes_\\varphi \\Z$ has rank 2;\n\\item $\\Z^d$ may be generated by a single $\\varphi$-orbit;\n\\item There exists $a\\in \\Z^d$ such that $\\{a,\\varphi(a),\\dots,\\varphi^{p-1}(a)\\}$ is a basis of $\\Z^d$. \n\\item $\\phi$ is conjugate to its companion matrix $M_\\varphi$ in $GL(d,\\Z)$.\n \\end{enumerate}\n \\item Suppose that the $\\varphi$-orbit of $a$ generates $\\Z^d$. Then the $\\varphi$-orbit of $b$ generates $\\Z^d$ if and only if $b=h(a)$ where $h\\in GL(d,\\Z)$ commutes with $\\varphi$. \n \\end{enumerate}\n \\end{lem}\n \n\n\n\\begin{proof} We already know that (a) is equivalent to (b). If $a$ is the first element of a basis of $\\Z^d$ in which $\\varphi$ is represented by the matrix $M_\\phi$, then the basis is $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$ and \nthe $\\varphi$-orbit of $a$ generates $\\Z^d$, so $(d)\\Rightarrow(c)\\Rightarrow(b)$.\n\nConversely, suppose that the $\\varphi$-orbit of $a$ generates $\\Z^d$. By the Cayley-Hamilton theorem, $\\Z^d$ is generated by $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$. This set is a basis of $\\Z^d$ in which $\\varphi$ is represented by $M_\\phi$. This proves 1.\n\nTo prove 2, suppose that $h$ commutes with $\\varphi$, and define $b=h(a)$. The image of the basis $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$ by $h$ is $\\{b,\\varphi(b),\\dots,\\varphi^{d-1}(b)\\}$, so the orbit of $b$ generates. Conversely, if the orbit of $b$ generates, define $h$ as the automorphism taking $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$ to $\\{b,\\varphi(b),\\dots,\\varphi^{d-1}(b)\\}$. It commutes with $\\varphi$ because $M_\\varphi$ represents $\\varphi$ in both bases.\n\\end{proof}\n\n\n\n\n \n\n\\begin{prop}\\label{compu}\n If \n $A$ is nilpotent, one can decide whether $G=A\\rtimes_\\varphi \\Z$ has rank 2 or not.\n\\end{prop}\n\n\\begin{proof} If $A=\\Z^d$, one has to decide whether $\\phi$ is conjugate to its companion matrix $M_\\varphi$ in $GL(d,\\Z)$. This is possible because the conjugacy problem is solvable in $GL(d,\\Z)$ by \\cite{grunewald}.\n\nWe now assume that $A$ is abelian. It fits in an exact sequence $0\\to T\\to A\\to \\Z^d\\to 0$ with $T$ finite. We denote by $a\\mapsto \\bar a$ the map $A\\to \\Z^d$, and by $h\\mapsto \\bar h$ the natural epimorphism $Aut(A)\\to Aut(\\Z^d)$. They each have finite kernel. \n\nWe have to decide whether $A$ may be generated by a single $\\varphi$-orbit. We first check whether the matrix of $\\bar\\varphi$ is conjugate to its companion matrix.\nIf not, the answer to our question is no. If yes, \\cite{grunewald} yields a conjugator and therefore an explicit $u\\in\\Z^d$ whose $\\bar\\varphi$-orbit generates $\\Z^d$. \n\nWe claim that $A$ may be generated by a single $\\varphi$-orbit if and only if there exist $a\\in A$ mapping onto $u$, and $\\psi\\in Aut(A)$ of the form $h\\varphi h\\m$ with $h\\in Aut(A)$ and $[\\bar h,\\bar\\varphi]=1$, such that the $\\psi$-orbit of $a$ generates $A$. \n\nThe ``if'' direction is clear. Conversely, suppose that the $\\varphi$-orbit of $b$ generates $A$. Then the $\\bar\\varphi$-orbit of $\\bar b$ generates $\\Z^d$, so by Lemma \\ref{comp} there exists $\\theta\\in Aut(\\Z^d)$ commuting with $\\bar \\varphi$ and mapping $\\bar b$ to $u$.\nLet $h$ be any lift of $\\theta$ to $Aut(A)$. Defining $a=h(b)$ and $\\psi=h\\varphi h\\m$, it is easy to check that the $\\psi$-orbit of $a$ generates $A$. This proves the claim.\n\nWe now explain how to decide whether $a$ and $\\psi$ as above exist. Note that $a$ and $\\psi$ must belong to explicit finite sets: $a$ belongs to the preimage $A_u$ of $u$, and \n$\\psi$ belongs to the preimage $X_\\varphi$ of $\\bar\\varphi$ in $Aut(A)$. \n\nBy Theorem C of \\cite{grunewald}, the centralizer of $\\bar\\varphi$ in $Aut(\\Z^d)$ is a finitely generated subgroup and one can compute a finite generating set. The same is true of \n $D=\\{h\\in Aut(A)\\mid [\\bar h,\\bar\\varphi]=1\\}$, so we can list the elements $\\psi$ in the orbit $D\\varphi$ of $\\varphi$ for the action of $D$ on $X_\\varphi$ by conjugation. \n \nBy the claim proved above, $A$ may be generated by a single $\\varphi$-orbit if and only if there exist \n $a \\in A_u$ and $\\psi\\in D\\varphi$ such that the $\\psi$-orbit of $a$ generates $A$. To decide this, we enumerate the pairs $(a,\\psi)$ with $a\\in A_u$ and $\\psi\\in D\\varphi$. For each pair, \n we consider the increasing sequence of subgroups $A_N=\\langle \\psi^{-N}(a), \\dots, \\psi^{-1}(a), a,\\psi(a), \\dots\\psi^{N}(a)\\rangle$. It stabilizes and we check whether $A_N=A$ for $N$ large. \n \n This completes the proof for $A$ abelian. If $A$ is nilpotent, let $B$ be its abelianization and let $\\rho:B\\to B$ be the automorphism induced by $\\varphi$. If $G_\\varphi=A\\rtimes_\\varphi\\Z$ has rank 2, so does its quotient $G_\\rho=B\\rtimes_\\rho\\Z$. Conversely, if $G_\\rho$ has rank 2, it is generated by $t$ and some $b\\in B$ whose $\\rho$-orbit generates $B$. Let $a$ be any lift of $b$ to $A$. The subgroup of $A$ generated by the $\\varphi$-orbit of $a$ maps surjectively to $B$, so equals $A$ by a classical fact about nilpotent groups (see e.g.\\ Theorem 2.2.3(d) of \\cite{k}). Thus $G_\\varphi$ has rank 2.\n \\end{proof}\n \n\\begin{cor} If $A=\\Z^2$ or $A=F_2$, one can compute the rank of $G$. \n\\end{cor}\n\n\\begin{proof} The rank is 2 or 3, so this is clear from the proposition if $A=\\Z^2$. \n\nRecall that the natural map $ Out(F_2)\\to Out(\\Z^2)=Aut(\\Z^2)$ is an isomorphism (both groups are isomorphic to $GL(2,\\Z)$).\nGiven $G=F_2\\rtimes_\\varphi \\Z$, let $\\rho$ be the image of $\\varphi$ in $Aut(\\Z^2)$. \nConsider $G_\\rho=\\Z^2\\rtimes_{\\rho} \\Z$. We prove that $G$ and $G_\\rho$ have the same rank. \n\nClearly $2\\le rk(G_\\rho)\\le rk(G)\\le3$. If $G_\\rho$ has rank 2, Lemma \\ref{comp} lets us assume that $\\rho$ is of the form $ \\left(\\begin{array}{rr} 0&\\pm1\\\\ 1&n \\end{array}\\right)$. Since $G$ only depends on the class of $\\varphi$ in $Out(F_2)$, it is isomorphic to $$\\langle a,b,t\\mid tat\\m=b,tbt\\m=a^{\\pm1}b^n\\rangle,$$ so has rank 2.\n\\end{proof} \n\n\\section{Nielsen equivalence}\n\n \\begin{prop} \\label {gp} Suppose \nthat $A$ is abelian and $G=A\\rtimes_\\varphi\\Z$ has rank 2. \n \\begin{enumerate}\n \n\\item Any generating pair of $G$ is Nielsen equivalent to a pair $(a,t)$ with $a\\in A$.\n \n\\item Two generating pairs $(a,t)$ and $(b,t)$, with $a,b\\in A$, are Nielsen equivalent if and only if $b$ belongs to the $\\varphi$-orbit of $a$ or $a\\m$.\n\\end{enumerate}\n \\end{prop}\n\n\\begin{proof} Given $x,y\\in A$, and $n$, write\n$$\n(x,ty)\\sim((ty)^nx(ty)^{-n},ty)=(\\varphi^n(x),ty)\n$$\nand\n$$\n(x,ty)\\sim (\\varphi^n(x),ty)\\sim(\\varphi^n(x),ty\\varphi^n(x))\\sim( x,ty\\varphi^n(x)).\n$$\n\nEvery generating pair is equivalent to some $(a,ty)$, with the $\\varphi$-orbit of $a$ generating $A$. But $(a,ty)\\sim ( a,ty\\varphi^n(a))$ so by an easy induction $(a,ty)\\sim (a,t)$. This proves 1.\n\nIf $b=\\varphi^n(a^\\varepsilon)$ with $\\varepsilon=\\pm1$, then $(b,t)=(\\varphi^n(a^\\varepsilon),t)=(t^na^\\varepsilon t^{-n},t)\\sim(a,t)$. The converse follows from Theorem 2.1 of \\cite{hw}.\nWe give a proof for completeness. If $(b,t)\\sim(a,t)$, we can write $b=w(a,t)$ with $w$ a primitive word with exponent sum 0 in $t$. Such a word is conjugate to $a^{\\pm1}$ in the free group $F(a,t)$, so $b$ is conjugate to $a^{\\pm1}$ in $G$. Since $A$ is abelian, $b$ belongs to the $\\varphi$-orbit of $a^{\\pm1}$. \n\\end{proof}\n\n\\begin{rem} More generally, if $A$ is abelian, any generating set of $G$ is Nielsen equivalent to a set of the form $\\{a_1,\\dots,a_k, t\\}$.\n\\end {rem}\n\n\\begin{rem} \\label{heis} The proposition does not extend to nilpotent groups. Let $A$ be the Heisenberg group $\\langle a,b,c\\mid [a,b]=c, [a,c]=[b,c]=1\\rangle$. Let $\\varphi$ map $a$ to $ab$ and $b$ to $b$. \nThe generating pairs $(a,t)$ and $(ac\\m,t)$ are Nielsen equivalent (even conjugate) but $ac\\m$ does not belong to the $\\varphi$-orbit of $a^{\\pm1}$.\nMoreover, $(a, tc)$ is a generating pair which is not Nielsen equivalent to a pair $(x,t)$ with $x\\in A$. \nIndeed, if it were, then $t$ would be conjugate to $tca^k$ for some $k\\in\\Z$ by \\cite{hw}. Counting exponent sum in $a$ yields $k=0$. But $t$ and $tc$ are not conjugate. \n\\end {rem}\n\n\n\\begin{cor} Let $A=\\Z^d$. If $G$ \nhas rank 2, the number of Nielsen classes of generating pairs is equal to the index of the group generated by $\\varphi$ and $-Id$ in the centralizer of $\\varphi$ in $GL(d,\\Z)$.\n\\end{cor}\n\n\\begin{proof} By Proposition \\ref{gp} we need only consider generating pairs of the form $(a,t)$. Fix one. To any $b\\in \\Z^d$ such that $(b,t)$ generates $G$ we associate the automorphism $\\psi_b$ of $\\Z^d$ taking the basis $\\{a,\\varphi(a),\\dots,\\varphi^{d-1}(a)\\}$ to the basis $\\{b,\\varphi(b),\\dots,\\varphi^{d-1}(b)\\}$. By Lemma \\ref{comp}, the image of this map $b\\mapsto \\psi_b$ is the centralizer of $\\phi$ in $GL(d,\\Z)$. By Proposition \\ref{gp}, $(b,t)\\sim(a,t)$ if and only if $\\psi_b$ is $\\pm \\varphi^n$ for some $n\\in \\Z$. \n\\end{proof}\n\n \\begin{example*} The number of Nielsen classes of generating pairs is always finite if $d=2$. If $\\phi=\n \\left(\\begin{array}{rrrr} 0&1&0&0\\\\ 1&1&0&0\\\\ 0 &0&0&1\\\\ 0&0&1&0 \\end{array}\\right)$, this number is infinite.\n \\end{example*}\n \n\\section{Powers}\n\nFix $\\varphi\\in GL(d,\\Z)$.\nSay that $v\\in \\Z^d$ is \\emph{$\\varphi$-cyclic} if \n its $\\varphi$-orbit generates $\\Z^d$, or equivalently if $\\{v,\\phi(v),\\dots,\\phi^{d-1}(v)\\}$ is a basis of $\\Z^d$. \n The existence of such a $v$ is equivalent to \n $\\phi$ being conjugate to its companion matrix, and also to $G$ having rank 2. If $v$ is $\\varphi^n$-cyclic for some $n\\ge2$, it is $\\varphi$-cyclic since its $\\varphi^n$-orbit is contained in its $\\varphi$-orbit. \n \n If $v$ is $\\varphi$-cyclic, we denote by $\\delta_n$ \n the index of the subgroup of $\\Z^d$ generated by the $\\varphi^n$-orbit of $v$.\n It does not depend on the choice of $v$ since $\\varphi$ always has matrix $M_\\phi$ in the basis $\\{v,\\varphi(v),\\dots,\\varphi^{d-1}(v)\\}$. Also note that $\\delta_1=1$. The group $G_n=\\Z^d\\rtimes_{\\varphi^n}\\Z$ has rank 2 (equivalently, $\\varphi^n$ is conjugate to its companion matrix) if and only if $\\delta_n=1$. \n\n\n\\begin{thm} \\label{po2}\nIf $\\varphi\\in GL(2,\\Z)$ has infinite order, the rank of $G_n=\\Z^2\\rtimes_{\\varphi^n}\\Z$ is 3 for all $n\\ge3$. \n\\end{thm}\n\n\\begin{proof}\n If $G_n$ has rank 2 for some $n$, there exists a $\\varphi^n$-cyclic element $v$. Such a $v$ is also $\\varphi$-cyclic. In the basis $\\{v,\\varphi(v)\\}$, the matrix of $\\varphi$ has the form \n $ M=\\left(\\begin{array}{rr} 0&\\varepsilon \\\\ 1&\\tau \\end{array}\\right)$ with $\\varepsilon=\\pm1$. If finite, the index \n$\\delta_n$ is the absolute value of the determinant $c_n$ of the matrix expressing the family $\\{v,\\varphi^n(v)\\}$ in the basis $\\{v,\\varphi(v)\\}$. We prove the theorem by showing $ | c_n | >1$ for $n\\ge3$.\n \n The number $c_n$ is determined by the equation $M^n=c_n M+ d_n I$. It follows from the Cayley-Hamilton theorem that the sequence $c_n$ satisfies the recurrence relation $c_{n+2}-\\tau c_{n+1}-\\varepsilon c_n=0$. \n \n If $\\varepsilon=-1$ one has $$c_n=\\prod_{k=1}^{n-1}(\\tau-2\\cos\\frac{k\\pi}n) $$\n because $c_n$ is a monic polynomial of degree $n-1$ in $\\tau$ which vanishes for $\\tau=2\\cos\\frac{k\\pi}n$ \n (one also has $c_n=U_{n-1}(\\tau\/2)$, with $U_{n-1}$ a Chebyshev polynomial of the second kind). \n \n \n If $\\varepsilon=1$ one has \n $$c_n=\\prod_{k=1}^{n-1}(\\tau-2i\\cos\\frac{k\\pi}n) .$$\n \n \n Since $\\varphi$ is assumed to have infinite order, one has $\\tau\\ne0$ if $\\varepsilon=1$, and $ | \\tau | \\ge2$ if $\\varepsilon=-1$. One checks that $ | c_n | >1$ for $n\\ge3$ (for $n\\ge 2$ if $\\varepsilon=-1$).\n \n \\end{proof}\n\n\n\n\\begin{thm} \\label{pow}\nSuppose that $\\varphi\\in GL(d,\\Z)$ has infinite order. \n\\begin{enumerate}\n\\item There exists $n_0$ such that $G_n=\\Z^d\\rtimes_{\\varphi^n}\\Z$ has rank $\\ge3$ for every $n\\ge n_0$. Equivalently: $\\varphi^n$ is not conjugate to its companion matrix for $n\\ge n_0$.\n\\item \nMore precisely, the minimum index of 2-generated subgroups of $G_n$ goes to infinity with $n$.\n\\end{enumerate}\n\\end{thm}\n\n\n Note that there are arbitrarily large values of $n$ for which the rank of $G_n$ is $d+1$ (whenever $\\varphi^n$ is the identity modulo some prime number). As already mentioned, it is proved in \\cite{AZ} that $n_0$ may be chosen to depend only on $d$.\n \n The key step in the proof of Theorem \\ref{pow} \n is the following result.\n \n \\begin{prop} \\label{key}\n If $\\varphi$ has infinite order and $v $ is $\\varphi$-cyclic, then the index $\\delta_n$ of the subgroup of $\\Z^d$ generated by the $\\varphi^n$-orbit of $v$ goes to infinity with $n$.\n \\end{prop} \n \n \\begin{proof}[Proof of the theorem from the proposition]\n As above, if $G_n$ has rank 2 for some $n$, there exists a $\\varphi $-cyclic element $v$. \n For $n$ large one has $\\delta_n>1$, so $G_n$ has rank $>2$. Assertion 1 is proved.\n \n For Assertion 2, suppose that there are arbitrarily large values of $n$ such that $G_n$ contains a 2-generated subgroup $H_n$ of index $\\le C$, for some fixed $C$. This subgroup has a generating pair of the form $(a_n,t_n)$ with $a_n \\in\\Z^d$, and the intersection of $H_n$ with $\\Z^d$ is generated by the $\\varphi^{nm_n}$-orbit of $a_n$ for some $m_n\\ge1$. It has index $\\le C$ in $\\Z^d$. \n \n \n The subgroup of $\\Z^d$ generated by the $\\varphi$-orbit of $a_n$ has index $\\le C$, so we can assume that it does not depend on $n$. Call it $J$. It is $\\varphi$-invariant so we can apply the proposition to the action of $\\varphi$ on $J$, with $v=a_n$. This gives the required contradiction.\n \\end{proof}\n \n \n\n \n \\begin{proof} [Proof of Proposition \\ref{key}] When $d=2$, one easily checks that $ c_n $, as computed above, goes to infinity with $n$. The proof in the general case is more involved. \n \n Define numbers $u_k(i)$, \n for $k=0,\\dots,d-1$ and $i\\ge0$, by $\\varphi^i(v)=\\sum_{k=0}^{d-1}u_k(i)\\varphi^k(v)$. The sequences $u_0,\\dots, u_{d-1}$ \n form a basis for the \n space $\\cals$ of sequences satisfying the linear recurrence associated to the characteristic polynomial of $\\varphi$ (the recurrence is $\\sum_{j=0}^d a_ju_k(i+j)=0$ if the characteristic polynomial is $\\sum _{j=0}^d a_jX^j$). \n \n The index $\\delta_n$ is the absolute value of the determinant $c_n$ of the matrix $(u_k(ni))_{0\\le i,k\\le d-1}$ (it is infinite if the determinant is $0$). We have to prove that, given $c\\ne0$, the set of $n$'s such that $c_n=c\n $ is finite. We assume it is not and we work towards a contradiction. \n \n A sequence satisfies a linear recurrence if and only if it is a finite sum of polynomials times exponentials, so $c_n$ also is a recurrent sequence. The Skolem-Mahler-Lech theorem \\cite{rs} then implies that $c_n=c\n $ for all $n$ in an arithmetic progression $\\N_0\\subset \\N$.\n \n \n We shall now replace the basis $u_k$ of $\\cals$ by another basis $w_k$ depending on the eigenvalues of $\\varphi$. We then assume that $D_n:=\\det (w_k(ni))_{0\\le i,k\\le d-1}=c'\\ne0$ for $n\\in\\N_0$. \n\nWe order the eigenvalues $\\lambda_k$ of $\\varphi$ so that $0<| \\lambda_1 |\\le | \\lambda_2 |\\le\\dots\\le | \\lambda_{d} | $.\nFirst suppose that the eigenvalues are all distinct. We then choose $w_k(i)= (\\lambda_{k+1})^i$. In this case $D_n$ is a Vandermonde determinant, for instance $$D_n=\n\\left | \\begin{matrix} \n1 & 1&1 \\cr\n(\\lambda_1)^{n}& (\\lambda_2)^{n} & (\\lambda_3)^{n} \\cr \n(\\lambda_1)^{2n} &(\\lambda_2)^{2n} &(\\lambda_3)^{2n} \n \\end{matrix}\\right | $$ for $d=3$, \n so $\\displaystyle D_n=\\prod_{1\\le k< m\\le d}\\bigl((\\lambda_m)^{n} -(\\lambda_k)^{n} \\bigr)$.\n \nIf all moduli $ | \\lambda_k | $ are distinct, \nthen $ | D_n | $ goes to infinity with $n$ because its diagonal term $$ (\\lambda_2)^{n} (\\lambda_3)^{2n} \\dots(\\lambda_{d})^{(d-1)n}=\\biggl( \\lambda_2 (\\lambda_3)^{2} \\dots(\\lambda_{d})^{(d-1)}\\biggr)^n$$ has modulus bigger than all others. \n\n If the $\\lambda_k$'s are distinct but their moduli are not, \n expand $D_n$ as a sum $\\sum_j \\varepsilon_j\\mu_j{}^ n$ (with $\\varepsilon_j=\\pm1$). \n Now \n there may be several (possibly cancelling) terms for which $ | \\mu_j | $ takes its maximal value $K= | \\lambda_2 (\\lambda_3)^{2} \\dots(\\lambda_{d})^{(d-1)} | $. Note that $K>1$ because otherwise all $\\lambda_k$'s have modulus 1, hence are roots of unity by a classical result, and $\\varphi$ has finite order. \n \n Since $D_n=c'$ for $n\\in \\N_0$ and $K>1$, one has $\\sum_{| \\mu_j | =K} \\varepsilon_j\\mu_j{}^ n=0$ for $n\\in\\N_0$. Call this sum $D_{n,K}$. Recall that $\\displaystyle D_n=\\prod_{1\\le k< m\\le d}\\bigl((\\lambda_m)^{n} -(\\lambda_k)^{n} \\bigr)$. To expand this product, one chooses one of $(\\lambda_m)^{n}$ or \n$(\\lambda_k)^{n} $ for each couple $k,m$. The corresponding term contributes to $D_{n,K}$ if and only if one always chooses a term of maximal modulus. In other words, $\\displaystyle D_{n,K}=\\prod_{1\\le k< m\\le p} E_{k,m}$ with $E_{k,m}=(\\lambda_m)^{n} -(\\lambda_k)^{n}$ if $ | \\lambda_m | = | \\lambda_k | $ and\n$E_{k,m}=(\\lambda_m)^{n} $ if $ | \\lambda_m | > | \\lambda_k | $. Since the $\\lambda_k$'s are non-zero, $D_{n,K}=0$ implies \n $(\\lambda_k)^n=(\\lambda_m)^n$ for some $k,m$ with $k\\ne m$, so that $D_n=0$, a contradiction. \n\n \n This completes the proof when the eigenvalues of $\\varphi$ are distinct. In the remaining case, the basis $w_k$ must have a different form: if $\\lambda$ is an eigenvalue of multiplicity $r$, we use the sequences $\\lambda^i, i\\lambda^i, \\dots, i^{r-1}\\lambda^i$. For instance,\n $$D_n=\n\\left | \\begin{matrix} \n1 & 0&0&1 \\cr\n(\\lambda_1)^{n}& n(\\lambda_1)^{n} & n^2(\\lambda_1)^{n} & (\\lambda_4)^{n} \\cr \n(\\lambda_1)^{2n} &2n(\\lambda_1)^{2n} &(2n)^2(\\lambda_1)^{2n} &(\\lambda_4)^{2n} \\cr\n(\\lambda_1)^{3n} &3n(\\lambda_1)^{3n} &(3n)^2(\\lambda_1)^{3n} &(\\lambda_4)^{3n} \n \n \\end{matrix}\\right | $$\n when $d=4$ and $\\lambda_1=\\lambda_2= \\lambda_3\\ne \\lambda_4$.\n \n Calling $\\nu_1,\\dots,\\nu_q$ the distinct eigenvalues of $\\varphi$, there exist integers $a,b,c_k,d_{mk}$ (depending only on the multiplicities of the eigenvalues) such that $$D_n=an^b\\prod _{k=1}^q(\\nu_k)^{nc_k}\\prod _{1\\le k< m\\le q}\\bigl((\\nu_m)^{n} -(\\nu_k)^{n}\\bigr)^{d_{mk}}$$ (see \\cite{fh} or Theorem 21 in \\cite{kr}). For instance, $D_n$ as displayed above equals $2n^3(\\lambda_1)^{3n}((\\lambda_4)^{n} -(\\lambda_1)^{n})^3$.\n \n If $K>1$, we conclude as in the previous case. \n If $K=1$,\n all eigenvalues are roots of unity and $D_n=n^bE_n$ where $E_n$ only takes finitely many values and $b>0$ (an eigenvalue $\\nu_j$ of multiplicity $r\\ge 2$ contributes $1+\\dots+(r-1)$ to $b$). \nSuch a product cannot take a non-zero value infinitely often.\n \\end{proof} \n \n \n \\begin{cor} If $A$ is abelian, and $\\varphi\\in Aut(A)$ has infinite order, then \n $G_n=A\\rtimes_{\\varphi^n}\\Z$ has rank $\\ge3$ for $n$ large. The minimum index of 2-generated subgroups of $G_n$ goes to infinity with $n$.\n \\end{cor}\n \n This follows readily from Theorem \\ref{pow}, writing $A\/T\\sim\\Z^d$ with $T$ finite.\n The analogous result for nilpotent groups is false, as the following example shows. \n Let $A$ be the Heisenberg group as in Remark \\ref{heis}. If $\\varphi$ maps $a$ to $bc$, $b$ to $ac^2$, and $c$ to $c\\m$, then $\\varphi^{2n+1}(a)=bc^{1-n}$, so $G_{2n+1}$ has rank 2 since $a$ and $\\varphi^{2n+1}(a)$ generate $A$. The automorphism induced by $\\varphi$ on the abelianization of $A$ has order 2.\n \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIntegrable quantum models are gaining an increasing role in modern physics. The access to an exact solution of a many-body interacting system gives an unprecedented opportunity to explore strongly correlated quantum systems beyond perturbative or numerical methods. Circumstances are especially encouraging in one-dimensional systems where some integrable models naturally appear and the theoretical predictions are of huge experimental value. For example the recent progresses in cold atoms experiments in optical lattices allow the manufacturing of different models in an almost ideally isolated environment. Many physical properties of some paradigmatic integrable models as the XXZ spin chain and the Lieb-Liniger model were indeed observed in such experiments\\cite{2004_Paredes_NATURE_429,2008_Amerongen_PRL_100,2014_Fabbri_arXiv} and similar observations were also possible in highly spatially anisotropic crystals \\cite{2013_Mourigal_NatPhys_9,2013_Lake_PRL_111}.\n\nThe standard technique to solve interacting integrable models is the Bethe Ansatz\\cite{KorepinBOOK}. This technique provides us a complete characterization of eigenstates and, at least in some cases, the matrix elements of physical operators between two eigenstates ({\\em form factors}). However these expressions are usually cumbersome since they depend on all the $N$ variables (\\emph{rapidities}) describing the eigenstates, where $N$ is the number of constituents in the system. Moreover in order to address the two-point correlation functions one has to perform a summation of form factors over the whole Hilbert space. Performing such summations is so far beyond our analytical abilities but they can be evaluated numerically \\cite{2005_Caux_PRL_95,2006_Caux_PRA_74,PhysRevA.89.033605}. In the conformal limit the summations simplifies allowing for an exact treatment \\cite{1742-5468-2012-09-P09001,1742-5468-2011-12-P12010}. This led to a derivation of the Luttinger liquid \\cite{1981_Haldane_PRL_47,GiamarchiBOOK} (an effective theory of gapless 1D models) results for correlation functions directly from a microscopic (integrable) theory. Similar results were also worked out in this direction: the phenomenological quantities entering the Luttinger liquid description of correlation functions were connected with microscopic data \\cite{2011_Shashi_PRB_84,2012_Shashi_PRB_85}. However the full determination of correlation functions resisted so far the best efforts.\n\nOne of the main difficulty arises from a complicated structure of form factors as they are highly non-trivial functions of rapidities of two eigenstates. The thermodynamic limit at fixed density however allows for some simplifications. On the physical grounds we can reason that form factors of local operators are non-zero only when evaluated between two very similar eigenstates. Indeed a local operator is not expected to modify a macroscopic number of degrees of freedom and consequently its form factors are functions of parameters specifying one of the states as an \\emph{excited state} over the other which we call an \\emph{averaging state}. The number of such excitations is a sub-extensive number $n$ such that in the thermodynamic limit $n\/N \\to 0$. We can then distinguish two different situations depending on a type of the averaging state. When the averaging state is the ground state of a gapless theory each excitation with a finite momentum and energy gets \"dressed\" by an infinite number of zero-energy excitations. This sign of the criticality in the system can be seen in a non-integer scaling behavior of the form factors with the size of the system (here the length $L$) as $L^{- \\alpha}$, with $\\alpha$ a rational number \\cite{1990_Slavnov_TMP_82}. This makes the evaluation of the spectral sum, required for computation of dynamical correlation functions, a daunting task. Still some progress was achieved in the aforementioned conformal limit \\cite{1742-5468-2012-09-P09001,1742-5468-2011-12-P12010}. Here we focus on another regime, when the averaging state is a finite entropy state. Similar averaging states were already considered, for example, in \\cite{1742-5468-2011-01-P01011}.\n\nThe excited states of a finite entropy averaging state contribute indeed \\emph{individually} to the whole sum over the Hilbert space. This can be seen again at the level of the form factors as their scaling with the system size is determined by number of excitations $n$ as $L^{-n}$. In particular as we restrict here to the (repulsive) Lieb-Liniger model and to the density operator form factors the only relevant excitations are \\emph{particle-hole} excitations.\n\nOur main result is the thermodynamic limit of the form factors of the density operator between the finite entropy averaging state and its excited states with a number $n$ of particle-hole excitations. Such form factors are the building blocks of the correlation functions for a general, non-critical situation. The results are applicable to compute correlation functions at finite temperatures (when the averaging state is the thermal state) and also for systems out of equilibrium with the averaging state being the steady state of the unitary time evolution \\cite{2013_Caux_PRL_110,PhysRevA.89.033601}.\n\n\\subsection{Structure of the article}\n\nIn section \\ref{1DBoseGas} we recall the Bethe Ansatz solution of the Lieb-Liniger model and we collect all the necessary ingredients to compute the thermodynamic limit of the density form factors. The bulk of the computation is shown in section \\ref{thermodynamiclimit}. In section \\ref{regularization} we show how to regularize divergences present in form factors. In section \\ref{expansion} we compute the dynamical structure factor (density-density correlation function) in the $1\/c$ expansion. In section~\\ref{numerics} we numerically evaluate the dynamical structure factor by including only the simplest (1 particle-hole) excitations.\n\n\n\\section{1D Bose gas} \\label{1DBoseGas}\n\nThe Hamiltonian of $N$ bosonic particles confined in a one spatial dimension is \\cite{1963_Lieb_PR_130_1,1998_Olshanii_PRL_81}\n\\begin{align}\\label{H}\n H = -\\sum_{j=1}^N \\partial_{x_j}^2 + 2c\\sum_{j0$) interactions and we set $\\hbar=1$ and $2m=1$. The wavefunctions are superpositions of plane waves \\cite{1963_Lieb_PR_130_1},\n\\begin{align} \\label{wv_fnc}\n \\langle {\\boldsymbol{x}}|{\\boldsymbol{\\lambda}}\\rangle = \\Psi({\\boldsymbol{x}}|{\\boldsymbol{\\lambda}}) = \\prod_{j>k}^N \\textrm{sgn}\\left(x_j-x_k\\right)\\sum_{P_N} \\mathcal{A}_P e^{i\\sum_{j=1}^N \\lambda_{P_j} x_j},\n\\end{align}\nwhere the summation extends over all permutations of $N$ particles. We also adopt a shorthand notation in which ${\\boldsymbol{x}} = \\{x_j\\}_{j=1}^N$ and ${\\boldsymbol{\\lambda}} = \\{\\lambda_j\\}_{j=1}^N$. The effect of interaction is encapsulated in the coefficients\n\\begin{align}\n \\mathcal{A}_P = (-1)^{[P]} e^{i\/2\\sum_{j>k} \\textrm{sgn}(x_j-x_k)\\theta(\\lambda_{P_j} - \\lambda_{P_k})},\n\\end{align}\nwith two-particle phase shift\n\\begin{align}\\label{phase_shift}\n \\theta(\\lambda) = 2\\arctan\\left(\\lambda\/c\\right).\n\\end{align}\nThe energy of an eigenstate $|{\\boldsymbol{\\lambda}}\\rangle$ is\n\\begin{align} \\label{energy_discrete}\n E({\\boldsymbol{\\lambda}}) = \\sum_{j=1}^N \\lambda_j^2.\n\\end{align}\nThe operator of the total momentum, $\\hat{P} = -i\\sum_{j=1}^N \\partial_{x_j}$, commutes with the Hamiltonian \\eqref{H} and its eigenvalues are simply\n\\begin{align} \\label{momentum_discrete}\n P({\\boldsymbol{\\lambda}}) = \\sum_{j=1}^N \\lambda_j.\n\\end{align}\nImposing the periodic boundary conditions constrains the set of rapidities ${\\boldsymbol{\\lambda}}$ to solutions of the Bethe equations\n\\begin{align} \\label{bethe}\n \\lambda_j = \\frac{2\\pi}{L}I_j - \\frac{1}{L} \\sum_{k=1}^N \\theta \\left(\\lambda_j - \\lambda_k\\right),\n\\end{align}\nwhere $L$ is the length of the system. Quantum numbers $I_j$ are integers (half-odd integers) for $N$ odd (even) and follow the Pauli principle - wave function vanishes identically if any two of them coincide. The Hilbert space is spanned by allowed choices of quantum numbers. It is customary to name the eigenstates of \\eqref{H} in the finite system the Bethe states. We follow this tradition.\n\nThe norm of the Bethe states admits a neat representation in the form of a determinant \\cite{1971_Gaudin_JMP_12_I,1982_Korepin_CMP_86}\n\\begin{align} \\label{norm}\n |{\\boldsymbol{\\lambda}}|^2 \\equiv \\langle{\\boldsymbol{\\lambda}}|{\\boldsymbol{\\lambda}}\\rangle = c^N \\prod_{j\\neq k}^N \\frac{\\lambda_j - \\lambda_k + ic}{\\lambda_j - \\lambda_k} \\det_N \\mathcal{G},\n\\end{align}\nwhere the Gaudin matrix is\n\\begin{align} \\label{gaudin}\n \\mathcal{G}_{jk} &= \\delta_{jk}\\left(L + \\sum_{m=1}^N K\\left(\\lambda_j - \\lambda_m\\right) \\right) - K\\left(\\lambda_j - \\lambda_k\\right),\\\\\n K(\\lambda) &= \\frac{2c}{\\lambda^2 + c^2}.\\label{K}\n\\end{align}\nNote also that the kernel $K(\\lambda)$ is a derivative of the two-particle phase shift $\\theta(\\lambda)$.\n\nWe define the density operator as\n\\begin{equation}\n\\hat{\\rho}(x) = \\sum_{j=1}^N \\delta( x - x_j)\n\\end{equation}\nwhere $\\{ x_j\\}_{j=1}^N$ are the positions of all the particles in the gas. The two-point correlation function of this operator is particularly relevant for both theory and experiment.\nThe form factors of the density operator are given by the Algebraic Bethe Ansatz approach \\cite{1990_Slavnov_TMP_82}\n\\begin{align} \\label{ff}\n\\langle {\\boldsymbol{\\mu}} | \\hat{\\rho}(0) | {\\boldsymbol{\\lambda}}\\rangle = \\left( \\sum_{j=1}^N (\\mu_j - \\lambda_j) \\right)\\prod_{j=1}^N \\left(V_j^+ - V_j^-\\right) \\prod_{j,k}^N \\left(\\frac{\\lambda_j - \\lambda_k + ic}{\\mu_j - \\lambda_k}\\right)\\frac{\\det_N \\left(\\delta_{jk} + U_{jk}\\right)}{V_p^+ - V_p^-},\n\\end{align}\nwhere both $|{\\boldsymbol{\\lambda}}\\rangle$ and $|{\\boldsymbol{\\mu}}\\rangle$ are Bethe states. Different factors appearing in \\eqref{ff} are\n\\begin{align} \\label{matrix_U}\n V_j^{\\pm} &= \\prod_{k=1}^N \\frac{\\mu_k - \\lambda_j \\pm ic}{\\lambda_k - \\lambda_j \\pm ic},\\\\\n U_{jk} &= i\\frac{\\mu_j - \\lambda_j}{V_j^+ - V_j^-} \\prod_{m\\neq j}^N \\left(\\frac{\\mu_m - \\lambda_j}{\\lambda_m - \\lambda_j}\\right) \\biggl(K\\left(\\lambda_j - \\lambda_k\\right) - K\\left(\\lambda_p - \\lambda_k\\right) \\biggl),\n\\end{align}\nand $\\lambda_p$ is an arbitrary number, not necessarily from the set ${\\boldsymbol{\\lambda}}$.\n\n\n\\subsection{Thermodynamic limit}\n\nWe consider now the thermodynamic limit $N\\rightarrow\\infty$ with fixed density $D = N\/L $. We denote such limit with $\\lim_{\\text{th}}$. The Bethe states can be then characterized by a filling function $\\vartheta(\\lambda)$ defined as a number of rapidities in an interval $(\\lambda,\\lambda+d\\lambda)$ divided by a maximal number of rapidities (in this interval). Due to an interacting nature of the gas the maximal number of the particles is not constant and the density of particles is connected with the filling function through an integral equation \\cite{1969_Yang_JMP_10}\n\\begin{align} \\label{rho_p}\n 2\\pi \\rho(\\lambda) &= \\vartheta(\\lambda)\\left(1 + \\int_{-\\infty}^{\\infty} d\\mu K(\\lambda -\\mu) \\rho(\\mu)\\right).\n\\end{align}\nThe filling function obeys\n\\begin{align}\\label{filling_condition}\n 0 \\leq \\vartheta(\\lambda) \\leq 1,\n\\end{align}\nwhich guarantees the existence of $\\rho(\\lambda)$ through \\eqref{rho_p}.\nThe filling function provides a complete macroscopic characterization of the Bethe states in the thermodynamic limit. For example the extensive part of the momentum and the energy is (c.f. with eqs. \\eqref{energy_discrete} and \\eqref{momentum_discrete})\n\\begin{align}\\label{energy_cont}\n P[\\vartheta] &= L\\int_{-\\infty}^{\\infty} d\\lambda\\, \\rho(\\lambda)\\, \\lambda.\\\\\n E[\\vartheta] &= L\\int_{-\\infty}^{\\infty} d\\lambda\\, \\rho(\\lambda)\\, \\lambda^2\\,.\n\\end{align}\nIn this work we focus on the regions of the Hilbert space that are characterized by a smooth (differentiable) filling function and are of the finite energy density: $E[\\vartheta]\/L < \\infty$.\n\nA given smooth filling function corresponds to many different microscopic eigenstates. The number of them is equal to the logarithm of the entropy $S[\\vartheta]$, the later is given by \\cite{1969_Yang_JMP_10}\n\\begin{align}\n S[\\vartheta] &= L\\int_{-\\infty}^{\\infty} {\\rm d}\\lambda\\ s[\\vartheta; \\lambda],\\\\\n s[\\vartheta;\\lambda] &= \\rho_t(\\lambda)\\log \\rho_t(\\lambda) - \\rho(\\lambda)\\log \\rho(\\lambda) - \\rho_h(\\lambda)\\log \\rho_h(\\lambda),\n\\end{align}\nwhere we introduced a shorthand notation\n\\begin{align}\n \\rho_t(\\lambda) &\\equiv \\rho(\\lambda)\/\\vartheta(\\lambda),\\\\\n \\rho_h(\\lambda) &\\equiv \\rho_t(\\lambda)\\left(1-\\vartheta(\\lambda)\\right),\n\\end{align}\nwhere $\\rho_t(\\lambda)$ has a meaning of the maximal density of the rapidities and $\\rho_h(\\lambda)$ denotes the density of holes.\n\n\n\nThe density operator is diagonal in the functional space of the filling functions. Its form factors are nonzero only when the two eigenstates are characterized by the same filling function $\\vartheta(\\lambda)$ and differ only by a number $n$ of excitations such that $\\lim_{\\text{th}} \\frac{n}{N} = 0$. The density form factors are zero for states containing different number of particles and therefore these excitations occur only as particle-hole pairs. We choose a set $\\{ \\lambda_j^-\\}_{j=1}^n$ of rapidities in the averaging state and we change their values to a new set $\\{ \\mu_j^+\\}_{j=1}^n$. We denote the pair of particle-hole as $\\{\\mu_j^+, \\mu_j^-\\}_{j=1}^n$, where $\\{ \\mu_j^+ \\}_{j=1}^n$ are particles: they are not present in the averaging state, while $\\{\\mu_j^-\\}_{j=1}^n$ are holes: they are related to the rapidities we have changed in the averaging state $\\{ \\lambda^-\\}_{j=1}^n$ by $1\/L$ corrections as $\\lambda_j^- = \\mu_j^- + \\left(\\frac{F(\\mu^-)}{L} \\right)$ for each $j=1, \\ldots , n$ and with the function $F(\\lambda)$ defined below. The rapidities $\\{ \\mu_j^- \\}_{j=1}^n$ are absent in the excited state. A single excitation will be denoted here with $\\mu^- \\to \\mu^+$.\n\nDue to the correlated nature of the gas, particle-hole excitations modify the density of particles not only in the vicinity of $\\mu_j^+$ and $\\mu_j^-$. In fact the density $\\rho(\\lambda)$ acquires a change of order of $1\/L$, as can be seen from studying the difference $\\mu_j - \\lambda_j$. This difference can be conveniently expressed as \\cite{KorepinBOOK}\n\\begin{align} \\label{back flow_def}\n F(\\lambda_j) = -L\\rho_t(\\lambda_j)\\left(\\mu_j - \\lambda_j\\right) ,\n\\end{align}\nwhere $F(\\lambda)$ is the back-flow function that fulfills the following linear integral equation (for a single particle-hole excitation)\n\\begin{align} \\label{backflow}\n 2\\pi F\\left(\\lambda\\,|\\, \\mu^+, \\mu^-\\right) =&\\; \\theta(\\lambda-\\mu^+) - \\theta(\\lambda-\\mu^-) \\nonumber\\\\\n&+ \\int_{-\\infty}^{\\infty} d\\mu K(\\lambda-\\mu) \\vartheta(\\mu) F\\left(\\mu\\,|\\, \\mu^+, \\mu^-\\right) .\n\\end{align}\nThe linearity of the back-flow implies that for multiple particle-hole excitations the total back-flow is the sum of individual contributions\n\\begin{equation} \\label{back-sum-flow}\n F\\left(\\lambda\\,|\\, \\{(\\mu_j^+, \\mu_j^-)\\}_{j=1}^n\\right) = \\sum_{j=1}^n F\\left(\\lambda\\,|\\, \\mu_j^+, \\mu_j^-\\right) .\n\\end{equation}\nIt also implies the back-flow can be further factorized in the particle and hole contributions. We define the back-flow for a single excitation as\n\\begin{align} \\label{back-single-flow}\n 2\\pi F\\left(\\lambda\\,|\\, \\mu \\right) &= \\theta(\\lambda-\\mu)\n+ \\int_{-\\infty}^{\\infty} d\\gamma K(\\lambda-\\gamma) \\vartheta(\\gamma) F\\left(\\gamma|\\, \\mu \\right) .\n\\end{align}\nThis allows to write the momentum and the energy of a single excitation as\n\\begin{align}\n & k[\\vartheta; \\mu] = \\mu - \\int_{-\\infty}^{\\infty} \\rmd\\lambda \\vartheta(\\lambda) F(\\lambda|\\mu)\\label{exc_momentum} ,\\\\\n & \\omega[\\vartheta; \\mu] = \\mu^2 - 2\\int_{-\\infty}^{\\infty} \\rmd \\lambda \\vartheta(\\lambda) \\lambda F(\\lambda|\\mu)\\label{exc_energy} ,\n\\end{align}\nwhich are the fundamental building blocks for the energy and momentum of a thermodynamic state with $n$ particle-holes\n\\begin{align}\n& \\Delta \\omega = \\sum_{j=1}^n \\omega[\\vartheta; \\mu^+_j] - \\omega[\\vartheta; \\mu^-_j] , \\nonumber \\\\&\n \\Delta k = \\sum_{j=1}^n k[\\vartheta; \\mu^+_j] - k[\\vartheta; \\mu^-_j] .\n\\end{align}\nThe excited states, due to the back-flow of their rapidities, have different entropy respect to the averaging state. The difference equals \\cite{1990_Korepin_NPB_340}\n\\begin{align}\n \\delta S[\\vartheta, \\mu^+, \\mu^-] = \\int_{-\\infty}^{\\infty} {\\rm d}\\lambda\\, s[\\vartheta;\\lambda]\\frac{\\partial}{\\partial \\lambda}\\left(\\frac{F(\\lambda| \\mu^+, \\mu^-)}{\\rho_t(\\lambda)}\\right) \\equiv \\delta S[\\vartheta, \\mu^+] - \\delta S[\\vartheta, \\mu^-] ,\\label{S_diff}\n\\end{align}\nwhere in the last step we used the back-flow function for a single excitation \\eqref{back-single-flow}. The differential entropy \\eqref{S_diff} corresponds to the number of microstates that share $n$ particle-hole excitations with the same thermodynamic energy and momentum (given by \\eqref{exc_energy} and \\eqref{exc_momentum}) but with different sub-leading corrections to them.\n\nFinally, let us introduce the form factors and relate them to the correlation functions. We consider an ensemble average, denoted by $<\\cdot>$ of the density-density correlation function. We assume that the ensemble has a saddle-point configuration uniquely specifying the filling function $\\vartheta(\\lambda)$ \\cite{1969_Yang_JMP_10}. In order to compute the form factors, it is useful to directly refer to a specific microscopic configuration that has $\\vartheta(\\lambda)$ as its thermodynamic limit. We will choose one such finite size configuration and call it a averaging state $|{\\boldsymbol{\\lambda}}\\rangle$. Any other state, with the same filling function but with microscopic differences can be viewed as an excitation (with a positive or negative energy) over the averaging state. The choice of the averaging state is not unique, indeed there is a number $e^{S[\\vartheta]}$ of possible choices, but the correlation functions in the thermodynamic limit are independent of this choice for most of the relevant operators \\cite{KorepinBOOK}. For simplicity we choose here the averaging state with rapidities distributed such that for each interval $[\\lambda, \\lambda + d\\lambda]$ there are $\\rho(\\lambda) d\\lambda$ uniformly distributed rapidities: $ (\\lambda_j- \\lambda_k)= \\left(\\frac{j-k}{L \\rho(\\lambda_j) } \\right)+ \\mathcal{O}(L^{-2})$ when $\\lambda_j \\sim \\lambda_k$.\n\nWe define then the density density correlation function in the thermodynamic limit~as\n\\begin{align}\n \\langle \\hat{\\rho}(x,t) \\hat{\\rho}(0,0)\\rangle = \\frac{\\langle \\vartheta|\\hat{\\rho}(x,t) \\hat{\\rho}(0,0)|\\vartheta\\rangle }{\\langle \\vartheta|\\vartheta\\rangle} = \\lim_{\\text{th}} \\frac{\\langle {\\boldsymbol{\\lambda}}|\\rho(x,t) \\rho(0,0)|{\\boldsymbol{\\lambda}}\\rangle }{\\langle {\\boldsymbol{\\lambda}}|{\\boldsymbol{\\lambda}}\\rangle} .\n\\end{align}\nThe correlation function in the finite system can be expanded using the complete basis of Bethe states\n\\begin{align} \\label{corr_func_finite_N}\n\\frac{\\langle {\\boldsymbol{\\lambda}}|\\hat{\\rho}(x,t) \\hat{\\rho}(0,0)|{\\boldsymbol{\\lambda}}\\rangle}{\\langle {\\boldsymbol{\\lambda}}|{\\boldsymbol{\\lambda}}\\rangle} = \\sum_{\\{\\mu_j\\}_{j=1}^N} e^{ix(P_{\\mu} - P_{\\lambda}) -it(E_{\\mu} -E_{\\lambda})} |\\mathcal{F}_N\\left( {\\boldsymbol{\\mu}}, {\\boldsymbol{\\lambda}}\\right)|^2.\n\\end{align}\nwhere we defined the microscopic form factors as\n\\begin{align} \\label{micro_ff}\n \\mathcal{F}_N\\left( {\\boldsymbol{\\mu}}, {\\boldsymbol{\\lambda}}\\right) = \\frac{\\langle{\\boldsymbol{\\mu}}| \\hat{\\rho}(0,0)|{\\boldsymbol{\\lambda}}\\rangle}{\\sqrt{\\langle {\\boldsymbol{\\lambda}}|{\\boldsymbol{\\lambda}}\\rangle \\langle {\\boldsymbol{\\mu}}|{\\boldsymbol{\\mu}}\\rangle}} ,\n\\end{align}\nand we used that\n\\begin{align}\n \\hat{\\rho}(x,t) = e^{i \\left(Ht - Px\\right)} \\hat{\\rho}(0,0)e^{-i \\left(Ht - Px\\right)} .\n\\end{align}\nTo proceed further in taking the thermodynamic limit it is important to note two things. First the summation in eq.~\\eqref{corr_func_finite_N} is constrained since the set of rapidities must be a solution to the Bethe equations \\eqref{bethe}. On the other hand the form factors become in the thermodynamic limit rather smooth functions of the filling $\\vartheta(\\lambda)$ and of the particles-holes momenta $\\{\\mu_j^+, \\mu_j^- \\}_{j=1}^n$. The only poles that appear are kinematic poles, when $\\mu_j^+\\rightarrow \\mu_k^-$, and they can be easily regularized (see Section~\\ref{regularization}). Therefore we do not need to evaluate the form factors precisely at the $\\{\\mu_j^+, \\mu_j^- \\}_{j=1}^n$ that follows from the solutions of the Bethe equations. In fact we can take now $\\{\\mu_j^+, \\mu_j^- \\}_{j=1}^n$ to be independent free parameters (macroscopic excitations) that we denote $\\{p_j, h_j\\}_{j=1}^n$. For each choice of $\\{p_j, h_j\\}_{j=1}^n$ there is a number $\\exp\\left(\\sum_{j=1}^n \\delta S[\\vartheta; p_j, h_j]\\right)$ (with $\\delta S$ defined in~\\eqref{S_diff}) of microscopic states which share the same form factor up to finite size corrections. In order then to use macroscopic variables we need to multiply the form factors at fixed $\\{p_j, h_j\\}_{j=1}^n$ times the number of microscopic states that are characterized by the same macroscopic excitations. This allows to define the thermodynamic limit of the form factors for smooth filling functions $\\vartheta$\n\\begin{equation}\n|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |= \\lim_{\\text{th}} \\Big( L^{n} |\\mathcal{F}(\\vartheta; \\{\\mu^-\\}, \\{\\mu^+\\})|\\Big) \\times \\exp\\left(\\sum_{j=1}^n \\delta S[\\vartheta; p_j, h_j]\\right) .\\label{FF_TL}\n\\end{equation}\nMoreover we can recast the sum over the macroscopic rapidites of the excitations into integrals by taking a special care of the divergences encountered (as is done in section~\\ref{regularization})\n\\begin{equation}\n\\sum_{\\mu_1^+< \\ldots < \\mu_n^+} \\sum_{\\mu_1^-< \\ldots < \\mu_n^-}= \\frac{1}{n!^2}\\sum_{\\{\\mu_j^+,\\mu_j^-\\}_{j=1}^n} = L^{2n} \\frac{1}{n!^2} \\left( \\int_{-\\infty}^{\\infty} dp_j \\rho_h(p_j) \\fint_{-\\infty}^{\\infty} d h_j \\rho(h_j) \\right).\n\\end{equation}\nwhere $ \\fint$ denotes the finite part of the integral, defined for a generic function $f(h)$ with a pole in $h=p$ as\n\\begin{equation}\n \\fint_{-\\infty}^{\\infty} d h f(h) = \\lim_{\\epsilon \\to 0^+} \\int_{-\\infty}^\\infty dh f(h + i \\epsilon) - \\pi i \\underset{h=p}{\\rm res} f(h).\n\\end{equation}\n\nWith this notation we can then write the correlation functions as a sum over all the possible excitations on the thermodynamic state~\\cite{1990_Korepin_NPB_340}\n\\begin{align}\n & \\langle\\hat{\\rho}(x,t) \\hat{\\rho}(0,0) \\rangle = \\sum_{n=0}^{\\infty} \\frac{1}{n!^2} \\left( \\int_{-\\infty}^{\\infty} d p_j \\rho_h(p_j)\\: \\fint_{-\\infty}^{\\infty} d h_j \\rho(h_j) \\right)|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j \\}_{j=1}^n \\rangle |^2 \\nonumber \\\\&\n\\times \\prod_{j=1}^n \\exp \\Big( {ix (k(p_j) -k(h_j) ) -it (\\omega(p_j) - \\omega(h_j))} \\Big) , \\label{corr_func_TL}\n\\end{align}\nwhere the momentum and energy of the excitation follows \\eqref{exc_momentum} and \\eqref{exc_energy} respectively.\n\nEq.~\\eqref{corr_func_TL} reminds the LeClair-Mussardo formula that appears in the context of integrable field theories~\\cite{1999_LeClair_NPB_552}\n\\begin{align}\n & \\langle\\hat{\\rho}(x,t) \\hat{\\rho}(0,0) \\rangle = \\sum_{n=0}^{\\infty} \\frac{1}{n!^2} \\left( \\int_{-\\infty}^{\\infty} d p_j \\rho_h(p_j) \\fint_{-\\infty}^{\\infty} d h_j \\rho(h_j) \\right)|\\langle 0 | \\hat{\\rho} | 0, \\{ h_j \\to p_j \\}_{j=1}^n \\rangle_{FT} |^2 \\nonumber \\\\&\n\\times \\prod_{j=1}^n \\exp \\Big( {ix (k(p_j) -k(h_j) ) -it (\\omega(p_j) - \\omega(h_j))} \\Big) \\label{corr_func_LM}.\n\\end{align}\nThe main difference between eqs.~\\eqref{corr_func_TL} and~\\eqref{corr_func_LM} are the form factors used. The field theoretic form factors simply come from excitations over a structure-less vacuum. Here, eq.~\\eqref{corr_func_TL} suggests that the concept of vacuum is not appropriate for the strongly correlated systems. The form factors still depend \\emph{explicitly} on the properties of the averaging state (through the filling function $\\vartheta$). Conceptually this difference is responsible for the insufficiency of the field theoretical approach to the two-point correlation function (contrary to the one-point functions where the field theory approach is correct)~\\cite{2002_Saleur_NPB_602, 2002_Alvaredo_NPB_636}. On the computational level this was explicitly shown in~\\cite{1742-5468-2010-11-P11012}.\n\n\n\\subsection{Finite size corrections}\n\nIn order to compute the thermodynamic limit of the form factors we need to characterize the density of the particles and the back-flow function up to order $1\/L$. This is due to existence of products in eq.~\\eqref{ff} which are of order $N$ and in the thermodynamic limit can yield finite contributions from order $1\/L$ terms. The derivation is similar to the one presented in~\\cite{2012_Shashi_PRB_85} for the ground state distribution. Therefore here we simply state the result highlighting few differences between the two cases and for the details we refer to the section IV.B of \\cite{2012_Shashi_PRB_85}.\n\nThe filling function $\\vartheta(\\lambda)$, as well as the particle density $\\rho(\\lambda)$, comes from the thermodynamic limit of a certain class of Bethe states. Let $\\{\\lambda\\}_{j=1}^N$ be the Bethe roots of one of these states. Let us consider Bethe equations \\eqref{bethe} and define a variable $x$ that satisfies\n\\begin{align}\n \\lambda(x) = 2\\pi x - \\frac{1}{L}\\sum_{k=1}^N \\theta(\\lambda(x) - \\lambda_k).\n\\end{align}\nClearly $\\lambda(I_j\/L) = \\lambda_j$ but we allow here $x$ to take any real value. Therefore $dx\/d\\lambda$ has a meaning of a number of possible quantum numbers in the range $d\\lambda$. Thus\n\\begin{align}\n \\frac{dx}{d\\lambda} = \\rho_t(\\lambda).\n\\end{align}\nand from the Euler-Maclaurin formula we have\n\\begin{align}\n \\rho_t(\\lambda) = \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{\\lambda_1}^{\\lambda_N} K(\\lambda-\\mu) \\rho(\\mu) d\\mu + \\frac{1}{2L}\\left(K(\\lambda-\\lambda_N) - K(\\lambda - \\lambda_1)\\right),\n\\end{align}\nwhere $\\lambda_{1,N}$ are the smallest and largest rapidities respectively. Incorporating the $1\/L$ in the boundaries of the integral yields\n\\begin{align}\n \\rho_t(\\lambda) = \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{-q}^{q} K(\\lambda-\\mu) \\rho(\\mu) d\\mu,\n\\end{align}\nwith $q_L = \\lambda_1 + 1\/(2L\\rho_p(\\lambda_1))$ and $q_R = \\lambda_N + 1\/(2L\\rho_p(\\lambda_N))$. In the thermodynamic limit we have $q_{R,L}\\rightarrow\\pm\\infty$ and it is convenient to separate the thermodynamic part from the finite size corrections. Before doing so, let us note that we can bound the finite-size corrections from above by choosing $q = \\textrm{min}(|q_L|, q_R)$. We have\n\\begin{align}\n \\rho_t(\\lambda) =& \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} K(\\lambda-\\mu) \\rho(\\mu) d\\mu - \\frac{1}{2\\pi}\\int_{q}^{\\infty} \\nonumber \\left(K(\\lambda-\\mu)+K(\\lambda+\\mu)\\right) \\rho(\\mu) d\\mu.\n\\\\ =& \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} K(\\lambda-\\mu) \\rho(\\mu) d\\mu\\nonumber\\\\ &- \\frac{1}{2\\pi}\\int_{0}^{\\infty} \\left(K(\\lambda-\\mu-q)+K(\\lambda+\\mu+q)\\right) \\rho(\\mu+q) d\\mu.\n\\end{align}\nThe last term that controls the finite-size corrections can be easily bounded (where $M$ is a positive constant)\n\\begin{align}\n \\frac{1}{2\\pi}\\int_{q}^{\\infty} \\left(K(\\lambda-\\mu)+K(\\lambda+\\mu)\\right) \\rho(\\mu) d\\mu \\leq M\\frac{\\rho(q)}{2\\pi},\n\\end{align}\nand the finite size corrections are proportional to $\\rho(q)$. For the energy of the state to be finite we require $\\rho(\\lambda) \\sim \\lambda^{-3-\\epsilon}$ for large $\\lambda$ (c.f. \\eqref{energy_cont}). The boundary $q$ itself is a monotonically increasing function of $N$. For the particle density to spread over the whole real line, rather than to accumulate in the final interval of it, we should have $q \\sim N^{1+\\delta}$ with $\\delta > 0$. Therefore $\\rho(q) \\sim N^{-3-\\gamma}$ with $\\gamma>0$ and the finite-size corrections are at least of order $1\/L^3$ and thus are negligible in the further analysis.\nWe have\n\\begin{align}\n \\rho_t(\\lambda) &= \\frac{1}{2\\pi} + \\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} K(\\lambda-\\mu) \\rho(\\mu) d\\mu + \\mathcal{O}(1\/L^3).\n\\end{align}\n\nThe finite-size corrections to the particle and hole rapidities follow from analogous computations as presented in \\cite{2012_Shashi_PRB_85} and are given by\n\\begin{align}\\label{finite_size_excitations}\n \\mu_0^+ &= 2\\pi \\frac{I^+}{L} - \\int_{-\\infty}^{\\infty} \\theta(\\mu_0^+ - \\lambda) \\rho(\\lambda) d\\lambda, \\;\\;\\;\\;\\;\\;\n \\mu_{1\/L}^+ = -\\frac{F(\\mu_0^+)}{L\\rho_t(\\mu_0^+)}, \\\\\n \\lambda_0^- &= 2\\pi \\frac{I^-}{L} - \\int_{-\\infty}^{\\infty} \\theta(\\lambda_0^- - \\lambda) \\rho(\\lambda) d\\lambda, \\;\\;\\;\\;\\;\\;\n \\lambda_{1\/L}^- = 0.\n\\end{align}\nFor the auxiliary rapidity $\\mu^-$ we have\n\\begin{align}\n \\mu_0^- = \\lambda_0^-, \\;\\;\\;\\;\\;\\;\n \\mu_{1\/L}^- = -\\frac{F(\\lambda_0^-)}{L\\rho_t(\\mu_0^-)},\n\\end{align}\nWe can consider now finite-size corrections to the back-flow function. The thermodynamic limit is given by eq. \\eqref{backflow}. The leading finite size corrections are\n\\begin{align}\n 2\\pi F_{1\/L}(\\lambda) &= \\left(\\int_{q_R}^{\\infty} + \\int_{-\\infty}^{q_L}\\right)d\\mu\\, K(\\lambda,\\mu) \\vartheta(\\mu) F(\\mu) \\nonumber\\\\\n &- \\frac{1}{L}\\left[ K(\\lambda,\\mu_0^+)\\left( \\frac{F(\\lambda)}{\\rho_t(\\lambda)}-\\frac{F(\\mu_0^+)}{\\rho_t(\\mu_0^+)}\\right) - K(\\lambda,\\mu_0^-)\\left( \\frac{F(\\lambda)}{\\rho_t(\\lambda)}-\\frac{F(\\mu_0^-)}{\\rho_t(\\mu_0^-)}\\right)\\right] \\nonumber\\\\\n & + \\frac{1}{2L} \\int_{-\\infty}^{\\infty} d\\mu\\, \\rho(\\mu)K'(\\lambda-\\mu) \\left(\\frac{F(\\lambda)}{\\rho_t(\\lambda)} - \\frac{F(\\mu)}{\\rho_t(\\mu)}\\right)^2.\n\\end{align}\nThe first integral can be estimated in the following way\n\\begin{align}\n\\left|\\int_{q_R}^{\\infty} d\\mu\\,K(\\lambda,\\mu) \\vartheta(\\mu) F(\\mu)\\right| &\\leq \\int_{q_R}^{\\infty} d\\mu \\left|\\,K(\\lambda,\\mu) \\vartheta(\\mu) F(\\mu)\\right| \\nonumber\\\\ &\\leq |\\vartheta(q_R)| \\int_{q_R}^{\\infty} d\\mu \\left|K(\\lambda,\\mu) F(\\mu)\\right|.\n\\end{align}\nBut $\\vartheta(q_R)$ is proportional to $\\rho(q_R)$ which is of order $1\/L^3$. The same holds for the other integral between $-\\infty$ and $q_L$. Therefore both integrals can be neglected in the leading order and\n\\begin{align}\n 2\\pi L F_{1\/L}(\\lambda) =& - \\left[ K(\\lambda,\\mu_0^+)\\left( \\frac{F(\\lambda)}{\\rho_t(\\lambda)}-\\frac{F(\\mu_0^+)}{\\rho_t(\\mu_0^+)}\\right) - K(\\lambda,\\mu_0^-)\\left( \\frac{F(\\lambda)}{\\rho_t(\\lambda)}-\\frac{F(\\mu_0^-)}{\\rho_t(\\mu_0^-)}\\right)\\right] \\nonumber\\\\\n&+ \\frac{1}{2} \\int_{-\\infty}^{\\infty} d\\mu\\, \\rho(\\mu)K'(\\lambda-\\mu) \\left(\\frac{F(\\lambda)}{\\rho_t(\\lambda)} - \\frac{F(\\mu)}{\\rho_t(\\mu)}\\right)^2.\n\\end{align}\nFinally the density of the excited state is\n\\begin{align}\n \\rho_{t,ex}(\\mu_j) = \\rho_t(\\lambda_j) + \\frac{1}{L}\\left(F'(\\lambda_j) - F(\\lambda_j)\\frac{\\rho_t'(\\lambda_j)}{\\rho_t(\\lambda_j)} \\right).\n\\end{align}\nThis completes the list of formulas required to take properly the thermodynamic limit of the form factors. This is achieved in the next section.\n\n\n\\section{Thermodynamic limit for smooth distribution of rapidities} \\label{thermodynamiclimit}\n\nIn this section we calculate the thermodynamic limit of the finite size (normalized) form factors~\\eqref{micro_ff} (see also~\\eqref{ff} and~\\eqref{norm} for explicit formulas). We are interested in the leading term in $1\/L$ so with the equivalence $\\sim$ we denote that we are neglecting extra sub-leading correction in $1\/L$. We proceed as in~\\cite{2012_Shashi_PRB_85} since the two calculations share many common steps. Introducing the short-hand notation\n\\begin{equation}\n\\mu^{}_{ij} = \\mu^{}_i - \\mu^{}_j ,\n\\end{equation}\nwe start from an intermediate expression for the thermodynamic limit of the form factors given by (c.f. eq. 88 in \\cite{2012_Shashi_PRB_85})\n\\begin{align}\\label{starting}\n|\\mathcal{F}_N| \\sim& \\prod_{i=1}^n\\left( \\frac{F_L(\\lambda^-_i)}{L(\\rho_L(\\lambda^-_i)\\rho_{ex}(\\mu^+_i))^{1\/2}(\\lambda_i^- - \\mu_i^+)}\\right)\\frac{i \\Delta k \\: {\\det_N}(\\delta_{jk} + U_{jk})}{(V_p^+ - V_p^-){\\rm Det}\\left(1-\\frac{\\hat{K \\vartheta}}{2\\pi} \\right)}\\nonumber\\\\\n&\\times \\left[\\prod_{j,k} \\left(\\frac{(\\lambda_j - \\mu_k +ic)(\\lambda_j - \\mu_k + ic)}{(\\lambda_{jk} + ic)(\\mu_{jk} + ic)}\\right)^{1\/2}\\right] \\nonumber\\\\ &\n\\times \\left\\{\\prod_j\\frac{{\\rm sin}(\\pi F_L(\\lambda_j))}{\\pi F_L(\\lambda_j)} \\prod_{j\\neq k}\\left(\\frac{\\lambda_{jk} \\mu_{jk}}{(\\mu_j - \\lambda_k)^2}\\right)^{1\/2}\\right\\}\\nonumber\\\\ \\times\n&{\\rm exp}\\left[\\int_{-\\infty}^\\infty d\\lambda \\vartheta(\\lambda) \\left(\\frac{\\pi F_L(\\lambda) {\\rm cos}(\\pi F_L(\\lambda))}{ {\\rm sin}(\\pi F_L(\\lambda))} - 1\\right) \\right] \\nonumber \\\\ & \\times \\exp \\left[ \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\left[ \\left( F_L'(\\lambda) - \\frac{F_L(\\lambda)\\rho_t'(\\lambda)}{2\\rho_t(\\lambda)}\\right) + \\frac{1}{2}F_L'(\\lambda) \\right]\\right] .\n\\end{align}\nwhere ${\\rm Det}( 1- \\frac{K \\vartheta }{2 \\pi})$ is the Fredholm determinant of the kernel\n\\begin{align}\n -\\left[\\frac{K \\vartheta }{2 \\pi}\\right](\\mu,\\nu) = -\\frac{1}{2 \\pi}\\frac{2c}{(\\mu - \\nu)^2 + c^2} \\vartheta(\\nu) ,\n\\end{align}\nIn eq. \\eqref{starting} there are three groups of elements which are still written for a finite system. We denote them as\n\\begin{align}\n M_1 &= \\prod_{j,k} \\left(\\frac{(\\lambda_j - \\mu_k +ic)(\\lambda_j - \\mu_k + ic)}{(\\lambda_{jk} + ic)(\\mu_{jk} + ic)}\\right)^{1\/2}, \\\\\n M_2 &= \\prod_j\\frac{{\\rm sin}(\\pi F_L(\\lambda_j))}{\\pi F_L(\\lambda_j)} \\prod_{j\\neq k}\\left(\\frac{\\lambda_{jk} \\mu_{jk}}{(\\mu_j - \\lambda_k)^2}\\right)^{1\/2}, \\\\\n \\Theta &= \\frac{\\det_N (\\delta_{jk} + U_{ij})}{V_p^+ - V_p^- }.\n\\end{align}\nThe thermodynamic limit of them requires some work. Calculation of $M_1$ is exactly the same as for the ground state form factors and thus we do not reproduce it here. For the details we refer again to \\cite{2012_Shashi_PRB_85}. On the other hand the term $M_2$ has a manifestly different thermodynamic limit and is responsible for different size dependence of the ground state (critical) form factors and the finite entropy state form factors. Computations are presented in the next section. The thermodynamic limit of $\\Theta$ was computed in \\cite{2012_Shashi_PRB_85} only for a specific type of excitations. As we require here the form factors for a generic particle-hole excitations we have to generalize the previous calculations. This is done in the subsequent section.\n\n\\subsection{\\texorpdfstring{Evalutation of $M_2$}{Evaluation of M(2)}}\nWe focus here on the evaluation of the double products given by\n\\begin{equation}\nM_2 = \\prod_j \\frac{\\sin(\\pi F_L(\\lambda_j))}{\\pi F_L(\\lambda_j)} \\prod_{j \\neq k =1}^N \\left(\\frac{\\lambda_{jk} \\mu_{kj}}{(\\mu_k - \\lambda_j)^2} \\right)^{1\/2} ,\n\\end{equation}\nwhich present formal differences in the thermodynamic limit when the states is described by a smooth distribution or when the distribution is discontinuous as for the ground state. Differently from the ground state situation this term is not expected to produce power law divergences in the system size as $1\/L^\\alpha$.\n\nFollowing \\cite{2012_Shashi_PRB_85} we decompose the product in three pieces\n\\begin{equation}\nM= T'' \\times T_{holes} \\times T_{particles} ,\n\\end{equation}\ndepending on which rapidities we let the sum run over.\n\\begin{align}\nT'' &=\\prod_{j \\neq k}'' \\left(\\frac{\\lambda_{jk} \\mu_{kj}}{(\\mu_k - \\lambda_j)^2} \\right)^{1\/2} \\nonumber \\\\\n&\\sim \\prod_{j \\neq k}'' \\left(1 + \\frac{F_L(\\lambda_k)}{L \\rho_t(\\lambda_k)(\\lambda_j - \\lambda_k)} \\right)^{-1\/2} \\left(1 - \\frac{F_L(\\lambda_k)}{L \\rho_t(\\lambda_k)(\\lambda_j - \\lambda_k)} \\right)^{-1\/2} \\nonumber\\\\\n&\\times \\left(1 + \\frac{F_L(\\lambda_k)}{L \\rho_t(\\lambda_k)(\\lambda_j - \\lambda_k)} -\\frac{F_L(\\lambda_k)}{L \\rho_t(\\lambda_j)(\\lambda_j - \\lambda_k)} \\right)^{-1\/2} ,\n\\end{align}\nwhere by $\\prod_{j \\neq k}''$ we denoted the product where we excluded the particles excitations $\\{ \\mu^+_j \\}_{j=1}^n$ but we included the holes $\\{ \\mu^-_j \\}_{j=1}^n$. When the two rapidities get closer, i.e.\\\nwhen $j \\in [k-n^*, k+ n^*]$ where $n$ is a given sub-extensive cut-off such that $ n \\propto L^{1- \\alpha}$ with $\\alpha<1\/2$, then we substitute for the difference between the two\n\\begin{equation}\\label{Expanded_rap}\nL \\rho_t(\\lambda_j )(\\lambda_j- \\lambda_k)= \\frac{j-k}{\\vartheta(\\lambda_j)} + \\frac{ (j-k)^2 \\partial_\\lambda \\rho_t(\\lambda_j)}{2 L \\rho(\\lambda_j)^2} ,\n\\end{equation}\nwhile for all the other $j$ we can just exchange the sum for an integral over the rapidities (see figure~\\ref{M2_regions}).\nThe cut-off $n^*$ delimits the region where the approximation \\eqref{Expanded_rap} start to break down, which corresponds to a distance in rapidities\n\\begin{equation}\n|\\lambda_j - \\lambda_k | \\equiv \\nu^*(\\lambda) = \\frac{n^*}{L \\rho(\\lambda)} + \\mathcal{O}(n^*\/L^2) .\n\\end{equation}\nWe denote the two regions in $\\lambda_j - \\lambda_k$ separated by the cut-off as the region $I$ (smooth part) and $II$ (discrete part) where the approximation \\eqref{Expanded_rap} is valid (IIa where $j < k$ and IIb where $k< j$ ). $T''$ is then given by the product of these three terms\n\\begin{equation}\nT'' = T_I \\times T_{IIa} \\times T_{IIb} ,\n\\end{equation}\nFor the fist term we have the following\n\\begin{align}\n \\log T_1 =& \\left[\\frac{1}{2} \\sum_{j \\neq [k - n^* , k+n^*]} \\frac{F_L(\\lambda_j) F_L(\\lambda_k)}{L^2 \\rho_t(\\lambda_j)\\rho_t(\\lambda_k)(\\lambda_j- \\lambda_j)} \\right] \\nonumber\\\\\n \\sim& \\frac{1}{2} \\left( \\int d\\lambda \\vartheta(\\lambda) \\int_{-\\infty}^{\\lambda - \\nu^*(\\lambda)} \\!\\!d \\mu \\vartheta(\\mu) \\frac{F(\\lambda) F(\\mu)}{(\\lambda - \\mu)^2} +\\int d\\lambda \\vartheta(\\lambda) \\int_{\\lambda + \\nu^*(\\lambda)}^{\\infty} \\!\\!d\\mu \\vartheta(\\mu) \\frac{F(\\lambda) F(\\mu)}{(\\lambda - \\mu)^2} \\right) \\nonumber \\\\\n =& -\\frac{1}{4} \\int d\\lambda \\int d \\mu \\frac{(F_L(\\lambda) \\vartheta(\\lambda)- F_L(\\mu) \\vartheta(\\mu))^2}{(\\lambda - \\mu)^2} \\nonumber \\\\\n &+ \\frac{1}{2} \\int d\\lambda \\vartheta(\\lambda)^2 F_L(\\lambda)^2\\left( \\int_{-\\infty}^{\\lambda - \\nu^*(\\lambda)} d \\mu \\frac{1}{(\\lambda - \\mu)^2} + \\int_{\\lambda + \\nu^*(\\lambda)}^{\\infty} d \\mu \\frac{1}{(\\lambda - \\mu)^2} \\right) \\nonumber \\\\\n =& -\\frac{1}{4} \\int d\\lambda \\int d \\mu \\frac{(F_L(\\lambda) \\vartheta(\\lambda)- F_L(\\mu) \\vartheta(\\mu))^2}{(\\lambda - \\mu)^2}\n + \\int d\\lambda \\vartheta(\\lambda)^2 F_L(\\lambda)^2 \\frac{1}{\\nu^*(\\lambda)} \\nonumber \\\\\n =& -\\frac{1}{4} \\int d\\lambda \\int d \\mu \\frac{(F_L(\\lambda) \\vartheta(\\lambda)- F_L(\\mu) \\vartheta(\\mu))^2}{(\\lambda - \\mu)^2} \\nonumber \\\\\n &+ \\frac{L}{n^*}\\int d\\lambda \\vartheta(\\lambda)^2 F_L(\\lambda)^2 \\rho(\\lambda) + \\mathcal{O}\\left(\\frac{L}{(n^*)^2} \\right).\n\\end{align}\nThe computation in the sector II can be done analogously as in \\cite{2012_Shashi_PRB_85} leading to\n\\begin{align}\n\\log T_{II}\n= & - \\int d\\lambda \\frac{\\rho'_t(\\lambda)}{\\rho_t(\\lambda)}\\frac{ F(\\lambda)} {2\\rho(\\lambda) \\partial_\\lambda(F(\\lambda)\\vartheta(\\lambda))} \\frac{\\partial }{\\partial \\lambda} \\log \\left( \\frac{\\pi F(\\lambda) \\vartheta(\\lambda)}{\\sin \\pi F(\\lambda) \\vartheta(\\lambda)} \\right) \\\\& - \\frac{L}{ n^*}\\int d\\lambda \\vartheta(\\lambda)^2 F_L(\\lambda)^2 \\rho(\\lambda) + \\mathcal{O}(L\/(n^*)^2) ,\n\\end{align}\nwhere the cut-off depended part cancels exactly the one in $T_I$ leading to a cut-off independent result.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{M2_regions.pdf}\\label{M2_regions}\n\\caption{Schematic of range of $\\lambda$'s. Cutoff parameter n* is used to separate regions I, II. Figure is taken from \\cite{2012_Shashi_PRB_85} and there are two extra regions (IIIa and IIIb) depicted. These are the regions in which the two rapidities are close two each other and simultaneously they are close to the edge of the distribution. For the smooth distribution of rapidities there is no edge and consequently the region III does not appear in the present calculations. Ultimately the existence or lack of region III determines the scaling of the form factors with the system size $L$. It is the region III that leads to the fractional power of the system size for the ground state form factors and consequently to the criticality of zero temperature 1D Bose gas. This shows an intimate connection between the distribution of rapidities and the form of the correlation function. (Figure reproduced from~\\cite{2012_Shashi_PRB_85}, courtesy of A. Shashi).}\n\\end{figure}\n\n\nNow we are left only with the computation of $T_{particle}$ and $T_{hole}$ defined as\n\\begin{equation}\nT_{hole} \\sim \\prod_{j}' \\prod_{k=1}^n \\left( 1 - \\frac{F_L(\\lambda_j)}{L \\rho(\\lambda_j)(\\lambda_j - \\lambda_k^-) } \\right)^{-1} ,\n\\end{equation}\n\\begin{equation}\nT_{particle} \\sim \\prod_{j}' \\prod_{k=1}^n \\left( 1 - \\frac{F_L(\\lambda_j)}{L \\rho(\\lambda_j)(\\lambda_j - \\lambda_k^-) } \\right) ,\n\\end{equation}\nwhere the product $\\prod_{j}'$ runs respectively over all the rapidities in the state but not over the holes(particle).\nProceeding as in \\cite{2012_Shashi_PRB_85} and using the fact that for a smooth distribution particle and holes are always arbitrarily close to an extensive number of rapidities of the state, we obtain\n\\begin{equation}\nT_{hole} = \\prod_{k=1}^n \\frac{\\sin \\pi F(\\mu_k^-) \\vartheta(\\mu_k^-)}{\\pi F(\\mu_k^-) \\vartheta(\\mu_k^-) } \\exp\\left( \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\: \\frac{F(\\lambda) - F(\\mu_k^-) }{(\\lambda - \\mu_k^-) } \\right) .\n\\end{equation}\nFor $T_{particle}$ we have the same result\n\\begin{equation}\nT_{particle} = \\prod_{k=1}^n \\frac{\\pi F(\\mu_k^+) \\vartheta(\\mu_k^+) }{ \\sin \\pi F(\\mu_k^+) \\vartheta(\\mu_k^+)} \\exp\\left( - \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\: \\frac{F(\\lambda) - F(\\mu_k^+) }{(\\lambda - \\mu_k^+) } \\right) .\n\\end{equation}\n\nFinally we can then write for the whole factor $M_2$ in the thermodynamic limit\n\\begin{align}\n M_2 & = \\exp\\left[ - \\int d\\lambda \\frac{\\rho'_t(\\lambda)}{\\rho_t(\\lambda)}\\frac{ F(\\lambda)} {2\\rho(\\lambda) \\partial_\\lambda(F(\\lambda)\\vartheta(\\lambda))} \\frac{\\partial }{\\partial \\lambda} \\log\\left( \\frac{\\pi F(\\lambda) \\vartheta(\\lambda)}{\\sin \\pi F(\\lambda) \\vartheta(\\lambda)} \\right) \\right] \\nonumber \\\\&\n \\times \\prod_{k=1}^n \\frac{\\sin \\pi F(\\mu_k^-) \\vartheta(\\mu_k^-)}{\\pi F(\\mu_k^-) \\vartheta(\\mu_k^-) } \\frac{\\pi F(\\mu_k^+) \\vartheta(\\mu_k^+) }{ \\sin \\pi F(\\mu_k^+) \\vartheta(\\mu_k^+)} \\nonumber \\\\& \\times\n \\exp\\left[ \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\: \\frac{F(\\lambda) - F(\\mu_k^-) }{(\\lambda - \\mu_k^-) } ) - \\int_{-\\infty}^{\\infty} d\\lambda \\vartheta(\\lambda) \\: \\frac{F(\\lambda) - F(\\mu_k^+) }{(\\lambda - \\mu_k^+) } \\right] \\nonumber \\\\& \\times\n \\exp\\left[ \\frac{-1}{4} \\int d\\lambda \\int d \\mu \\frac{(F_L(\\lambda) \\vartheta(\\lambda)- F_L(\\mu) \\vartheta(\\mu))^2}{(\\lambda - \\mu)^2} \\right] ,\\label{M2_final}\n\\end{align}\n\n\n\\subsection{Fredholm determinant}\nWe are left with the problem of computing the thermodynamic limit of the determinant\n\\begin{equation}\\label{theta}\n\\Theta = \\frac{\\det_N (\\delta_{jk} + U_{jk})}{V_p^+ - V_p^- }\n\\end{equation}\nwhere the matrix $U$ is given in \\eqref{matrix_U} and $\\lambda_p$ is an arbitrary number. Analogously as is done in \\cite{2012_Shashi_PRB_85} we can take the limit $\\lambda_p \\to \\infty$ leading to\n\\begin{equation}\n\\Theta = \\frac{i c}{2 \\Delta k } \\det \\left(\\delta_{jk} + \\frac{i (\\mu_j - \\lambda_j)}{V_j^+ - V_j^- } \\prod_{m \\neq j} \\frac{\\mu_m - \\lambda_j}{\\lambda_m - \\lambda_j} \\left( \\frac{2 c}{(\\lambda_j - \\lambda_k)^2 + c^2} - \\frac{2}{c} \\right) \\right)\n\\end{equation}\nIt is useful to consider a vector\n\\begin{equation}\na_j = \\frac{i (\\mu_j - \\lambda_j)}{V_j^+ - V_j^-} \\prod_{m \\neq j} \\frac{\\mu_m - \\lambda_j}{\\lambda_m - \\lambda_j}.\n\\end{equation}\nWith this notation the determinant in \\eqref{theta} is expressed as\n\\begin{align}\n \\det_N \\left( \\delta_{ij}+ A_{ij}\\right),\n\\end{align}\nwith the matrix $A$ given by\n\\begin{align}\n A_{jk} = a_j \\left(K(\\lambda_j-\\lambda_k) - \\frac{2}{c} \\right)\n\\end{align}\nDepending on $j$ the vector $a_j$ has a different scaling behaviors with the system size\n \\begin{align}\n a_j \\sim \\begin{cases}\n \\mathcal{O}(1\/L), &\\lambda_j \\notin \\{\\lambda_j^-\\}_{j=1}^n\\\\\n \\mathcal{O}(1), &\\lambda_j \\in \\{\\lambda_j^-\\}_{j=1}^n\n \\end{cases}.\n\\end{align}\nWe can use this property to simplify the computation of the determinant.\nWe denote by $\\tilde{a}_j$ a vector $a_j$ in which we substitute $\\mu_j$ by $\\mu_j^-$. Correspondingly we define a matrix $\\tilde{A}_{jk}$ with $\\tilde{a}_j$ as a prefactor instead of $a_j$. Thus the matrix elements of $\\tilde{A}_{jk}$ are all of $\\mathcal{O}(1\/L)$. By $B_{jk}$ we denote the difference of the two matrices: $B_{jk} = A_{jk} - \\tilde{A}_{jk}$. Note that the matrix $B_{jk}$ has only $n$ non-zero rows corresponding to the excited rapidites $\\{ \\lambda_j^-\\}_{j=1}^n$. Using standard proprieties of the determinant and assuming the matrix $\\delta_{ij} +\\tilde{A}_{ij} $ is invertible we can recast the determinant in a product of the determinant of an $N\\times N$ matrix and the determinant of an $n\\times n$ one\n\\begin{equation}\\label{two_det}\n \\det_N (\\delta_{ij}+ \\tilde{A}_{ij}+ B_{ij} ) = \\det_N( \\delta_{ij} +\\tilde{A}_{ij} ) \\times \\det_n \\left( \\delta_{ij} + \\sum_{k=1}^N B_{ik} \\left(1+ \\tilde{A}\\right)^{-1}_{kj}\\right),\n\\end{equation}\nwhere the indices $i,j$ in the second determinant run only over the $n$ excited rapidities\nand $1$ denotes here the identity matrix. We can now take the thermodynamic limit. The first determinant becomes a Fredholm determinant ${\\rm Det}(1 + \\hat{A})$ with the kernel\n\\begin{equation} \\label{kernelA}\n \\hat{A}(\\lambda, \\mu) = \\tilde{a}(\\lambda)\\left( K(\\lambda-\\mu) - \\frac{2}{c}\\right),\n\\end{equation}\nwhere $\\tilde{a}(\\lambda_j)$ is the thermodynamic limit of $\\rho(\\lambda_j){a}_j$. Standard computations give \\cite{2012_Shashi_PRB_85}\n\\begin{align}\n \\tilde{a}(\\lambda ) = & \\frac{\\vartheta(\\lambda)F(\\lambda)}{\\Gamma[1 + \\vartheta(\\lambda)F(\\lambda)] \\Gamma[1 - \\vartheta(\\lambda)F(\\lambda)]} \\underset{\\mu^-_k \\neq \\lambda}{\\prod_{k=1}^n} \\frac{ \\mu_k^+ - \\lambda}{\\mu_k^- - \\lambda}\\exp \\left[ - {\\rm PV}\\! \\int d\\mu \\frac{\\vartheta(\\mu) F(\\mu)}{\\mu - \\lambda} \\right]\n \\nonumber \\\\& \\times\n\\left[2 {\\rm Im} \\left( \\prod_{k=1}^n \\frac{\\mu_k^+ - \\lambda + i c}{\\mu_k^- - \\lambda+ ic} \\exp\\left[- \\int d\\mu \\frac{\\vartheta(\\mu)F(\\mu)}{\\mu - \\lambda + ic} \\right] \\right)\\right]^{-1}. \\label{a_tilde_derivation}\n\\end{align}\nwhere the product $\\underset{\\mu^-_k \\neq \\lambda}{\\prod_{k=1}^n} $ runs over all the holes (particles) except the $j-$th one when $\\lambda = \\mu_j^{-}$ for any $j=1,\\ldots, n$. With ${\\rm PV} \\int$ we denoted the principal value of the integral\n\\begin{equation}\n{\\rm PV} \\int dx \\frac{f(x)}{x} = \\lim_{\\epsilon \\to 0} \\left( \\int_{-\\infty}^{-\\epsilon}dx \\frac{f(x)}{x} + \\int_{\\epsilon}^{\\infty} dx \\frac{f(x)}{x} \\right)\n\\end{equation}\n The second line of eq.~\\eqref{a_tilde_derivation} can be still simplified. We separate the exponential term into real and imaginary parts and for the imaginary part use the integral equation for the backflow \\eqref{backflow} to obtain\n\\begin{align}\n &\\exp \\left(-\\int d\\mu \\frac{\\vartheta(\\mu) F(\\mu)}{\\mu - \\lambda +ic}\\right) = \\exp\\left(-\\frac{1}{2c} \\int d\\mu (\\mu-\\lambda)\\vartheta(\\mu) F(\\mu) K(\\lambda - \\mu)\\right) \\nonumber\\\\ &\\times \\exp\\left(i\\pi F(\\lambda) -\\frac{i}{2}\\sum_{k=1}^n \\left(\\theta(\\lambda-\\mu_k^+) - \\theta(\\lambda - \\mu_k^-) \\right) \\right).\n\\end{align}\nFrom the definition of $\\theta(\\lambda) = 2{\\rm atan}(\\lambda\/c)$~\\eqref{phase_shift} and an identity \\\n\\begin{align}\n \\exp\\left(2i{\\rm atan}(x)\\right) = \\frac{1 + ix}{1 - ix},\n\\end{align}\nit follows that\n\\begin{align}\n &{\\rm Im}\\left[ \\prod_k^n\\frac{\\mu_k^+ - \\lambda +ic}{\\mu_k^- - \\lambda + ic}\\exp\\left(-\\int d\\mu \\frac{\\vartheta(\\mu) F(\\mu)}{\\mu - \\lambda +ic}\\right) \\right] = \\sin \\pi F(\\lambda) \\prod_{k=1}^n\\left(\\frac{K(\\mu_k^- - \\lambda)}{K(\\mu_k^+ -\\lambda)} \\right)^{1\/2}\n\\nonumber\\\\\n&\\times\n\\exp\\left(-\\frac{1}{2c} \\int d\\mu (\\mu-\\lambda)\\vartheta(\\mu) F(\\mu) K(\\lambda - \\mu)\\right) .\n\\end{align}\nUsing Euler's reflection formula for $\\Gamma$ functions\n\\begin{align}\n \\Gamma(1-z)\\Gamma(1+z) = \\frac{\\pi z}{\\sin \\pi z},\n\\end{align}\nwe obtain\n\\begin{align}\n &\\tilde{a}(\\lambda) = \\frac{\\sin[ \\pi \\vartheta(\\lambda) F(\\lambda)]}{2\\pi \\sin [ \\pi F(\\lambda)]} \\nonumber\\\\& \\times \\frac{\\prod_{k=1}^n \\mu_k^+ - \\lambda}{\\underset{\\mu^-_k \\neq \\lambda}{\\prod_{k=1}^n}\\mu_k^- - \\lambda} \\prod_{k=1}^n\\left(\\frac{K(\\mu_k^+ -\\lambda)}{K(\\mu_k^- - \\lambda)} \\right)^{1\/2}\n \\exp \\left[ \\frac{c}{2}{\\rm PV}\\! \\int d\\mu \\frac{\\vartheta(\\mu)F(\\mu) K(\\mu - \\lambda)}{\\mu - \\lambda} \\right] .\n\\end{align}\n\n\nIn the $n \\times n$ determinant in \\eqref{two_det} we note that we can neglect the $1\/L$ corrections to $B$ since $\\lim_{\\text{th}} \\frac{n}{L}=0$. We introduce the matrix $W = A (1+\\tilde{A})^{-1}$ as the solution of the following equation\n\\begin{equation}\nW_{ij} + \\sum_{k=1}^N W_{ik} \\tilde{A}_{kj}= A_{ij},\\;\\;\\;i,j=1,\\dots,n,\n\\end{equation}\nwhich in the thermodynamic limit becomes a linear integral equations for the function $W(\\lambda , \\mu)$\n\\begin{equation}\\label{kernelW}\n W(\\lambda, \\mu) + \\int_{-\\infty}^\\infty d\\alpha W(\\lambda, \\alpha) \\tilde{a}(\\alpha) \\left(K(\\alpha - \\mu) - \\frac{2}{c} \\right) = b( \\lambda)\\left(K(\\lambda - \\mu) - \\frac{2}{c} \\right) ,\n\\end{equation}\nwith the vector $b(\\lambda)$ given by\n\\begin{align}\n b(\\lambda) & =- \\frac{\\sin [\\pi \\vartheta(\\lambda) F(\\lambda)]}{2\\pi \\vartheta(\\lambda) F(\\lambda) \\sin [\\pi F(\\lambda)]} \\nonumber\\\\\n &\\times \\frac{\\prod_{k=1}^n \\mu_k^+ - \\lambda}{\\underset{\\mu^-_k \\neq \\lambda}{\\prod_{k=1}^n}\\mu_k^- - \\lambda} \\prod_{k=1}^n\\left(\\frac{K(\\mu_k^+ -\\lambda)}{K(\\mu_k^- - \\lambda)} \\right)^{1\/2}\n\\exp \\left[ \\frac{c}{2}{ \\rm PV} \\int d\\mu\\frac{\\vartheta(\\mu) F(\\mu) K(\\lambda - \\mu)}{\\lambda - \\mu}\\right],\n\\end{align}\n\nPutting everything together we have then\n\\begin{equation}\n \\lim\\nolimits_\\text{th} \\det_N (\\delta_{ij}+ \\tilde{A}_{ij}+ B_{ij} ) = {\\rm Det}( 1 +\\hat{A} ) \\det_n \\left( \\delta_{ij} + W(\\mu_i^-,\\mu_j^-) \\right),\n\\end{equation}\nand the determinant part of the form factors is then expressed as\n\\begin{equation}\n \\Theta = \\frac{i c}{2 \\Delta k} {\\rm Det}( 1+ \\hat{A}) {\\rm det}_n \\left( \\delta_{ij} + W(\\mu_i^-,\\mu_j^-) \\right). \\label{theta_final}\n\\end{equation}\nNote that the Fredholm determinant is still a function of the excitations and not only of the the averaging state. This unfortunately poses still serious problems to the computation of correlation functions. However from the numerical point of view its evaluation can be effectively approximated with the very first terms of its expansion in powers of the trace.\n\n\\subsection{Final result}\nWe report here the final expression for the thermodynamic limit of the form factors of the density operator between the representative state given by a smooth distribution $\\rho(\\lambda)$ and a number $n$ of particle-hole excitations. Most of the remaining computations can be carried out exactly as is done in \\cite{2012_Shashi_PRB_85} by simply rescalling the shift function as $\\tilde{F}(\\lambda) = \\vartheta (\\lambda)F(\\lambda)$.\nCombining the partial thermodynamic limit of the form factors from eq. \\eqref{starting} with the results for $M_1$, $M_2$ (\\eqref{M2_final} and $\\Theta$ (\\eqref{theta_final} we find the form factors between a thermodynamic state $|\\vartheta\\rangle$ and one of its excited states with $n$ particle-holes, as defined in \\eqref{FF_TL}, to be\n\\begin{align}\\label{FFd_final_expression}\n & |\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |= \\nonumber \\\\&\n \\frac{c}{2} \\left[\\prod_{k=1}^n \\frac{F(h_k)}{ (\\rho_t(p_k) \\rho_t(h_k))^{1\/2} } \\frac{\\pi \\tilde{F}(p_k) }{ \\sin \\pi \\tilde{F}(p_k) } \\: \\frac{\\sin \\pi \\tilde{F}(h_k)}{\\pi \\tilde{F}(h_k) } \\right] \\nonumber \\\\& \\times\n \\prod_{i,j=1}^n \\left[\\frac{(p_i - h_j + i c)^2}{(h_{i,j} + ic)(p_{i,j} + ic)} \\right]^{1\/2} \\frac{\\prod_{i< j =1}^n h_{ij} p_{ij}}{\\prod_{i , j} (p_i - h_j)}\n \\det_n \\left( \\delta_{ij} + W(h_i,h_j) \\right) \\nonumber \\\\& \\times \\exp\\left(- \\frac{1}{4} \\int d\\lambda \\int d\\mu \\left( \\frac{\\tilde{F}(\\lambda) - \\tilde{F}(\\mu)}{\\lambda - \\mu}\\right)^2 - \\frac{1}{2} \\int d\\mu d \\lambda \\left( \\frac{\\tilde{F}(\\lambda)\\tilde{F}(\\mu)}{(\\lambda - \\mu + i c)^2}\\right) \\right) \\nonumber \\\\& \\times\n \\exp\\left( \\sum_{k=1}^n {\\rm PV} \\int_{-\\infty}^{\\infty} d\\lambda \\: \\frac{\\tilde{F}(\\lambda) (h_k - p_k) }{(\\lambda - h_k) ( \\lambda - p_k) }+ \\int d\\lambda \\frac{\\tilde{F}(\\lambda) (p_k - h_k)}{(\\lambda - h_k + i c) (\\lambda - p_k + i c)} \\right)\n \\nonumber \\\\& \\times \\frac{{\\rm Det}\\left(1 + \\hat{A} \\right)}{{\\rm Det}\\left(1 - \\frac{K \\vartheta}{2 \\pi }\\right)} \\exp\\left(\\sum_{j=1}^n \\delta S[\\vartheta; p_j, h_j]\\right),\n\\end{align}\nwith the kernels $\\hat{A}$ and $W$ given respectively in \\eqref{kernelA} and \\eqref{kernelW}. The form factors are now completely characterized by thermodynamic data. Knowing the $\\vartheta(\\lambda)$ function we can find the density $\\rho_t(\\lambda)$. Specifying the rapidities of the excitations $\\{h_j\\rightarrow p_j\\}_{j=1}^n$ the back-flow function $F(\\lambda|\\{h_j\\rightarrow p_j\\}_{j=1}^n)$ and the form factor itself follows. Note that in order to have a complete resolution of identity we need to include also the diagonal form factor with $n=0$\n\\begin{equation} \\label{diagonal}\n|\\langle \\vartheta | \\hat{\\rho} | \\vartheta \\rangle |= D\n\\end{equation}\nwhere the density of particles can be chosen to be unitary $D=1$.\n\nThe expression \\eqref{FFd_final_expression} is complicated and the meaning of many terms is rather obscure. The main difficulty is hidden in the Fredholm determinant which depends on the excitations and a factorization of it is still not possible. In order to have some insight on the structure of the form factors it is interesting to consider the small density limit. That is we let $\\vartheta(\\lambda) \\approx 0$ and obtain\n\\begin{align}\n & |\\langle 0 | \\hat{\\rho} | 0, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |=\n\\frac{c}{2} \\left[\\prod_{k=1}^n \\sum_{l=1}^n (\\theta(p_l - h_k) - \\theta(h_l - h_k) )\\right] \\nonumber \\\\& \\times\n\\prod_{i,j=1}^n \\left[\\frac{(p_i - h_j + i c)^2}{(h_{i,j} + ic)(p_{i,j} + ic)} \\right]^{1\/2} \\frac{\\prod_{i< j =1}^n h_{ij} p_{ij}}{\\prod_{i , j} (p_i - h_j)}\n \\det_n \\left( \\delta_{ij} + W(h_i,h_j) \\right)\\label{FFd_zero_density},\n\\end{align}\nwhere we used that $\\rho_t(\\lambda) = 1\/(2\\pi) + \\mathcal{O}(\\vartheta(\\lambda))$ and\n\\begin{align}\n F(\\lambda) = \\frac{1}{2\\pi} \\sum_{k=1}^n \\left(\\theta(p_k - \\lambda) - \\theta(h_k - \\lambda)\\right) + \\mathcal{O}(\\vartheta(\\lambda)).\n\\end{align}\nThe matrix $W(h,p)$ \\eqref{kernelW} also simplifies. The kernel $\\hat{A}$ becomes small and we obtain an explicit expression for $W(\\mu, \\lambda)$\n\\begin{align}\n \n W(\\lambda, \\mu) = b(\\lambda)\\left(K(\\lambda - \\mu) - \\frac{2}{c} \\right) + \\mathcal{O}(\\vartheta(\\lambda))\n\\end{align}\nwith\n\\begin{align}\n b(\\lambda) = \\frac{-1}{2\\sin\\left[\\frac{1}{2} \\sum_{k=1}^n\\left(\\theta(p_k -\\lambda)-\\theta(h_k - \\lambda) \\right) \\right]} \\prod_{k=1}^n \\frac{K^{1\/2}(h_k-\\lambda)}{K^{1\/2}(p_k-\\lambda)} \\frac{\\prod_{k=1}^n (p_k - \\lambda)}{\\underset{h_k \\neq \\lambda}{\\prod_{k=1}^n} (h_k - \\lambda)}.\n\\end{align}\nIn the case of 1 particle-hole excitation the form factor simplifies to\n\\begin{equation} \\label{1ph_small_density}\n |\\langle 0 | \\hat{\\rho} | 0, h \\to p \\rangle | = \\frac{1}{2}\\frac{\\theta(p-h)}{ (p-h)} \\left((p-h)^2 + c^2\\right)^{1\/2} ,\n\\end{equation}\nNote that the form factor \\eqref{1ph_small_density} describes a process of creating a particle-hole excitation in a low density state. Therefore is very different from the form factor \\eqref{ff} for $N=1$. The later equals\n\\begin{equation}\n |\\langle \\mu| \\hat{\\rho} |\\lambda\\rangle| = c,\n\\end{equation}\nand describes the process of exciting a single particle state with momentum $\\lambda$ to momentum $\\mu$ (Since these are single particle states the momentum is equal to the rapidity.). This shows that particle-hole excitations over the averaging state cannot be identified with particle creation over the vacuum in the field theory. Note that contrary to the relativistic field theory \\cite{MussardoBOOK} there is no crossing symmetry that would allow to transform the hole in the ket state into a particle in the bra state in eq.~\\eqref{1ph_small_density}.\n\n\n\n\\section{Regularization of the divergences} \\label{regularization}\nTo compute correlation functions we need to perform an integration over all possible values of the rapidites of the excitations.\nThe form factors \\eqref{FFd_final_expression} have however a singularity whenever $h_j = p_k$ and they are finite only when we consider only one single particle-hole $n=1$ with $p \\to h$, when the form factor becomes indeed diagonal. Therefore we need to be careful while rewriting the sums as integrals. The aim of this section is to show how this can be done. Let us start with the finite size form of the correlation function where we already neglect sub-leading corrections \\eqref{corr_func_TL}\n\\begin{align}\n& \\langle \\hat{\\rho}(x,t) \\hat{\\rho}(0) \\rangle = \\sum_{n=0}^\\infty \\frac{1}{n!^2} \\prod_{j=1}^n \\left[ \\frac{1}{L} \\sum_{p_j} \\frac{1}{L}\\sum_{h_j} \\right] \\\\&\n\\times |\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |^2 e^{\\sum_{j=1}^n \\left[ - i x (k(p_j) - k(h_j)) - i t ( \\omega(p_j) - \\omega(h_j))\\right]}.\n\\end{align}\nThe sum over particle and holes rapidites transforms into a product of integrals under a proper regularization. The idea, already introduced to regularize the field theory form factors in \\cite{1742-5468-2010-11-P11012}, is to write the sum over the holes as a complex integral over all the values that the holes rapidites can take for a finite (but large) $L$ using \\eqref{finite_size_excitations}\n\\begin{equation}\nL Q(h) =L \\left( h + \\int d\\lambda \\theta(h - \\lambda) \\rho(\\lambda) \\right) = 2 \\pi I_j,\n\\end{equation}\nwhere $\\{ I_j \\}$ are all the quantum numbers of the averaging state at some large fixed system size $L$.\nWith a help of $Q(h)$ we can write the sum of a function $f(z)$ over all the values of hole rapidity $h$ as\n\\begin{align}\n \\frac{1}{L}\\sum_{h} f(h) =& \\sum_{I_j} \\oint_{I_j} \\frac{dz}{2 \\pi} \\frac{f(z) Q'(z)}{ e^{i L Q(z)} -1} \\nonumber\\\\\n=& \\left(\\int_{\\mathbb{R} - i \\epsilon} -\\int_{\\mathbb{R} + i \\epsilon} \\right)\\frac{f(z) Q'(z)}{ e^{i L Q(z)} -1} \\frac{dz}{2 \\pi} - \\sum_{\\text{poles(f)} \\in \\Gamma_\\epsilon} \\oint dz \\frac{f(z) Q'(z)}{ e^{i L Q(z)}-1} - \\sum_{r_j \\not \\in \\{ I_j \\} } f(z_j),\n\\end{align}\nwhere the first integrals are taken on a single contour including the poles in $Q(z) = 2 \\pi I_j$ where $I_j$ are all the possible quantum numbers of the hole. In the second step we modified the sum over all these contours in the integral over the line above and below the real axes. In order to do that we need to subtract extra poles that we do not want to include. One type of them are the poles of $f(z)$ in the stripe $\\Gamma_\\epsilon$ delimited by the two imaginary lines. Other poles are located at the values $z$ such that $Q(z)= 2 \\pi r_j$ with $r_j$ not a quantum number of the averaging state (where holes cannot be created). When $L \\to \\infty$ only the integral above the real line survives the limit (since $Q(z)$ is monotonic in $z$) leading to\n\\begin{equation}\n\\frac{1}{L}\\sum_{h} f(h) = \\int_{\\mathbb{R} + i \\epsilon} {f(z) \\rho(z)}{} dz - \\pi i \\sum_{\\text{res(f)} \\in \\Gamma_\\epsilon} {f(z) \\rho(z)}{ }.\n\\end{equation}\n If now we impose that $f(z)$ has only a double pole in $z=p$ we can then rewrite the sum in terms of the finite part of the integral over $h$\n \\begin{equation}\n\\frac{1}{L}\\sum_{h} f(h) = \\lim_{\\epsilon \\to 0^+} \\int_{-\\infty}^\\infty dh f(h + i \\epsilon) - \\pi i \\underset{h=p}{\\rm res} f(h) = \\: \\fint_{-\\infty}^{\\infty} d h f(h).\n\\end{equation}\nIn order to compute the finite part is then useful to compute the limit $p_j \\to h_j$ of the form factors \\eqref{FFd_final_expression}\n\\begin{align} \\label{FF_recursion}\n \\frac{|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |}{|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^{n-1} \\rangle |} {=} \\frac{F(h_n)}{\\rho_t(h_n)(p_n - h_n)} + \\mathcal{O}(p_n - h_n).\n\\end{align}\nwhere the back-flow is now computed as the sum of the other back-flows for the residual excitation \\eqref{back-sum-flow}\n\\begin{equation}\n F\\left(\\lambda\\,|\\, \\{(\\mu_j^+, \\mu_j^-)\\}_{j=1}^n\\right) = \\sum_{j=1}^{n-1} F\\left(\\lambda\\,|\\, \\mu_j^+, \\mu_j^-\\right) .\n\\end{equation}\n\n\n\\section{\\texorpdfstring{Dynamical structure factor in $1\/c$ expansion}{Dynamical structure factor in 1\/c expansion}} \\label{expansion}\nWe consider here the expansion in $1\/c$ of the dynamical structure factor, defined as the Fourier transform of the density-density correlation\n\\begin{align}\\label{dsf}\n&S(q, \\omega) =\\int dx dt \\: e^{i q x - i \\omega t} \\langle \\rho(\\lambda) | \\hat{\\rho}(x,t) \\hat{\\rho}(0,0) | \\rho(\\lambda) \\rangle\n\\nonumber \\\\&\n=(2 \\pi)^2 \\sum_{n=0}^\\infty \\frac{1}{n!^2}\\left[ \\prod_{j=1}^n \\int_{-\\infty}^{\\infty} d p_j \\rho_h(p_j) \\fint_{-\\infty}^{\\infty} d h_j \\rho(h_j) \\right] \\delta\\left(q-\\sum_{j=1}^n (k(p_j) - k(h_j))\\right) \\nonumber\\\\\n& \\times \\delta\\left(\\omega - \\sum_{j=1}^n (\\omega(p_j) - \\omega(h_j) \\right) |\\langle \\vartheta | \\hat{\\rho} | \\vartheta, \\{ h_j \\to p_j\\}_{j=1}^n \\rangle |^2 , \\nonumber \\\\\n\\end{align}\nfor a generic thermal state at temperature $T=\\beta^{-1}$ and density $D=1$.\nExpanding at the first order in $1\/c$ the only relevant form factors are the ones with only 1 particle-hole excitation $p,h$\n\\begin{align}\n & |\\langle \\vartheta | \\hat{\\rho} | \\vartheta, h \\to p \\rangle | \\nonumber \\\\&\n =\n \\frac{1}{2 \\pi } \\frac{1 + \\frac{2}{ c} }{(\\rho_t(h) \\rho_t(p))^{1\/2}}\n \\left[1 - \\frac{(p - h)^2}{\\pi c} {\\rm PV} \\int d\\lambda \\frac{\\vartheta(\\lambda)}{(\\lambda - p)(\\lambda - h)} \\right] + \\mathcal{O}(1\/c^2),\n\\end{align}\nsince the ones with two or more particle-hole excitations contribute at the order $1\/c^2$ or higher. The filling fraction for a thermal state at temperature $\\beta$, including the $1\/c$ correction, is given by\n\\begin{align}\n \\vartheta(\\lambda) = \\frac{1 + \\frac{2}{c}}{1 + e^{\\beta(\\lambda^2 - h)}} ,\n\\end{align}\nwith $h$ the chemical potential fixing the density $D=1$ of the gas.\nIn the 1 particle-hole spectrum dynamical structure factor at $S(q,\\omega)$ is given in terms of a single form factor with energy $\\omega$ and momentum $q$ times the density of states, which is simply the Jacobian of the transformation from the rapidities of the excitations to the energy and momentum variable\n\\begin{align}\np^2 - h^2 & = \\omega, \\\\\np - h & = q \\Big( 1 + \\frac{2}{c} \\Big)^{-1},\n\\end{align}\nwhich gives a Jacobian factor $\\Big|\\det \\frac{\\partial ( \\omega, q)}{\\partial( p, h ) } \\Big|= 2 q (1+2\/c)^{-1}$ with the rapidities of the excitations given by\n\\begin{align}\n& p = \\frac{q}{2 (1 + 2\/c)} + \\frac{\\omega (1 + 2\/c)}{2 q} , \\\\\n& h = - \\frac{q}{2 (1 + 2\/c)}+ \\frac{\\omega (1 + 2\/c)}{2 q}.\n\\end{align}\n\nWe obtain then an expression for the thermal dynamical structure factor up to $1\/c^2$ corrections\n\\begin{align}\\label{dynamical_final}\nS(q, \\omega) &= (2\\pi)^2 \\frac{1 + \\frac{2}{c}}{2 q } \\Big[\\rho_h( p) \\rho( h)|\\langle \\vartheta | \\hat{\\rho} | \\vartheta, h \\to p \\rangle |\\Big]\\nonumber\\\\\n&=\\frac{2 }{\\pi} \\left( \\pi \\frac{1 + \\frac{6}{c}}{4 q} + \\frac{1}{2 c} {\\rm PV}\\int \\frac{\\vartheta(\\lambda + p) - \\vartheta(\\lambda + h)}{\\lambda } \\right) {\\vartheta(h)\\left( 1 - \\vartheta(p)\\right)} + \\mathcal{O}(1\/c^2).\n\\end{align}\nUsing\n\\begin{align}\n 1-e^{-\\beta\\omega}=\\frac{\\vartheta(h) - \\vartheta(p)}{\\vartheta(h)(1- \\vartheta(p))}.\n\\end{align}\nwe obtain the same result as in \\cite{2005_Brand_PRA_72} (where here we have chosen unitary density $D=1$). Note that the limit $T \\to 0$ can be easily recovered from \\eqref{dynamical_final}. The same is believed to be true for all the orders in $1\/c$ of the correlation functions. This is a non-trivial statement since the procedure to obtain the form factors when the averaging state is the ground state and when is a thermal state are manifestly different.\n\n\nThe example here is carried on for a thermal state, however this result can be extended to any filling fraction $\\vartheta(\\lambda)$ including for example the saddle point state after a quench in the Lieb-Liniger model \\cite{2014_DeNardis_PRA_89}\n\n\n\\section{\\texorpdfstring{Numerical evaluation of the dynamical structure factor}{Numerical evaluation}} \\label{numerics}\n\nThe dynamical structure factor \\eqref{dsf} can be computed through numerical evaluations of the exact formula \\eqref{FFd_final_expression}. The sum over all the possible number of excitations $n = 1, 2 , \\ldots$ requires a great numerical effort, mainly due to the complicated structure of the form factors. To simplify the problem we focus here only on the simplest excitations ($n=1$) consisting of a single particle-hole pair. This leads to an approximate expression for the correlation function which is shown in figures~\\ref{fig2} and~\\ref{fig3}. As in the $1\/c$ section, for concreteness we limit ourselves to thermal equilibrium correlations.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.52]{fig2_new.pdf}\n\\caption{Correlation function $S(k,\\omega)$ at $T=1$, unitary density $D=1$ and with $c=16$ (\\emph{on the left}) and with $c=4$ (\\emph{on the right}). We plot it as a function of energy $\\omega$ for fixed value of momentum $k=k_F$. The 1 particle-hole approximation $S_{1ph}(q, \\omega)$ (blue) fits well the ABACUS results (green) and misses only on the correlation weight which is expected to come from multiple particle-hole excitations. For $c=16$ we plot also the results of the $1\/c$ expansion (red). Inset in the right figure shows the percentage of the f-sum rule \\eqref{fsumrule} saturation for some values of the interaction parameter $c$.}\n\\label{fig2}\n\\end{figure}\n\n\nThe approximation of the correlation function to a single particle-hole pair becomes exact in the limit of large interactions (c.f. previous section) and in the limit of very small momentum $k$.\\footnote{This is the same idea as presented in study of the dynamic structure factor of the XXZ spin chain at small momentum~\\cite{2007_Pereira_JSTAT_8}.} As the interaction is decreased and momentum is increased we expect the approximation to become worse. To quantify how far the resulting correlation function is from the true one we compare our results with an exact numerical evaluation of the correlation function in a finite system~\\cite{PhysRevA.89.033605} via the ABACUS algorithm~\\cite{2009_Caux_JMP_50}. This shows that even for values of $c\\sim 1$, which go well beyond the $1\/c$ expansion and at finite momentum $k=k_F$, the 1 particle-hole contribution captures the essential features of the dynamic structure factor. For $c\\approx 4$ the kinetic and potential energy \\eqref{H} of the ground state of the system are equal \\cite{JCS_Comment} and thus the correlation function is the most difficult to compute.\n\nAdditionally we consider the f-sum rule~\\cite{LL_StatPhys2_BOOK}, an exact equality obeyed by the dynamic structure factor for any fixed momentum $q$\n\\begin{equation} \\label{fsumrule}\n\\int_{-\\infty}^\\infty \\frac{d\\omega}{2 \\pi} \\omega S(q,\\omega) = D q^2.\n\\end{equation}\nIn the limit $c\\to \\infty$ or $k\\rightarrow 0$ the 1 particle-hole spectrum is the full excitation spectrum for the density operator and consequently the f-sum rule is completely saturated by including only these types of excitations in the sum \\eqref{dsf}. However as $c$ decreases with $k$ finite we observe that the f-sum rules is saturated only up to a certain precision and more excitations have to be taken into account in order to obtain the full correlation function. Again, even at values of $c \\sim 1$ and $k\\sim k_F$ the contribution of the 1 particle-hole excitations remains very significant (see insets of figures~\\ref{fig2} and~\\ref{fig3}).\n\nThe results of this section confirm that the form factors~\\eqref{FFd_final_expression} can be directly used to compute the dynamic structure factor or in general the density-density correlation on a generic state with non-zero entropy. Moreover it asserts that the expansion in particle-hole excitation numbers is an effective method to compute the correlation function.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.52]{fig3_new.pdf}\n\\caption{Correlation function $S(k,\\omega)$ at $T=1$, unitary density $D=1$ and with $c=16$ (\\emph{on the left}) and with $c=4$ (\\emph{on the right}) as a function of energy $\\omega$ for fixed value of momentum $k=k_F\/10$. For $c=16$ we plot also the result of the $1\/c$ expansion. The f-sum rule (as plotted inside the inset) is saturated exactly (up to a numerical precision of $\\sim 1\\%$) since the 1 particle-hole approximation becomes exact at small momenta. Lower value of momentum allows for development of a second peak, at negative energies, due to the detailed balance relation $S(k,-\\omega) = S(k, \\omega) \\exp(-\\omega\/T)$~\\cite{LL_StatPhys2_BOOK}. Note that at higher momenta the correlation function is shifted towards higher energies and the negative energy peak becomes practically invisible (See figure~\\ref{fig2}).}\n\\label{fig3}\n\\end{figure}\n\n\n\n\\section{Conclusions} \\label{discussion}\n\nIn this work we studied the thermodynamic limit of the particle-hole form factors for the density operator of the 1D Bose gas. The computations presented here can be generalized to different operators (like the bosonic field operator $\\Psi$) but also to other Bethe Ansatz solvable models for which the microscopic matrix elements are known such as the XXZ spin chain. These problems will be addressed in the future.\nThese form factors constitute the building blocks to compute thermal or post-quench equilibrium correlation functions in the thermodynamic limit at fixed density of particles. They also provide a first step towards the post-quench time evolution as recently done in \\cite{me} for the Tonks-Girardeau ($c=\\infty$) regime.\n\nThe final formula \\eqref{FFd_final_expression} is valid in the thermodynamic limit and it is considerably simpler than its finite size version but still it is not suitable to obtain close-form expressions of correlation functions. The Fredholm determinant of the kernel $\\hat{A}$ is a non-trivial functions of the excitations parameters $\\{ p_j, h_j\\}_{j=1}^n$ and we were not able to obtain further simplifications. A fully factorized expression of the form factors involving a simple almost factorized part depending only on the excitation parameters is still under research.\n\nWe computed the exact thermodynamic dynamical structure factor including only the 1 particle-hole excitations over a thermal state. This approximation is qualitatively different from the usual perturbative one or the low energy limit. For example the perturbation theory in $1\/c$ breaks at $c\\sim 10$ yielding unphysical, negative values of the correlation~\\cite{2005_Brand_PRA_72}, while the (non-linear) Luttinger liquid theory is not able to reproduce the exact shape of the correlation function~\\cite{2012_Imambekov_RMP_84}. We showed that a thermodynamic Bethe Ansatz approach with only single particle-hole excitations produces a good estimate of the density correlations of the system for a wide range of values of the interaction parameter and momentum. Therefore the effect of the extra particle-hole excitations is mainly to increase the weight of the correlation at large momentum.\n\n\nAnother interesting point is to compare our result with similar ones for the thermodynamic limit of one-point functions of the Lieb-Liniger model obtained from the non-relativistic limit of the sinh-Gordon model \\cite{2009_Kormos_PRA_81}. As shown in \\cite{1742-5468-2011-11-P11017} the large volume limit of the diagonal form factor obtained by Bethe Ansatz \\eqref{diagonal} can be expressed as a LeClair-Mussardo series \\cite{1999_LeClair_NPB_552} of the elementary form factors obtained via the bootstrap program \\cite{MussardoBOOK,2009_Kormos_PRA_81}. How to extend this relation to two-point functions remains to be clarified. The Bethe Ansatz approach, presented in this work, might shed a light on this important problem of the Quantum Integrable Field Theories.\n\nFinally, following \\cite{2011_Shashi_PRB_84} where a relation between the form factors and the prefactors of the Lutinger liquid correlation functions at zero temperature was established, it would be interesting to see whether such simple relations also exist at finite temperature or even out-of-equilibrium. The result of \\cite{2011_Kozlowski_JSTAT_P03019} where low temperature correlation functions were studied seem to suggest that such relations might exists. This will be also a subject of a further research.\n\n\n\n\\ack\nWe are very grateful to J.-S. Caux for his support and critical comments and to R. Konik for a stimulating and encouraging discussion.\nWe acknowledge useful and inspiring discussions with S. Eli\\\"{e}ns, G. Mussardo and H. Saleur.\n\\noindent J. De Nardis acknowledges support from the Netherlands Organisation for Scientific Research (NWO). M. Panfil acknowledges support from the Foundation for Fundamental Research on Matter (FOM) at the early stage of this work.\n\n\\noindent This work was supported by ERC under the Starting Grant n. 279391 EDEQS.\n\n\\section*{References}\n\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Missing Details of Section~\\ref{sec:flow}}~\\label{sec:appx_flow}\n\\section{Recap: Flow for Additive Valuations}\\label{sec:flow_additive}\nWhen the valuations are additive, we simply view $t_{ij}$ as bidder $i$'s value for receiving item $j$. Although there are many possible ways to define a flow, we focus on a class of simple ones. Every flow in this class $\\lambda^{(\\beta)}$ is parametrized by a set of parameters $\\beta=\\{\\beta_{ij}\\}_{i\\in[n], j\\in[m]}\\in\\R^{nm}$. Based on $\\beta_i=\\{\\beta_{ij}\\}_{j\\in[m]}$, we first partition the type space $T_i$ for each bidder $i$ into $m+1$ regions:\n\\begin{itemize}[leftmargin=0.7cm]\n\t\\item $R_{0}^{(\\beta_i)}$ contains all types $t_i$ such that $t_{ij}<\\beta_{ij}$ for all $j\\in[m]$.\n\t\\item $R_j^{(\\beta_i)}$ contains all types $t_i$ such that $t_{ij}-\\beta_{ij}\\geq 0$ and $j$ is the smallest index in $\\argmax_k\\{t_{ik}-\\beta_{ik}\\}$.\n\\end{itemize}\n\nWe use essentially the same flow as in~\\cite{CaiDW16}. Here we provide a partial specification and state some desirable properties of the flow. See Figure~\\ref{fig:multiflow} for an example with $2$ items and~\\cite{CaiDW16} for a complete description of the flow.\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth}\n{\\bf Partial Specification of the flow $\\lambda^{(\\beta)}$:}\n\\begin{enumerate}[leftmargin=0.7cm]\n \\item For every type $t_{i}$ in region $R^{(\\beta_{i})}_{0}$, the flow goes directly to $\\varnothing$ (the super sink).\n \\item For all $j>0$, any flow entering $R^{(\\beta_{i})}_{j}$ is from $s$ (the super source) and any flow leaving $R^{(\\beta_{i})}_{j}$ is to $\\varnothing$.\n \\item For all $t_{i}$ and $t_{i}'$ in $R^{(\\beta_{i})}_{j}$ ($j>0$), {$\\lambda^{(\\beta)}_{i}(t_{i},t_{i}')>0$} only if $t_{i}$ and $t_{i}'$ only differ in the $j$-th coordinate.\n\\end{enumerate}\n\\end{minipage}}\n\\caption{Partial Specification of the flow $\\lambda^{(\\beta)}$.}\n\\label{fig:flow specification}\n\\end{figure}\n\n\\begin{figure}\n \\centering{\\includegraphics[width=0.5\\linewidth]{multi_flow.png}}\n \\caption{An example of $\\lambda^{(\\beta)}_{i}$ for additive bidders with two items.}\n \\label{fig:multiflow}\n\\end{figure}\n\n\n\\begin{lemma}[\\cite{CaiDW16}\\footnote{Note that this Lemma is a special case of Lemma 3 in~\\cite{CaiDW16} when the valuations are additive. }]\\label{lem:additive flow properties}\n\tFor any $\\beta$, there exists a flow $\\lambda^{(\\beta)}_{i}$ such that the corresponding virtual value function $\\Phi_{i}(t_{i}, \\cdot)$ satisfies the following properties:\n\t\\begin{itemize}[leftmargin=0.5cm]\n\t\t\\item For any $t_{i}\\in R^{(\\beta_i)}_{0}$, $\\Phi_{i}(t_{i},S) = \\sum_{k\\in S} t_{ik}$.\n\t\t\\item For any $j>0$, $t_{i}\\in R^{(\\beta_i)}_{j}$, $$\\Phi_{i}(t_{i},S)\\leq \\sum_{k\\in S \\land k\\neq j} t_{ik}+{\\tilde{\\varphi}}_{ij}(t_{ij})\\cdot\\mathds{1}[j\\in S],$$ where ${\\tilde{\\varphi}}_{ij}(\\cdot)$ is Myerson's ironed virtual value function for $D_{ij}$.\n\t\\end{itemize}\n\\end{lemma}\n\nThe properties above are crucial for showing the approximation results for simple mechanisms in~\\cite{CaiDW16}. One of the key challenges in approximating the optimal revenue is how to provide a tight upper bound. A trivial upper bound is the social welfare, which may be arbitrarily bad in the worst case. By plugging the virtual value functions in Lemma~\\ref{lem:additive flow properties} into the partial Lagrangian, we obtain a new upper bound that replaces the value of the buyer's favorite item with the corresponding Myerson's ironed virtual value. As demonstrated in~\\cite{CaiDW16}, this new upper bound is at most $8$ times larger than the optimal revenue when the buyers are additive, and its appealing structure allows the authors to compare the revenue of simple mechanisms to it. In \\notshow{the next section} {Section~\\ref{sec:flow}}, we identify some difficulties in directly applying this flow to subadditive valuations. Then we show how to overcome these difficulties by relaxing the subadditive valuations and obtain a similar upper bound.\n\n\\section{Proof of Lemma~\\ref{lem:subadditive flow properties}}\\label{sec:proof_virtual_relaxation}\n\\begin{lemma}\\label{lem:separate the favorite out in virtual value}\n\tFor any flow $\\lambda^{(\\beta)}_i$ that respects the partial specification in Figure~\\ref{fig:flow specification}, the corresponding virtual valuation function $\\Phi_i^{(\\beta_i)}$ of $v_i^{(\\beta_i)}$ for any buyer $i$ is:\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\n \\left(V_i(t'_{ij})-V_i(t_{ij})\\right)$, if $t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S$.\n\\item $v_i(t_i,S)$, otherwise.\n\\end{itemize}\n\n\\begin{comment}\n\t \\begin{equation*}\n\\begin{aligned}\n\\Phi_i^{(\\beta_i)}(t_i, S)=\n\\begin{cases}\nv_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right) &\\text{if }t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S\\\\\nv_i(t_i,S) & \\text{o.w.}\n\\end{cases}\n\\end{aligned}\n\\end{equation*}\n\\end{comment}\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:separate the favorite out in virtual value}\n\tThe proof follows the definitions of the virtual valuation function (Definition~\\ref{def:virtual value}) and relaxed valuation (Definition~\\ref{def:relaxed valuation}). We use $t_{i,-j}=\\langle t_{i{j'}}\\rangle_{j'\\not=j}$ to denote bidder $i$'s information for all items except item $j$. If $t_i\\in R_j^{(\\beta_i)}$ and $j\\in S$, $v_i^{(\\beta_i)}(t_i,S) = v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})$. Since $\\lambda(t_i,t_i')>0$ only when $t_{i,-j}=t_{i,-j}'$ and $t_i'\\in R_j^{(\\beta_i)}$, $v_i^{(\\beta_i)}(t'_i,S) = v_i(t'_i, S\\backslash \\{j\\})+V_i(t'_{ij})= v_i(t_i, S\\backslash \\{j\\})+V_i(t'_{ij})$. Therefore,\n\\begin{align*}\n \\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})\n -\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)\n\\end{align*}\n\t\n\tIf $t_i\\in R_j^{(\\beta_i)}$ and $j\\notin S$ or $t_i\\in R_0^{(\\beta_i)}$, then $v_i^{(\\beta_i)}(t_i,S) =v_i(t_i, S)$. If $t_i\\in R_0^{(\\beta_i)}$, there is no flow entering $t_i$ except from the source, so clearly $\\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S)$. If $t_i\\in R_j^{(\\beta_i)}$, then for any $t'_i$ that only differs from $t_i$ in the $j$-th coordinate, we have $v_i(t'_i, S)=v_i(t_i,S)$, because {$j\\not\\in S$}. Hence, $\\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S)$.\n\\end{prevproof}\n\n\n\\begin{prevproof}{Lemma}{lem:subadditive flow properties}\n\nLet $\\Psi_{ij}^{(\\beta_i)}(t_i)=V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)$. According to Lemma~\\ref{lem:separate the favorite out in virtual value}, it suffices to prove that for any $j>0$, any $t_{i}\\in R^{(\\beta_i)}_{j}$, $\\Psi_{ij}^{(\\beta_i)}(t_i)\\leq {\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$.\n\n\\begin{claim}\nFor any type $t_{i}\\in R^{(\\beta_i)}_{j}$, if we only allow flow from type $t'_{i}$ to $t_{i}$, where $t_{ik}=t'_{ik}$ for all $k\\neq j$ and $t'_{ij}\\in \\argmin_{s\\in T_{ij} \\land V_i(s)> V_i(t_{ij})} V_i(s)$, and the flow $\\lambda(t_i',t_i)$ equals $\\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}$ fraction of the total in flow to $t_i'$, then there exists a flow $\\lambda$ such that\n\\begin{align*}\n\\Psi_{ij}^{(\\beta_i)}(t_i)=\\varphi_{ij}(V_i(t_{ij}))\n=V_i(t_{ij})-\\frac{\\left(V_i({t'_{ij}})-V_i(t_{ij})\\right)\\cdot\\Pr_{v\\sim F_{ij}}[v>V_i(t_{ij})]}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]},\n\\end{align*} where $\\varphi_{ij}(V_i(t_{ij}))$ is the Myerson virtual value for $V_i(t_{ij})$ with respect to $F_{ij}$. \\end{claim}\n\\begin{proof}\n{As the flow only goes from $t_i'$ and $t_i$, where $t_i'$ and $t_i$ only differs in the $j$-th coordinate, and \\\\\n\\noindent$t_{ij}\\in \\argmax_{s\\in T_{ij} \\land V_i(s)< V_i(t_{ij}')} V_i(s)$. If $t_{ij}$ is a type with the largest $V_i(t_{ij})$ value in $T_{ij}$, then there is no flow coming into it except the one from the source, so $\\Psi_{ij}^{(\\beta_i)}(t_i)=V_i(t_{ij})$. For every other value of $t_{ij}$, the in flow is exactly\n\\begin{align*} \\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}\\prod_{k\\neq j} f_{ik}(t_{ik})\\cdot \\sum_{x\\in T_{ij}:V_i({x})>V_i(t_{ij})} f_{ij}(x) \n=\\prod_{k} f_{ik}(t_{ik})\\cdot \\frac{\\Pr_{v\\sim F_{ij}}[v>V_i(t_{ij})]}{\\Pr_{v\\sim F_{ij}}[v=V_i(t_{ij})]}.\\end{align*}\n {This is because each type of the form $(x,t_{i,-j})$ with $V_i(x) > V_i(t_{ij})$ is also in $R^{(\\beta_i)}_{j}$. So $\\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}$ of all flow that enters these types will be passed down to $t_{i}$ (and possibly further, before going to the sink), and the total amount of flow entering all of these types from the source is exactly {$\\prod_{k\\neq j} f_{ik}(t_{ik})\\cdot \\sum_{x\\in T_{ij}:V_i({x})>V_i(t_{ij})} f_{ij}(x) $}.} Therefore, $\\Psi_{ij}^{(\\beta_i)}(t_i)=\\varphi_{ij}(V_i(t_{ij}))$. Whenever there is no more type $t_i\\in R_j^{(\\beta_i)}$ with smaller $V_i(t_{ij})$ value, we push all the flow to the sink.}\n\\end{proof}\n\n\nIf $F_{ij}$ is regular, this completes our proof. When $F_{ij}$ is not regular, we can iron the virtual value function in the same way as in \\cite{CaiDW16}. Basically, for two types $t_i,t_i'\\in R^{(\\beta_i)}_{j}$ that only differ in the $j$-th coordinate, if $\\Psi_{ij}^{(\\beta_i)}(t_i)>\\Psi_{ij}^{(\\beta_i)}(t_i')$ but $V_i(t_{ij})0$ if and only if\n\\begin{itemize}\n\\item $t_i\\in R_j^{\\pi}$\n\\item $t_{ik}'=t_{ik},\\forall k\\not=j$\n\\item $t_{ij}'>t_{ij}$\n\\end{itemize}\n\nThen after ironing, the Language function becomes:\n\\begin{equation}\nL(\\lambda^{\\pi},\\pi,p)=\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\bigg(t_{ij}\\cdot \\mathds{1}\\big[t_i\\not\\in R_j^{\\pi}\\big]+\\tilde{\\phi_{ij}}(t_{ij})\\cdot \\mathds{1}\\big[t_i\\in R_j^{\\pi}\\big]\\bigg)\n\\end{equation}\n\nBy the Strong Duality Theorem,\n\\begin{equation}\nREV=\\min_{\\lambda}\\max_{\\pi\\in P(\\mathcal{F},D),p}L(\\lambda,\\pi,p)\\leq \\max_{\\pi\\in P(\\mathcal{F},D),p}L(\\lambda^{\\pi},\\pi,p)\n\\end{equation}\n\nIt's enough to bound $L(\\lambda^{\\pi},\\pi,p)$ for all feasible $\\pi$ to obtain an upper bound for $REV$.\n\n\\subsection{Sequential Mechanism and c-Selectability}\nWe will use a new Sequential mechanism. The mechanism posts a price $\\theta_{ij}$ of each item $j$ for each bidder $i$ and determine an order $\\sigma$ for bidders. Each bidder comes in order $\\sigma$. When bidder $i$ with type $t_i$ comes, suppose the set of items left is $S_i$. The mechanism will let bidder $i$ know $S_i$ and charge him an entry fee $\\delta_i$. If he chooses to join the auction, he pays the entry fee and then takes his favorite bundle $S_i^{*}=\\arg\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}(t_{ij}-\\theta_{ij})$ and pay $\\sum_{j\\in S_i^{*}}\\theta_{ij}$. Let the optimal revenue that such a mechanism can get is SEQ.\n\nWhen bidder $i$ is facing the remaining item set $S_i$, we have no idea on which item he will take. c-Selectability from [\\ref{feldman}] actually take control of it. The following lemma directly comes from the definition of c-Selectability:\n\\begin{lemma}\\label{feldman2015}\n(Feldman 2015) For a downward close $\\mathcal{F}$, if there exists a $c$-selectable greedy OCRS, then in the Sequential mechanism, when it's bidder $i$'s turn, as long as $j\\in S_i$ and $t_{ij}>\\theta_{ij}$, item $j$ is in $S_i^{*}$ with at least probability $c$. The probability is taken over the randomness of $S_i$.\n\\end{lemma}\n\nIf $\\mathcal{F}$ is a matroid, the paper shows that $c=1-b$.\n\n\n\\subsection{Separating into Pieces}\n\\begin{equation}\n\\begin{aligned}\n&L(\\lambda^{\\pi},\\pi,p) \\\\\n=&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\tilde{\\phi_{ij}}(t_{ij})\\cdot \\mathds{1}\\big[t_i\\in R_j^{\\pi}\\big] \\text{(SINGLE)}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot t_{ij}\\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big]\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot t_{ij}\\cdot \\mathds{1}\\big[t_{ij}< \\beta_{ij}^{\\pi}\\big]\\\\\n\\leq& \\text{ SINGLE}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big] \\text{(SURPLUS)}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\cdot \\mathds{1}\\big[t_{ij}\\geq \\beta_{ij}^{\\pi}\\big]+\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\cdot \\mathds{1}\\big[t_{ij}< \\beta_{ij}^{\\pi}\\big]\\\\\n\\leq& \\text{ SINGLE + SURPLUS }+\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\text{(PROPHET)}\n\\end{aligned}\n\\end{equation}\n\n~\\\\\n\n\\begin{equation}\n\\begin{aligned}\nSURPLUS&=\\sum_i\\sum_{t_i: \\exists k,t_{ik}>\\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big] \\\\\n&+\\sum_i\\sum_{t_i: \\forall k,t_{ik}\\leq \\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\Pr_{t_{i,-j}\\sim T_{i,-j}}\\big[\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\text{ (TAIL)}\\\\\n&+\\sum_i\\sum_{t_i: \\forall k,t_{ik}\\leq \\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\cdot \\mathds{1}\\big[t_{ij}\\geq \\beta_{ij}^{\\pi}\\big]\\text{ (CORE)}\\\\\n\\end{aligned}\n\\end{equation}\n\nHere, $\\tau_i^{\\pi}$ is chosen such that\n\\begin{equation}\n\\sum_j \\Pr_{t_{ij}}[t_{ij}\\geq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}]=\\frac{1}{2}\n\\end{equation}\n\n~\\\\\n\n\\subsection{Bounding SINGLE}\n\\begin{lemma}\nFor any downward close $\\mathcal{F}$, $\\text{SINGLE}\\leq \\text{OPT}^{\\text{COPIES}}$.\n\\end{lemma}\n\\begin{proof}\nWe build a new mechanism $M'$ in the Copies setting based on $M$. Whenever $M$ allocates item $j$ to bidder $i$ and $t_i\\in R_j^{\\pi}$, $M'$ serves the agent $(i,j)$. Then $M'$ is feasible in the Copies setting and SINGLE is the ironed virtual welfare of $M'$ with respect to $\\tilde{\\phi}(\\cdot)$. Since the maximum revenue in the Copies setting is achieved by the maximum virtual welfare, thus $\\text{OPT}^{\\text{COPIES}}$ is no less than SINGLE.\n\\end{proof}\n\nThe $\\text{OPT}^{\\text{COPIES}}$ can be achieved $\\frac{1}{6}$-approximately by a post-price mechanism.\n\n~\\\\\n\n\\subsection{Bounding PROPHET}\n\\begin{equation}\n\\begin{aligned}\n\\text{PROPHET }&=\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi}\\\\\n&=\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}(t_i)\\\\\n&=\\frac{1}{b}\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\\label{prophet}\nFor $\\mathcal{F}$, if there exists a $c$-selectable greedy OCRS,\n\\[\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq \\frac{1}{(1-b)\\cdot c}\\cdot \\text{SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider a Sequential mechanism without entry fee and post price $\\theta_{ij}=\\beta_{ij}^{\\pi}$. Then according to Lemma \\ref{feldman2015},\n\\begin{equation}\n\\begin{aligned}\nSEQ&\\geq \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot \\Pr_{t_{-i}}[j\\in S_i]\\cdot \\Pr_{t_{ij}}[t_{ij}>\\beta_{ij}^{\\pi}]\\cdot c\\\\\n&\\geq \\sum_i\\sum_j \\beta_{ij}\\cdot (\\sum_{l=1}^{i-1}q_{lj}^{\\pi})\\cdot q_{ij}^{\\pi}\\cdot c\\\\\n&\\geq (1-b)\\cdot c\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nSpecifically, if $\\mathcal{F}$ is a matroid,\n\\begin{equation}\n\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq \\frac{1}{\\cdot (1-b)^2}\\cdot SEQ\n\\end{equation}\n\n~\\\\\n\n\\subsection{Bounding TAIL}\nWe will bound TAIL for any downward close $\\mathcal{F}$. Let $P_{ij}=\\arg\\max_{x\\geq \\tau_i^{\\pi}}x\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq x]$, and $r_{ij}=P_{ij}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq P_{ij}]$, $r_i=\\sum_j r_{ij}$, $r=\\sum_i r_i$. We have the following relationship between $r_i$ and $\\tau_i^{\\pi}$:\n\\begin{lemma}\\label{tail0}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot \\tau_i^{\\pi}$.\n\\end{lemma}\n\\begin{proof}\n\\begin{equation}\n\\begin{aligned}\nr_i&= \\sum_j P_{ij}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq P_{ij}]\\\\\n&\\geq \\sum_j \\tau_i^{\\pi}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq \\tau_i^{\\pi}]\\\\\n&=\\frac{1}{2}\\cdot \\tau_i^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n~\\\\\n\nWe notice by the definition of $r_{ij}$,\n\\begin{equation}\\label{tail1}\n\\begin{aligned}\n\\text{TAIL }&=\\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\Pr_{t_{i,-j}\\sim T_{i,-j}}\\big[\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\sum_{k\\not=j}\\Pr_{t_{ik}\\sim T_{ik}}\\big[t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}\\\\\n&\\leq \\sum_i r_i\\cdot \\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\\\\n&=\\frac{1}{2}\\cdot r\n\\end{aligned}\n\\end{equation}\n\nThe following lemma shows that $r$ can be approximately achieved by the Sequential mechanism.\n\\begin{lemma}\\label{tail2}\n\\[r\\leq \\frac{2}{(1-b)}\\text{ SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider the Sequential mechanism with item price $\\theta_{ij}=P_{ij}+\\beta_{ij}^{\\pi}$, $\\sigma=(1,2,...,n)$ and without entry fee. When bidder $i$ comes, he will definitely take only item $j$ and pay $P_{ij}+\\beta_{ij}^{\\pi}$ if:\n\\begin{itemize}\n\\item $j\\in S_i$\n\\item $t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}$\n\\item $\\forall k\\not=j, t_{ij}\\leq P_{ij}+\\beta_{ij}^{\\pi}$\n\\end{itemize}\n\nNotice that due to the second condition, every bidder will take item $j$ with at most $q_{ij}$ probability. Thus we have\n\\begin{equation}\n\\begin{aligned}\nSEQ&\\geq \\sum_i\\sum_j (P_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{-i}}[j\\in S_i]\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\Pr_{t_{i,-j}}[\\forall k\\not=j, t_{ij}\\leq P_{ij}+\\beta_{ij}^{\\pi}]\\\\\n&\\geq \\sum_i\\sum_j P_{ij}\\cdot \\big(\\sum_{l=1}^{i-1}q_{lj}^{\\pi}\\big)\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\Pr_{t_{i,-j}}[\\forall k\\not=j, t_{ij}\\leq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}]\\\\\n&\\geq (1-b)\\cdot \\sum_i\\sum_j P_{ij}\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\big(1-\\sum_{k\\not=j}\\Pr_{t_{ik}}[t_{ik}\\leq P_{ik}+\\tau_i^{\\pi}]\\big)\\\\\n&\\geq \\frac{1}{2}(1-b)\\cdot \\sum_i\\sum_j r_{ij}\\\\\n&\\geq \\frac{1}{2}(1-b)\\cdot r\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nCombining Equation (\\ref{tail1}) and Lemma \\ref{tail2}, we have\n\\begin{equation}\nTAIL\\leq \\frac{1}{1-b}\\cdot SEQ\n\\end{equation}\n\n\\subsection{Bounding CORE}\nWe will bound CORE for matroid $\\mathcal{F}$. Define $t_{ij}'=(t_{ij}-\\beta_{ij}^{\\pi})\\cdot \\mathds{1}\\big[\\beta_{ij}^{\\pi}\\leq t_{ij}\\leq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}\\big]\\in [0,\\tau_i^{\\pi}]$. Then since $\\pi(\\cdot)$ is feasible,\n\\begin{equation}\nCORE=\\sum_i\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]\n\\end{equation}\n\nConsider the Sequential mechanism with item price $\\theta_{ij}=\\beta_{ij}^{\\pi}$ and order $\\sigma=(1,2,...,n)$. When it's bidder $i$'s turn, suppose the set of items left is $S_i$. Define the entry fee $\\delta_i$ for bidder $i$ as:\n\\[\\Pr_{t_i'}[\\max_{S \\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\leq \\delta_i]=\\frac{2}{3}\\]\n\nNotice that if bidder $i$ enters the auction, the profit he gets is\n\\[\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}(t_{ij}-\\theta_{ij})\\geq \\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\]\nThus with probability at least $\\frac{1}{3}$, bidder $i$ will pay the entry fee $\\delta_i$. Besides, with the same argument in Lemma \\ref{prophet},\n\\begin{equation}\n\\text{SEQ}\\geq \\sum_i\\frac{1}{3}\\big(\\delta_i+(1-b)^2\\cdot \\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\big)\n\\end{equation}\n\n~\\\\\n\nThe following result from [Schechtman 1999] can be applied to bound $\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]$ for all $i$:\n\n\\begin{lemma}\\label{core2}\n(Schechtman, 1999)Let $f(t,S)$ be a value function that is additive under a downward close constaint, drawn from a distribution D over support $[0,\\tau]$ for some $\\tau\\geq 0$. Let $\\Delta$ be a value such that $\\Pr_{t\\sim D}\\big[f(t,[m])\\leq \\Delta\\big]=\\frac{2}{3}$. Then for all $k>0$,\n\\begin{equation}\n\\Pr_{t\\sim D}\\big[f(t,[m])\\geq 3\\Delta+k\\cdot \\tau\\big]\\leq \\frac{9}{4}\\cdot 2^{-k}\n\\end{equation}\n\\end{lemma}\n\n\\begin{corollary}\\label{core3}\nFor each $i$,\n\\[\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]\\leq 3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4\\ln(2)}\\]\n\\end{corollary}\n\\begin{proof}\nLet $g(t_i')=\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'$. Directly from Lemma \\ref{core2},\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E}_{t_i'}\\big[g(t')\\big]&= \\mathbf{E}_{t_i'}\\big[g(t_i');g(t_i')<3\\delta_i\\big]+ \\mathbf{E}_{t_i'}\\big[g(t');g(t_i')\\geq 3\\delta_i\\big]\\\\\n&\\leq 3\\delta_i+\\int_0^{\\infty}Pr[g(t_i')>3\\delta_i+y]dy\\\\\n&\\leq 3\\delta_i+\\int_0^{\\infty}\\frac{9}{4}\\cdot 2^{-y\/\\tau}dy\\\\\n&\\leq 3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4\\ln(2)}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nNow we can finish bounding the CORE with the following lemma.\n\n\\begin{lemma}\n\\[\\text{CORE }+3(1-b)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq (\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider the mechanism above. Assume $S^{*}=\\arg\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'$. Notice for all $i,j$, $\\Pr_{t_{-i}}[j\\in S_i]\\geq 1-\\sum_i q_{ij}=1-b$. With Lemma \\ref{tail0}, \\ref{tail2}, and \\ref{core2}, we have\n\\begin{equation}\n\\begin{aligned}\nCORE&=\\sum_i\\mathbf{E}_{t_i'}\\big[\\sum_{j\\in S^{*}} t_{ij}'\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i\\mathbf{E}_{t'}\\big[\\sum_{j\\in S^{*}} t_{ij}'\\cdot \\mathds{1}[j\\in S_i]\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i\\mathbf{E}_{t'}\\big[\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i (3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4ln(2)})\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\bigg(9\\cdot \\text{SEQ }-3(1-b)^2\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}+\\frac{9r}{2\\cdot ln(2)}\\bigg)\\\\\n&\\leq (\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}-3(1-b)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n~\\\\\n\n\\subsection{Optimizing the Constant}\nPut all pieces together:\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda^{\\pi},\\pi,p)&\\leq \\text{SINGLE}+\\text{TAIL}+\\text{PROPHET}+\\text{CORE} \\\\\n&\\leq 6\\cdot \\text{SREV}+\\frac{1}{1-b}\\cdot \\text{SEQ}+\\big(\\frac{1}{b}-3(1-b)\\big)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}+(\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}\\\\\n&\\leq 6\\cdot \\text{SREV}+\\bigg(\\frac{7}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2}+\\frac{1}{b(1-b)^2}\\bigg)\\cdot \\text{SEQ}\n\\end{aligned}\n\\end{equation}\n\nWhen $b=\\frac{1}{7}$. The larger one of SREV and SEQ gets a $1\/42$-approximately mechanism.\n\n~\\\\\n\n\n\\section{Single Bidder, Subadditive Valuation}\n\\subsection{Problem Setting\\textsuperscript{[\\ref{subadditive}]}}\nLet $m$ be the number of items, $I=[m]$ be the set of items. For each item $j$, let $\\Omega_j$ be a compact space. Each type $t_j\\in \\Omega_j$ represent the information for item $j$. For all $S\\subseteq I$, denote $\\Omega_S=\\times_{j\\in S}\\Omega_j$. $\\Omega=\\Omega_{I}$. The bidder's type $t=\\langle t_j \\rangle_{j\\in I}$ is drawn from a distribution $D$ on $\\Omega$. After $t$ is drawn, the bidder's valuation for a set of items S is only determined by the vector $\\langle t_j \\rangle_{j\\in S}$. Formally, denote $\\Omega^{*}=\\bigcup_{S\\subseteq I}\\Omega_S$. There exists a valuation function $V:\\Omega^{*}\\to \\mathcal{R}$ such that $\\forall t\\in \\Omega, S\\subseteq I$,\n\\begin{equation}\nv(t,S)=V(\\langle t_j \\rangle_{j\\in S},S)\n\\end{equation}\n\nIn this problem, we assume that the bidder's valuation is subadditive, i.e., $\\forall t\\in \\Omega$, $\\forall P,Q\\subseteq I$,\n\\begin{equation}\nv(t,P\\cup Q)\\leq v(t,P)+v(t,Q)\n\\end{equation}\n\n~\\\\\n\n\\subsection{Duality}\nWe use the following functions(variables) to describe the mechanism:\n\\begin{itemize}\n\\item $p(t)$, $t\\in \\Omega^{+}$: the prize that the bidder should pay when his type is $t$.\n\\item $\\phi(t,t')$, $t\\in \\Omega, t'\\in \\Omega^{+}$: the expect valuation if the bidder pretends to be $t'$ when his type is $t$.\n\\end{itemize}\n\nwhere $\\Omega^{+}=\\Omega\\cup \\{\\emptyset\\}$ which allows the bidder not to participate in the auction. When $t=\\emptyset$, $p(t)=\\phi(\\cdot,t)=0$.\n\n\\textbf{Remark:} For a mechanism, let $\\pi_S(t)$ be the probability for bidder with type $t$ to obtain a set $S$ of item. Then $\\phi(t,t')$ can be written as\n\\begin{equation}\n\\phi(t,t')=\\sum_{S\\subseteq I}\\pi_S(t)v_{t'}(S)\n\\end{equation}\n\nHere we use $\\phi(t,t')$ to replace the original variables $\\pi_j(t)=\\sum_{S:j\\in S}\\pi_S(t)$. The function in fact includes both the probability and the valuation. Like $\\pi$, all the $\\phi(t,t')$'s should stay in some feasible region to avoid over-allocation. We use $\\phi\\in \\mathcal{O}$ to represent it.\n\n~\\\\\n\nOur primal is:\n\\begin{itemize}\n\\item \\textbf{Variables}: $p(t),\\phi(t,t'),\\quad t\\in \\Omega, t'\\in \\Omega^{+}$\n\\item \\textbf{Constraint}:\n\n$\\quad(1)\\phi(t,t)-p(t)\\geq \\phi(t,t')-p(t'),\\quad \\forall t\\in \\Omega, t'\\in \\Omega^{+}$\n\n$\\quad(2)\\phi\\in \\mathcal{O}$\n\n\\item \\textbf{Objective}: \\text{min} $\\sum_{t\\in \\Omega}f(t)p(t)$\n\\end{itemize}\n\n~\\\\\n\nThe Language dual function $L(\\lambda,\\phi,p)$ is\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda,\\phi,p)&=\\sum_{t\\in T}f(t)p(t)+\\sum_{t\\in \\Omega,t'\\in \\Omega^{+}}\\lambda(t,t')\\bigg((\\phi(t,t)-p(t))-(\\phi(t,t')-p(t'))\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}\\bigg(f(t)+\\sum_{t'\\in \\Omega}\\lambda(t',t)-\\sum_{t'\\in \\Omega^{+}}\\lambda(t,t')\\bigg)+\\sum_{t\\in \\Omega}\\bigg(\\phi(t,t)\\cdot\\sum_{t'\\in \\Omega^{+}}\\lambda(t,t')-\\sum_{t'\\in \\Omega}\\lambda(t',t)\\cdot \\phi(t',t)\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}f(t)\\bigg(\\phi(t,t)-\\frac{1}{f(t)}\\sum_{t'\\in \\Omega}\\lambda(t',t)\\big(\\phi(t',t)-\\phi(t,t)\\big)\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}f(t)\\Phi(t)\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Phi(t)=\\phi(t,t)-\\frac{1}{f(t)}\\sum_{t'\\in \\Omega}\\lambda(t',t)\\big(\\phi(t',t)-\\phi(t,t)\\big)\n\\end{equation}\n\nHere we need $\\forall t\\in \\Omega$,\n\\begin{equation}\nf(t)+\\sum_{t'\\in \\Omega}\\lambda(t',t)-\\sum_{t'\\in T^{+}}\\lambda(t,t')=0\n\\end{equation}\nto avoid the unlimited optimal value.\n\n\\subsection{Change the Valuation}\n\nWe will change the valuation and apply the duality to the new valuation. Let\n\n\\begin{equation}\nR_j=\\bigg\\{t\\in\\Omega: j=\\arg\\max_k V(t_k,\\{k\\})\\bigg\\}\n\\end{equation}\n\nWe define a new valuation $\\hat{v}(\\cdot)$. For a type $t\\in \\Omega$, if $t\\in R_j$, then\n\\begin{equation}\n\\begin{aligned}\n\\hat{v}(t,S)=\n\\begin{cases}\nv(t,S\\backslash\\{j\\})+v(t,\\{j\\}), &j\\in S\\\\\nv(t,S), &j\\not\\in S\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\nSince $v(t,\\cdot)$ is subadditive, $\\hat{v}(t,S)\\geq v(t,S)$ for all subset S.\n\nLet $REV(v)$ and $REV(\\hat{v})$ be the optimal revenue for the two valuations. We would like to see that the two values are not far so that we can bound $REV(v)$ by bounding $REV(\\hat{v})$. The following two lemmas give the proof of this fact:\n\n\\begin{lemma}\n\\textsuperscript{[\\ref{subadditive}]}Consider a distributions D with two different valuations $v$ and $\\hat{v}$. Let $\\delta(t,S)=\\hat{v}(t,S)-v(t,S)$. If $\\delta(t,S)\\geq 0$ for all $t\\in T, S\\subseteq I$, then for any $\\epsilon\\in (0,1)$,\n\\begin{equation}\nREV(\\hat{v})\\geq (1-\\epsilon)(REV(v)-\\frac{\\bar\\delta}{\\epsilon})\n\\end{equation}\nwhere $\\bar\\delta=\\mathbb{E}_{t\\sim D}[\\max_{S\\subseteq I}\\delta(t,S)]$.\n\nIf we choose $\\epsilon=\\frac{1}{2}$,\n\\begin{equation}\nREV(v)\\leq 2REV(\\hat{v})+2\\bar\\delta\n\\end{equation}\n\n\\end{lemma}\n\n~\\\\\n\n\\begin{lemma}\nLet $\\bar\\delta'=\\mathbb{E}_{t\\sim D}\\bigg[v(t,I\\backslash\\{j\\}), j=\\arg\\max_k V(t_k,\\{k\\})\\bigg]$, then\n\\begin{equation}\n\\bar\\delta\\leq \\bar\\delta'\\leq 6BREV+9.2SREV\n\\end{equation}\n\\end{lemma}\n\n~\\\\\n\nFor a fixed type $t$, we make a partition $I=C_t\\cup T_t$ based on some cutoff $r$, where $C_t=\\{j\\in I: v(t,\\{j\\})V(t_j,\\{j\\})$, there is at least one bundle with positive profit. The mechanism will definitely sell some item, obtaining expected revenue at least $V(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]$. Thus,\n\\begin{equation}\nV(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]\\leq SREV\n\\end{equation}\n\n\\begin{equation}\\label{e2}\n(TAIL)\\leq SREV\\cdot \\sum_j\\sum_{t_j:V(t_j,\\{j\\})\\geq r}f_j(t_j)=SREV\n\\end{equation}\n\n~\\\\\n\nNow we can finish the prove of Lemma 2.\n\n\\begin{equation}\n\\begin{aligned}\nPr_t[\\exists j, V(t_j,\\{j\\})\\geq r]&\\geq 1-\\prod_{j}F_j(r)\\\\\n&\\geq 1-(\\frac{\\sum_j F_j(r)}{n})^n\\\\\n&\\geq 1-(1-1\/n)^n\\\\\n&=1-\\frac{1}{e}\n\\end{aligned}\n\\end{equation}\n\nConsider the auction that sells every item with price $r$. Then with probability at least $1-\\frac{1}{e}$, the mechanism will sell at least one item at price at least $r$, obtaining revenue at least $(1-\\frac{1}{e})r$. Thus,\n\\begin{equation}\\label{e3}\nr\\leq \\frac{1}{1-1\/e}\\cdot SREV\n\\end{equation}\n\nCombining Equation (\\ref{e1})(\\ref{e2})(\\ref{e3}), we have\n\\begin{equation}\n\\bar\\delta\\leq (CORE)+(TAIL)\n\\leq 6BREV+\\frac{4r}{ln(2)}+SREV\n\\leq 6BREV+9.2SREV\n\\end{equation}\nwhich finishes Lemma 2.\n\n~\\\\\n\n\\subsection{Construction of the Flow}\nWe apply duality on the new valuation $\\hat{v}$. For $t,t'\\in \\Omega$, $\\lambda(t',t)>0$ if and only if\n\\begin{itemize}\n\\item $t$ and $t'$ only differ on the $j$-th coordinate.\n\\item $t\\in R_j$.\n\\item $V(t_j',\\{j\\})>V(t_j,\\{j\\})$.\n\\end{itemize}\n\nWe now consider $\\hat{v}(t',S)-\\hat{v}(t,S)$ when $\\lambda(t',t)>0$. There are two conditions for subset S:\n\n(1)$j\\not\\in S$, notice that only the $j-$th coordinate is different,\n\\begin{equation}\n\\hat{v}(t',S)=V'(\\langle t_k' \\rangle_{k\\in S},S)=V'(\\langle t_k \\rangle_{k\\in S},S)=\\hat{v}(t,S)\n\\end{equation}\n\n(2)$j\\in S$, we have\n\\begin{equation}\n\\begin{aligned}\n\\hat{v}(t',S)-\\hat{v}(t,S)&=\\bigg(V'(\\langle t_k' \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})+V'(t_j',\\{j\\})\\bigg)-\n\\bigg(V'(\\langle t_k \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})+V'(t_j,\\{j\\})\\bigg)\\\\\n&=\\bigg(V'(\\langle t_k' \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})-V'(\\langle t_k \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})\\bigg)+\\bigg(V'(t_j',\\{j\\})-V'(t_j,\\{j\\})\\bigg)\\\\\n&=V'(t_j',\\{j\\})-V'(t_j,\\{j\\})\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Bound the Language Function}\nFor all type $t$, the valuation when bidder tells the truth is\n\n\\begin{equation}\n\\begin{aligned}\n\\phi'(t,t)&=\\sum_{S\\subseteq I}\\pi_S(t)\\hat{v}(t,S)\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\sum_{S:j\\in S}\\pi_S(t)\\big(\\hat{v}(t,S\\backslash\\{j\\})+\\hat{v}(t,\\{j\\})\\big)+\\sum_{S:j\\not\\in S}\\pi_S(t)\\hat{v}(t,S)\\bigg)\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\pi_j(t)\\hat{v}(t,\\{j\\})+\\sum_{S\\subseteq I\\backslash\\{j\\}}\\pi_S(t)\\hat{v}(t,S)\\bigg)\\\\\n&\\leq \\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\pi_j(t)\\hat{v}(t,\\{j\\})+\\hat{v}(t,I\\backslash\\{j\\})\\bigg)\n\\end{aligned}\n\\end{equation}\n\nThe virtual valuation $\\Phi(t)$:\n\n\\begin{equation}\n\\begin{aligned}\n\\Phi(t)&=\\phi'(t,t)-\\frac{1}{f(t)}\\sum_{S\\subseteq I}\\pi_S(t)\\sum_{t'\\in \\Omega}\\lambda(t',t)(\\hat{v}(t',S)-\\hat{v}(t,S))\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\hat{v}(t,I\\backslash\\{j\\})+\\pi_i(t)\\big(\\hat{v}(t,\\{j\\})-\\sum_{t_j':V'(t_j',\\{j\\})>V'(t_j,\\{j\\})}\\frac{f_j(t_j')}{f_j(t_j)}\\big)\\bigg)\\\\\n&\\leq\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\hat{v}(t,I\\backslash\\{j\\})+\\pi_i(t)\\tilde{\\phi}_j(t_j)\\bigg)\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{\\phi}_j(t_j)$ is the Myerson Virtual Value for item j with the marginal distribution on $\\Omega_j$ and valuation $V'(t_j,\\{j\\})$.\n\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda,\\phi',p)&=\\sum_{t\\in \\Omega}f(t)\\Phi(t)\\\\\n&=\\bar\\delta'+\\sum_t\\sum_jf(t)\\pi_j(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\\\\\n&\\leq \\bar\\delta'+\\sum_t\\sum_jf(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\n\\begin{equation}\n\\sum_t\\sum_jf(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\\leq 6SREV\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nConsider the optimal revenue $OneRev$ when the mechanism is only allowed to sell one item. Assume a mechanism M allocate item j only if $t\\in R_j$. Then LHS is the virtual welfare of this mechanism, which is less than $OneRev$. Besides, this allocation rule is same as assuming bidder's valuation is unit-demand. Thus selling separately can reach a $\\frac{1}{6}$-approximation of this revenue.\n\\end{proof}\n\n~\\\\\n\n\\subsection{Conclusion}\nPut everything together we have\n\\begin{equation}\nREV(v)\\leq 2REV(\\hat{v})+2\\bar\\delta \\leq 2L(\\lambda,\\phi',p)+2\\bar\\delta \\leq 4\\bar\\delta'+12SREV \\leq 24BREV+49SREV\n\\end{equation}\n\n\n\n\n\\section{Duality}\\label{sec:duality}\n\n\n The focus of \\cite{CaiDW16} was on additive and unit-demand valuations and their respective dual was derived from an LP that is only meaningful for constrained additive valuations. In order to tackle general valuations, we need to apply the duality framework to an LP that is meaningful for general valuations. Instead of using the ``implicit forms'' LP from~\\cite{CaiDW13b, CaiDW16}, we choose a slightly different and more intuitive LP formulation (see Figure~\\ref{fig:LPRevenue}). For all bidders $i$ and types $t_i \\in T_i$, we use $p_i(t_i)$ as the interim price paid by bidder $i$ and $\\sigma_{iS}(t_i)$ as the interim probability of receiving the exact bundle $S$. To ease the notation, we use a special type $\\varnothing$ to represent the choice of not participating in the mechanism. More specifically, ${\\sigma}_{iS}(\\varnothing)=0$ for any $S$ and $p_{i}(\\varnothing)=0$. Now a Bayesian IR (BIR) constraint is simply another BIC constraint: for any type $t_{i}$, bidder $i$ will not want to lie to type $\\varnothing$. We let $T_{i}^{+}=T_{i}\\cup \\{\\varnothing\\}$.\n\nFollowing the recipe provided by~\\cite{CaiDW16}, we take the partial Lagrangian dual of the LP in Figure~\\ref{fig:LPRevenue} by lagrangifying the BIC constraints. Let $\\lambda_i(t_i,t_i')$ be the Lagrange multiplier associated with the BIC constraint that if bidder $i$'s true type is $t_i$ she will not prefer to lie to type $t_i'$\n (see Figure~\\ref{fig:Lagrangian} and Definition~\\ref{def:Lagrangian}). As shown in~\\cite{CaiDW16}, the dual solution has finite value if and only if the dual variables $\\lambda_i$ form a valid flow for every bidder $i$. The reason is that the payments $p_i(t_i)$ are unconstrained variables, therefore the corresponding coefficients must be $0$ in order for the dual solution to have finite value. It turns out when all these coefficients are $0$, the dual variables $\\lambda$ can be interpreted as a flow described in Lemma~\\ref{lem:useful dual}. We refer the readers to~\\cite{CaiDW16} for a complete proof. From now on, we only consider $\\lambda$ that corresponds to a flow.\n\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth} {\n\\noindent\\textbf{Variables:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $p_i(t_i)$, for all bidders $i$ and types $t_i \\in T_i$, denoting the expected price paid by bidder $i$ when reporting type $t_i$ over the randomness of the mechanism and the other bidders' types.\n\\item $\\sigma_{iS}(t_i)$, for all bidders $i$, all bundles of items $S\\subseteq[m]$, and types $t_i \\in T_i$, denoting the probability that bidder $i$ receives \\textbf{exactly} the bundle $S$ when reporting type $t_i$ over the randomness of the mechanism and the other bidders' types.\n\\end{itemize}\n\\textbf{Constraints:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\sum_{S\\subseteq[m]} {\\sigma}_{iS}(t_i) \\cdot v_i(t_i,S) - p_i(t_i) \\geq\\sum_{S\\subseteq[m]}{\\sigma}_{iS}(t'_i) \\cdot v_i(t_i, S) - p_i(t'_i) $, for all bidders $i$, and types $t_i \\in T_i, t'_i \\in T_i^+$, guaranteeing that the reduced form mechanism $({\\bf{\\sigma}},{p})$ is BIC and Bayesian IR.\n\\item ${\\bf{\\sigma}} \\in {P(D)}$, guaranteeing ${\\sigma}$ is feasible.\n\\end{itemize}\n\\textbf{Objective:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\displaystyle\\max \\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)$, the expected revenue.\\\\\n\\end{itemize}}\n\\end{minipage}}\n\\caption{A Linear Program (LP) for Revenue Optimization.}\n\\label{fig:LPRevenue}\n\\end{figure}\n\n\\begin{definition}\\label{def:Lagrangian}\nLet ${\\mathcal{L}}(\\lambda, \\sigma, p)$ be the partial Lagrangian defined as follows:\n\\begin{align*}\n& {\\mathcal{L}}(\\lambda, \\sigma, p)\\\\\\stepcounter{equation}\\tag{\\theequation} \\label{eq:primal lagrangian}\n=&\\sum_{i=1}^{n} \\left(\\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)+\\sum_{t_{i}\\in T_{i},t_{i}'\\in T_i^{+}} \\lambda_{i}(t_{i},t_{i}')\\cdot \\left(\\sum_{S\\subseteq[m]} v_i(t_{i},S)\\cdot\\left(\\sigma_{iS}(t_{i})-\\sigma_{iS}({t_{i}'})\\right)-\\left((p_{i}(t_{i})-p_{i}(t_{i}')\\right)\\right)\\right)\\\\\n=& \\sum_{i=1}^{n}\\left(\\sum_{t_{i}\\in T_{i}} p_{i}(t_{i})\\cdot\\left(f_{i}(t_{i})+\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})-\\sum_{t_{i}'\\in T_{i}^{+}} \\lambda_{i}(t_{i},t_{i}')\\right)\\right)\\\\\n&+\\sum_{i=1}^{n}\\left(\\sum_{t_{i}\\in T_{i}}\\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot \\left(v_i(t_{i},S)\\cdot \\sum_{t_{i}'\\in T_{i}^{+}}\\lambda_{i}(t_{i},t_{i}')-\\sum_{t'_i\\in T_{i}}\\left(v_i(t'_{i},S)\\cdot \\lambda_{i}(t_{i}',t_{i})\\right)\\right)\\right)~~ ({\\sigma}_i(\\varnothing)=\\textbf{0},\\ p_{i}(\\varnothing)=0)\\stepcounter{equation}\\tag{\\theequation} \\label{eq:dual lagrangian}\n\\end{align*}\n\\end{definition}\n\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth} {\n\\noindent\\textbf{Variables:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\lambda_i(t_{i},t_{i}')$ for all $i,t_{i}\\in T_{i},t_{i}' \\in T_i^{+}$, the Lagrangian multipliers for Bayesian IC and IR constraints.\n\\end{itemize}\n\\textbf{Constraints:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\lambda_i(t_{i},t_{i}')\\geq 0$ for all $i,t_{i}\\in T_{i},t_{i}' \\in T_i^{+}$, guaranteeing that the Lagrangian multipliers are non-negative.\n\\end{itemize}\n\\textbf{Objective:}\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $\\displaystyle\\min_{\\lambda}\\max_{\\sigma\\in {P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)$.\\\\\n\\end{itemize}}\n\\end{minipage}}\n\\caption{Partial Lagrangian of the Revenue Maximization LP.}\n\\label{fig:Lagrangian}\n\\end{figure}\n\n\n\\notshow{ \\begin{definition}[Useful Dual Variables~\\cite{CaiDW16}]\nA set of feasible duals $\\lambda$ is \\textbf{useful} if $\\max_{\\sigma\\in{P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)< \\infty$.\n\\end{definition}}\n\n\n\\begin{lemma}[Useful Dual Variables~\\cite{CaiDW16}]\\label{lem:useful dual}\nA set of feasible duals $\\lambda$ is \\textbf{useful} if $\\max_{\\sigma\\in{P(D)}, p} {\\mathcal{L}}(\\lambda, \\sigma, p)< \\infty$. $\\lambda$ is useful iff for each bidder $i$, $\\lambda_{i}$ forms a valid flow, i.e., iff the following satisfies flow conservation at all nodes except the source and the sink:\n\n \\noindent\\textbf{\\emph{1.}} Nodes: A super source $s$ and a super sink $\\varnothing$, along with a node $t_{i}$ for every type $t_{i}\\in T_{i}$.\n\n\\noindent \\textbf{\\emph{2.}} An edge from $s$ to $t_{i}$ with flow $f_i(t_{i})$, for all $t_{i}\\in T_{i}$.\n\n\\noindent \\textbf{\\emph{3.}} An edge from $t_i$ to $t_i'$ with flow $\\lambda_i(t_i,t_i')$ for all $t_i\\in T_i$, and $t_i'\\in T_{i}^{+}$ (including the sink).\n\n\\end{lemma}\n\n\\begin{definition}[Virtual Value Function]\\label{def:virtual value}\nFor each flow $\\lambda$, we define a corresponding virtual value function $\\Phi(\\cdot)$, such that for every bidder $i$, every type $t_{i}\\in T_{i}$ and every set $S\\subseteq[m]$,\n$$\\Phi_{i}(t_{i}, S)=v_i(t_{i},S)-{1\\over f_{i}(t_{i})}\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})\\left(v_i(t_{i}',S)-v_i(t_{i},S)\\right)$$.\n\\notshow{\n\\vspace{.1in}\\noindent$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\Phi_{i}(t_{i}, S)=v_i(t_{i},S)-$\n\n$\\hspace{1.3cm}{1\\over f_{i}(t_{i})}\\sum_{t_{i}'\\in T_{i}} \\lambda_{i}(t_{i}',t_{i})\\left(v_i(t_{i}',S)-v_i(t_{i},S)\\right).$\t\n}\n\\end{definition}\nThe proof of Theorem~\\ref{thm:revenue less than virtual welfare} is essentially the same as in~\\cite{CaiDW16}. We include it in Appendix~\\ref{sec:proof_duality} for completeness.\n\\begin{theorem}[Virtual Welfare $\\geq$ Revenue~\\cite{CaiDW16}]\\label{thm:revenue less than virtual welfare}\nFor any flow $\\lambda$ and any BIC mechanism $M=(\\sigma,p)$, the revenue of $M$ is $\\leq$ the virtual welfare of {$\\sigma$} w.r.t. the virtual valuation $\\Phi(\\cdot)$ corresponding to $\\lambda$.\n$$\\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)\\leq \\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i}) \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot\\Phi_{i}(t_{i},S)$$\n\n\\notshow{\n\\vspace{.05in} $\\sum_{i=1}^{n} \\sum_{t_i \\in T_i} f_{i}(t_{i})\\cdot p_i(t_i)\\leq $\n\n$\\hspace{1cm}\\sum_{i=1}^{n} \\sum_{t_{i}\\in T_{i}} f_{i}(t_{i}) \\sum_{S\\subseteq[m]}\\sigma_{iS}(t_{i})\\cdot\\Phi_{i}(t_{i},S)$\n}\nLet $\\lambda^{*}$ be the optimal dual variables and $M^{*}=(\\sigma^{*},p^{*})$ be the revenue optimal BIC mechanism, then the expected virtual welfare with respect to $\\Phi^{*}$ (induced by $\\lambda^{*}$) under $\\sigma^{*}$ equals to the expected revenue of $M^{*}$.\n\\end{theorem}\n\n\\section{Canonical Flow and Properties of the Virtual Valuations}\\label{sec:flow}\n\n\nIn this section, we present a canonical way of setting the dual variables\/flow that induces our benchmarks. A recap of the flow for additive valuations and the appealing properties of the corresponding virtual valuation functions can be found in Appendix~\\ref{sec:flow_additive}. We refer readers to that Section for more intuition about the flow.\n\nAlthough any flow can provide a finite upper bound of the optimal revenue, we focus on a particular class of flows, in which every flow $\\lambda^{(\\beta)}$ is parametrized by a set of parameters $\\beta=\\{\\beta_{ij}\\}_{i\\in[n],j\\in[m]}\\in\\R^{nm}_{\\geq 0}$. Based on $\\beta$, we partition the type set $T_i$ of each buyer $i$ into $m+1$ regions: \\textbf{(i)} $R_{0}^{(\\beta_i)}$ contains all types $t_i$ such that $V_i(t_{ij})<\\beta_{ij}$ for all $j\\in[m]$. \\textbf{(ii)} $R_j^{(\\beta_i)}$ contains all types $t_i$ such that $V_i(t_{ij})-\\beta_{ij}\\geq 0$ and $j$ is the smallest index in $\\argmax_k\\{V_i(t_{ik})-\\beta_{ik}\\}$. Intuitively, if we view $\\beta_{ij}$ as the price of item $j$ for bidder $i$, then $R^{(\\beta_i)}_0$ contains all types in $T_i$ that cannot afford any item, and any $R^{(\\beta_i)}_j$ with $j>0$ contains all types in $T_i$ whose ``favorite'' item is $j$. We first provide a {\\bf Partial Specification of the flow $\\lambda^{(\\beta)}$:}\n\n\\noindent\\textbf{1.} For every type $t_{i}$ in region $R^{(\\beta_{i})}_{0}$, the flow goes directly to $\\varnothing$ (the super sink).\n\n\\noindent \\textbf{2.} For all $j>0$, any flow entering $R^{(\\beta_{i})}_{j}$ is from $s$ (the super source) and any flow leaving $R^{(\\beta_{i})}_{j}$ is to $\\varnothing$.\n\n\\noindent \\textbf{3.} For all $t_{i}$ and $t_{i}'$ in $R^{(\\beta_{i})}_{j}$ ($j>0$), {$\\lambda^{(\\beta)}_{i}(t_{i},t_{i}')>0$} only if $t_{i}$ and $t_{i}'$ only differ in the $j$-th coordinate.\n\n\\notshow{\n\\begin{figure}\n \\centering{\\includegraphics[width=0.5\\linewidth]{multi_flow.png}}\n \\caption{An example of $\\lambda^{(\\beta)}_{i}$ for additive bidders with two items.}\n \\label{fig:multiflow}\n\\end{figure}\n}\n\nFor additive valuations and any type $t_i \\in R_j^{(\\beta_i)}$ , the contribution to the virtual value function $\\Phi(t_i,S)$ from any type $t_i'\\in R_j^{(\\beta_i)}$ is either $0$ if $j\\notin S$, or {$\\lambda_i^{(\\beta)}(t_i', t_i)(v_i(t_i',S)-v_i(t_i,S))=\\lambda_i^{(\\beta)}(t_i', t_i)(t_{ij}'-t_{ij})$} if $t_i$, $t_i'$ only differs on the $j$-th coordinate and $j\\in S$. In either case, the contribution does not depend on $t_{ik}$ for any $k\\neq j$. This is the key property that allows~\\cite{CaiDW16} to choose a flow such that the value of the favorite item is replaced by the corresponding Myerson's ironed virtual value in the virtual value function $\\Phi_i(t_i,\\cdot)$.\nUnfortunately, this property no longer holds for subadditive valuations. When $j\\in S$ and $\\lambda_i^{(\\beta)}(t_i',t_i)>0$, the contribution {$\\lambda_i^{(\\beta)}(t_i', t_i)(v_i(t_i',S)-v_i(t_i,S))$} heavily depends on $t_{ik}$ of all the other item $k\\in S$. All we can conclude is that the contribution lies in the range {$[-\\lambda_i^{(\\beta)}(t_i', t_i)\\cdot V_{i}(t_{ij}), \\lambda_i^{(\\beta)}(t_i', t_i)\\cdot V_{i}(t_{ij}')]$}\\footnote{$v_i(t,\\cdot)$ is subadditive and monotone for every type $t\\in T_i$, therefore $v_i(t_i,S)\\in[v_i(t_i, S\\backslash\\{j\\}),v_i(t_i, S\\backslash\\{j\\})+V_{i}(t_{ij})]$ and $v_i(t'_i,S)\\in[v_i(t'_i, S\\backslash\\{j\\}),v_i(t'_i, S\\backslash\\{j\\})+V_{i}(t'_{ij})]$.}, but this is not sufficient for us to convert the value of item $j$ into the corresponding Myerson's ironed virtual value.\n\n\\subsection{Valuation Relaxation}\\label{sec:valuation relaxation}\nThis is the first major barrier for extending the duality framework to accommodate subadditive valuations. We overcome it by considering a relaxation of the valuation functions. More specifically, for any $\\beta$, we construct another function $v_i^{(\\beta_i)}(\\cdot,\\cdot): T_i\\times 2^{[m]}\\mapsto {\\mathbb{R}_{\\geq 0}}$ for every buyer $i$ such that: (i) for any $t_i$, $v_i^{(\\beta_i)}(t_i,\\cdot)$ is subadditive and monotone, and for every bundle $S$ the new value $v_i^{(\\beta_i)}(t_i,S)$ is no smaller than the original value $v_i(t_i,S)$; (ii) for any BIC mechanism $M$ with respect to the original valuations, there exists another mechanism $M^{(\\beta)}$ that is BIC with respect to the new valuations and its revenue is comparable to the revenue of $M$; (iii) for the new valuations $v^{(\\beta)}$, there exists a flow whose induced virtual value functions have properties similar to those in the additive case.\nProperty (ii) implies that the optimal revenue with respect to $v^{(\\beta)}$ can serve as a proxy for the original optimal revenue. Moreover, due to Theorem~\\ref{thm:revenue less than virtual welfare}, the optimal revenue for $v^{(\\beta)}$ is upper bounded by the partial Lagrangian dual with respect to $v^{(\\beta)}$, which has an appealing format similar to the additive case by property (iii). Thus, we obtain a benchmark for subadditive bidders that resembles the benchmark for additive bidders in~\\cite{CaiDW16}\n\n\\begin{definition}[Relaxed Valuation]\\label{def:relaxed valuation}\n\tGiven $\\beta$, for any buyer $i$, define $v_i^{(\\beta_i)}(t_i,S)=v_i(t_i,S\\backslash\\{j\\})+V_i(t_{ij})$, if the ``favorite'' item is in $S$, i.e., $t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S$. Otherwise, define $v_i^{(\\beta_i)}(t_i,S)=v_i(t_i,S)$.\n\n\n\\begin{comment}\n\\begin{equation}\n\\begin{aligned}\nv_i^{(\\beta_i)}(t_i,S)=\n\\begin{cases}\nv_i(t_i,S\\backslash\\{j\\})+V_i(t_{ij}) &\\text{if the ``favorite'' item is in $S$, i.e., }t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S\\\\\nv_i(t_i,S) & \\text{o.w.}\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\\end{definition}\n\nIn the next Lemma, we show that for any BIC mechanism $M$ for $v$, there exists a BIC mechanism $M^{(\\beta)}$ for $v^{(\\beta)}$ such that its revenue is comparable to the revenue of $M$ (property (ii)). Moreover, the ex-ante probability for any buyer $i$ to receive any item $j$ in $M^{(\\beta)}$ is no greater than in $M$ (property (i)). We will see later that this is an important property for our analysis. The proof of Lemma~\\ref{lem:relaxed valuation} is similar to the $\\epsilon$-BIC to BIC reduction in~\\cite{HartlineKM11, BeiH11,DaskalakisW12} and can be found in Appendix~\\ref{sec:proof_relaxed_valuation}.\n\n\n\\begin{lemma}\\label{lem:relaxed valuation}\n\n\tFor any $\\beta$ and any BIC mechanism $M$ for subadditive valuation $\\{v_i(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, there exists a BIC mechanism $M^{(\\beta)}$ for valuations $\\{v_i^{(\\beta_i)}(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, such that\n\n \\vspace{.1in}\n \\noindent \\emph{\\textbf{(i)}} $\\displaystyle\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\n \\vspace{.1in}\n \\noindent \\emph{\\textbf{(ii)}} $\\displaystyle\\textsc{Rev}(M, v, D)\\leq2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}\\displaystyle+2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right).$\n\n\\vspace{0.05in}\n\\noindent$\\textsc{Rev}(M, v, D)$ (or $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$) is the revenue of the mechanism $M$ (or $M^{(\\beta)}$) while the buyers' types are drawn from $D$ and buyer $i$'s valuation is $v_i(t_i,\\cdot)$ (or $v_i^{(\\beta_i)}(t_i,\\cdot)$). $\\sigma_{iS}(t_i)$ (or $\\sigma^{(\\beta)}_{iS}(t_i)$) is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i$ in mechanism $M$ (or $M^{(\\beta)}$)\n\\end{lemma}\n\\notshow{\nFrom now on, we fix $M^{(\\beta)}$ to be the mechanism that is constructed by setting $\\eta$ to be $1\/2$ and $\\epsilon$ be a extremely tiny positive constant $\\epsilon_o$ in Lemma~\\ref{lem:relaxed valuation}.\n\\begin{corollary}\n\tFor any $\\beta$, there exists a mechanism $M^{(\\beta)}$ such that\n\t$$\\textsc{Rev}(M, v, D)\\leq 2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\epsilon_0.$$\n\\end{corollary}}\n\\subsection{Virtual Valuation for the Relaxed Valuation}\\label{sec:virtual for relaxed}\nFor any $\\beta$, based on the same partition of the type sets as in the beginning of Section~\\ref{sec:flow}, we construct a flow $\\lambda^{(\\beta)}$ that respects the partial specification, such that the corresponding virtual valuation function for $v^{(\\beta)}$ has the same appealing properties as in the additive case.\nFor the relaxed valuation, as {$\\lambda_i^{(\\beta)}(t_i, t_i')$} is only positive for types $t_i$, $t_i'\\in R_j^{(\\beta_i)}$ that only differ in the $j$-th coordinate, the contribution from item $j$ to the virtual valuation solely depends on $t_{ij}$ and $t'_{ij}$ but not $t_{ik}$ for any other item $k\\in S$\n. Notice that this property does not hold for the original valuation, and it is the main reason why we choose the relaxed valuation as in Definition~\\ref{def:relaxed valuation}. Moreover, we can choose $\\lambda_i^{(\\beta)}$ carefully so that the virtual valuation of $v^{(\\beta)}$ has the following format:\n\n\n\n\n\\begin{lemma}\\label{lem:subadditive flow properties}\n\tLet $F_{ij}$ be the distribution of $V_i(t_{ij})$ when $t_{ij}$ is drawn from $D_{ij}$. For any $\\beta$, there exists a flow $\\lambda^{(\\beta)}_i$ such that the corresponding virtual value function $\\Phi^{(\\beta_i)}_{i}(t_{i}, \\cdot)$ of valuation $v_i^{(\\beta_i)}(t_i,\\cdot)$ satisfies the following properties:\n\n\\vspace{.05in}\t\n\\noindent 1. For any $t_{i}\\in R^{(\\beta_i)}_{0}$, $\\Phi^{(\\beta_i)}_{i}(t_{i},S) = v_i(t_i, S)$.\n\n\\vspace{.05in}\n\\noindent 2. For any $j>0$, $t_{i}\\in R^{(\\beta_i)}_{j}$, $\\Phi_{i}^{(\\beta_i)}(t_{i},S)\\leq v_i (t_{i}, S)\\cdot\\mathds{1}[j\\notin S]+\\left(v_i (t_{i}, S\\backslash\\{j\\})+{\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))\\right)\\cdot\\mathds{1}[j\\in S],$ where ${\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$ is the Myerson's ironed virtual value for $V_i(t_{ij})$ with respect to $F_{ij}$.\n\\end{lemma}\n\nThe proof of Lemma \\ref{lem:subadditive flow properties} is postponed to Appendix~\\ref{sec:proof_virtual_relaxation}.\nNext, we use the virtual welfare of the allocation $\\sigma^{(\\beta)}$ to bound the revenue of $M^{(\\beta)}$.\n\n\\begin{lemma}\\label{lem:upper bound the revenue of the relaxed mechanism}\n\tFor any $\\beta$, \\begin{align*} &\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\leq \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot\\Phi^{(\\beta_i)}_i(t_i,S)\\\\\n \\leq &\t\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S)\\\\\n &+ \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in[m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\backslash\\{j\\})+\\sum_{S:j\\notin S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S)\\right)\\\\\n &+ \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in[m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\pi^{(\\beta)}_{ij}(t_i)\\cdot {\\tilde{\\varphi}}_{ij}(t_{ij}),\\end{align*}\n where $ \\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$. {\\bf \\textsc{Non-Favorite}$(M, \\beta)$} denotes the sum of the first two terms. {\\bf \\textsc{Single}$(M, \\beta)$} denotes the last term. \\end{lemma}\n\n\\begin{proof}\nThe Lemma follows easily from the properties in Lemma~\\ref{lem:subadditive flow properties} and Theorem~\\ref{thm:revenue less than virtual welfare}.\n\\end{proof}\n\nWe obtain Theorem~\\ref{thm:revenue upperbound for subadditive} by combining Lemma~\\ref{lem:relaxed valuation} and~\\ref{lem:upper bound the revenue of the relaxed mechanism}.\n \\begin{theorem}\\label{thm:revenue upperbound for subadditive}\nFor any mechanism $M$ and any $\\beta$,\n$$\\textsc{Rev}{(M,v,D)}\\leq 4\\cdot\\textsc{Non-Favorite}(M, \\beta)+2\\cdot\\textsc{Single}(M,\\beta).$$\n\\end{theorem}\n\n\\begin{prevproof}{Theorem}{thm:revenue upperbound for subadditive}\nFirst, let's look at the value of $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)$. When $t_i\\in R_j^{(\\beta_i)}$ for some $j>0$ and $j\\in S$, $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)= v_i(t_i, S\\backslash\\{j\\})+V_i(t_{ij})-v_i(t_i,S)\\leq v_i(t_i, S\\backslash\\{j\\}),$ because $V_i(t_{ij})\\leq v_i(t_i,S)$. For the other cases, $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)=0$. Therefore,\n\\begin{align*}\\label{eq:bounding delta}\n\t&\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\nonumber\\\\\n\\leq &\\sum_i\\sum_{t_i} f_i(t_i)\\sum_{j} \\mathds{1}[t_i\\in R_j^{(\\beta_i)}]\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\cdot v_i(t_i, S\\backslash\\{j\\})\\nonumber\\\\\n\\leq &\\textsc{Non-Favorite}(M,\\beta)~~~~~~~~~\\text{(Definition of $\\textsc{Non-Favorite}(M,\\beta)$)}\n\\end{align*}\n\nOur statement follows from combining Lemma~\\ref{lem:relaxed valuation}, Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism} with the inequality above.\n\\end{prevproof}\n\\begin{comment}\nNow we bound $\\textsc{Rev}(M, v, D)$. By Lemma~\\ref{lem:relaxed valuation},\n\t\\begin{align*}&\\textsc{Rev}(M, v, D)\\\\\n\t\\leq& 2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}\\\\\n+&2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\\\\t\n\\leq &2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+2\\cdot\\textsc{Non-Favorite}(M,\\beta)~~~~\\text{(Equation~(\\ref{eq:bounding delta}))}\\\\\n\\leq & 4\\cdot\\textsc{Non-Favorite}(M,\\beta)+2\\cdot\\textsc{Single}(M,\\beta)~~~~\\text{(Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism})}.\n\t\\end{align*}\n\\end{comment}\n\n\n\n\\subsection{Upper Bound for the Revenue of Subadditive Buyers}~\\label{sec:choice of beta}\n In Section~\\ref{sec:valuation relaxation}, we have argued that for any $\\beta$, there exists a mechanism $M^{(\\beta)}$ such that its revenue with respect to the relaxed valuation $v^{(\\beta)}$ is comparable to the revenue of $M$ with respect to the original valuation. In Section~\\ref{sec:virtual for relaxed}, we have shown for any $\\beta$ how to choose a flow to obtain an upper bound for $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)$ and also an upper bound for $\\textsc{Rev}(M,v,D)$. Now we specify our choice of $\\beta$.\n\nIn~\\cite{CaiDW16}, the authors fixed a particular $\\beta$, and shown that under any allocation rule, the corresponding benchmark can be bounded by the sum of the revenue of a few simple mechanisms. However, for valuations beyond additive and unit-demand, the benchmark becomes much more challenging to analyze\\footnote{Indeed, the difficulties already arise for valuations as simple as $k$-demand. A bidder's valuation is $k$-demand if her valuation is additive subject to a uniform matroid with rank $k$.}. We adopt an alternative and more flexible approach to obtain a new upper bound. Instead of fixing a single $\\beta$ for all mechanisms, we customize a different $\\beta$ for every different mechanism $M$. Next, we relax the valuation and design the flow based on the chosen $\\beta$ as specified in Section~\\ref{sec:valuation relaxation} and \\ref{sec:virtual for relaxed}.\n Then we upper bound the revenue of $M$ with the benchmark in Theorem~\\ref{thm:revenue upperbound for subadditive} and argue that for any mechanism $M$, the corresponding benchmark can be upper bounded by the sum of the revenue of a few simple mechanisms. As we allow $\\beta$, in other words the flow $\\lambda^{(\\beta)}$, to depend on the mechanism, our new approach may provide a better upper bound. As it turns out, our new upper bound is indeed easier to analyze.\n\n Lemma~\\ref{lem:requirement for beta} specifies the two properties of our $\\beta$ that play the most crucial roles in our analysis. We construct such a $\\beta$ in the proof of Lemma~\\ref{lem:requirement for beta}, however the construction is not necessarily unique and any $\\beta$ satisfying these two properties suffices. Note that our construction heavily relies on property \\textbf{(i)} of Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:requirement for beta}\n\tFor any constant $b\\in (0,1)$ and any mechanism $M$, there exists a $\\beta$ such that: for the mechanism $M^{(\\beta)} $ constructed in Lemma~\\ref{lem:relaxed valuation} according to $\\beta$, any $i\\in[n]$ and $j\\in[m]$,\n\n\\noindent\\emph{\\textbf{(i)}} $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]\\leq b$;\n\n\\noindent\\emph{\\textbf{(ii)}} $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\leq \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\/ b$, where $\\pi_{ij}^{(\\beta)}(t_i) = \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)$.\n\\end{lemma}\n\nBefore proving Lemma~\\ref{lem:requirement for beta}, we provide some intuition behind the two required properties.\nProperty \\textbf{(i)} is used to guarantee that if item $j$'s price for bidder $i$ is higher than $\\beta_{ij}$ for all $i$ and $j$ in an RSPM, for any item $j'$ and any bidder $i'$, $j'$ is still available with probability at least $(1-b)$ when $i'$ is visited. As for any bidder $k\\neq i'$ to purchase item $j'$, $V_k(t_{kj'})$ must be greater than her price for item $j'$. By the union bound, the probability that there exists such a bidder is upper bounded by the LHS of property (i), and therefore is at most $b$. With this guarantee, we can easily show that the RSPM achieves good revenue (Lemma~\\ref{lem:neprev}). Property \\textbf{(ii)} states that the ex-ante probability for bidder $i$ to receive an item $j$ in $M^{(\\beta)}$ is not much bigger than the probability that bidder $i$'s value is larger than item $j$. This is crucial for proving our key Lemma~\\ref{lem:hat Q}, in which we argue that two different valuations provide comparable welfare under the same allocation rule $\\sigma^{(\\beta)}$. With Lemma~\\ref{lem:hat Q}, we can show that the ASPE obtains good revenue.\n\n\\begin{prevproof}{Lemma}{lem:requirement for beta}\n\tWhen there is only one buyer, we can simply set every $\\beta_j$ to be $0$ and both conditions are satisfied.\n\tWhen there are multiple players, we let $$\\beta_{ij}:=\\inf\\{{x\\geq 0}: \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x\\right] \\leq b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\},$$ where $ \\pi_{ij}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}(t_i)$. Clearly, when the distribution of $V_i(t_{ij})$ is continuous, then\n\\begin{equation}\\label{equ:beta_second_condition}\n\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i),\n\\end{equation}\nand therefore for any $j$, $$\\sum_i\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=b\\cdot\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\leq b.$$\n\nSo the first condition is satisfied. The second condition holds because by the first property in Lemma~\\ref{lem:relaxed valuation}, $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)$.\n\nWhen the distribution for $V_i(t_{ij})$ is discrete, it is possible that Equation~\\ref{equ:beta_second_condition} does not hold, but this is essentially a tie breaking issue and not hard to fix. Let $\\epsilon>0$ be an extremely small constant that is smaller than $\\left|V_i(t_{ij})-V_i(t'_{ij})\\right|$ for any $t_{ij}, t'_{ij}\\in T_{ij}$, any $i$ and any $j$. Let $\\zeta_{ij}$ be a random variable uniformly distributed on $[0,\\epsilon]$, and think of it as a random rebate that the seller gives to bidder $i$ when she purchases item $j$. Now we modify the definition of $\\beta_{ij}$ as $\\beta_{ij}:=\\inf\\{{x\\geq 0}: \\Pr_{t_{ij},\\zeta_{ij}}[V_i(t_{ij})+\\zeta_{ij}\\geq x] \\leq b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\}.$\n\n\\notshow{\\begin{equation}\n\\epsilon_1=\\epsilon\\cdot \\frac{\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]-b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)}{\\Pr_{t_{ij}}\\left[V_i(t_{ij})= \\beta_{ij}\\right]}\n\\end{equation}\n\nBy the definition of $\\beta_{ij}$, $\\epsilon_1\\in [0,\\epsilon)$. Let $\\zeta_{ij}$ be a random variable uniformly distributed on $[\\epsilon_1-\\epsilon,\\epsilon_1]$. It is not hard to verify that $\\Pr_{t_{ij},\\zeta_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+\\zeta_{ij}\\right]=b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)$. For those probabilities related to $\\beta_{ij}$ shown in the proofs below, which is written for simplification, we refer to this definition. With regard to this definition, we make essentially small changes for the mechanism described below. Instead of using fixed price, add a small disturbance $\\zeta_{ij}$ on item $j$'s price for bidder $i$. Since $\\epsilon$ can be chosen as small as possible, the revenue will only be affected by a small constant. All the argument maintain to be true.}\n\nBoth of the two properties in Lemma~\\ref{lem:requirement for beta} hold if we replace $V_i(t_{ij})$ with $V_i(t_{ij})+\\zeta_{ij}$. The only change we need to make in the mechanism is to actually give the bidders $\\zeta_{ij}$ as the corresponding rebate. Since we can choose $\\epsilon$ to be arbitrarily small, the sum of the rebate is also arbitrarily small. For the simplicity of the presentation, we will omit $\\zeta_{ij}$ and $\\epsilon$ in the rest of the paper.\nThe random rebate indeed makes our mechanism randomized(according to the random variable $\\zeta_{ij}\\sim [0,\\epsilon]$). However, the randomized mechanism is a uniform distribution of deterministic DSIC mechanisms (after determining all $\\zeta_{ij}$), and the expected revenue of the randomized mechanism is simply the average revenue of all these deterministic mechanisms. Therefore, there must be one realization of the rebates such that the corresponding deterministic mechanism has revenue above the expectation, i.e., the expected revenue of the randomized one. Thus if the randomized mechanism is proved to achieve some approximation ratio, there must exist a deterministic one that achieves the same ratio. The deterministic mechanism will use a fixed value $z_{ij}\\in [0,\\epsilon]$ as the rebate.\n\nSimilarly, the same issue about discrete distributions arises when we define some other crucial parameters later, e.g., in the Definition of $c$, $c_i$ and $\\tau_i$. We can resolve all of them together using the trick (adding a random rebate) described above, and we will not include a detailed proof for those cases.\n\\end{prevproof}\n\n\n\n\n\\subsubsection{Bad Example for Chawla and Miller's Approach}\\label{sec:cs_ocrs}\nLet bidders be constrained additive and $\\mathcal{F}_i$ be bidder $i$ feasibility constraint. We use $P_{\\mathcal{F}_i}=conv(\\{1^S | S\\in \\mathcal{F}_i\\})$ to denote the feasibility polytope of bidder $i$. Let $\\{q_{ij}\\}_{i\\in[n],j\\in[m]}$ be a collection of probabilities that satisfy $\\sum_i q_{ij}\\leq 1\/2$ for all item $j$ and $\\boldsymbol{q}_i = (q_{i1},\\ldots, q_{im})\\in b\\cdot P_{\\mathcal{F}_i}$. Let $\\beta_{ij}=F_{ij}^{-1}(q_{ij})$. The analysis by Chawla and Miller~\\cite{ChawlaM16} needs to upper bound $\\sum_{i, j}\\beta_{ij}\\cdot q_{ij}$ using the revenue of some BIC mechanism. When $\\mathcal{F}_i$ is a matroid for every bidder $i$, this expression can be upper bounded by the revenue of a sequential posted price mechanism constructed using OCRS from~\\cite{FeldmanSZ16}. Here we show that if the bidders have general downward closed feasibility constraints, this expression is gigantic. More specifically, we prove that even when there is only one bidder, the expression could be $\\Omega\\left(\\frac{\\sqrt{m}}{\\log m}\\right)$ times larger than the optimal social welfare.\n\nConsider the following example.\n\\begin{example}\\label{ex:counterexample ocrs}\n\tThe seller is selling $m=k^2$ items to a single bidder. The bidder's value for each item is drawn i.i.d. from distribution $F$, which is the equal revenue distribution truncated at $k$, i.e.,\n\t\\[F(x)=\n\\begin{cases}\n1-\\frac{1}{x},&\\text{if}~~x0$, there exists $q\\in b\\cdot P_{\\mathcal{F}}$ such that for sufficiently large $k$, $$\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=\\Theta(\\frac{k}{\\log k })\\cdot SW$$.\n\\end{lemma}\n\n\\begin{proof}\n\nFor any $b>0$, consider the following feasible allocation rule: w.p. $(1-b)$, don't allocate anything, and w.p. $b$, give the buyer one of the sets $A_i$ uniformly at random. The corresponding ex-ante probability vector $q$ satisfies $q_j=\\frac{b}{k}, \\forall j\\in [m]$. Thus $q\\in b\\cdot P_{\\mathcal{F}}$. Since $q_j<\\frac{1}{k}$, $F^{-1}(1-q_j)=k$ for all $j\\in [m]$. We have $\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=k^2\\cdot \\frac{b}{k}\\cdot k=b\\cdot k^2$. We use $V_i$ to denote the random variable of the bidder's value for set $A_i$. It is not hard to see that $SW={\\mathbb{E}}[\\max_{i\\in[k]} V_i]$. \n\n\\begin{lemma}\nFor any $i\\in [k]$,\n\\[\\Pr\\left[V_i>3\\cdot k\\log(k)\\right]\\leq k^{-3}\\]\n\\end{lemma}\n\\begin{proof}\nLet $X$ be random variable with cdf $F$. Notice $E[X]=\\log(k)$, $E[X^2]=2k$, and $|X|\\leq k$.\nFor every $i$, by the Bernstein concentration inequality, for any $t>0$,\n\\[\\Pr\\left[V_i-k\\log(k)>t\\right]\\leq \\exp\\left(-\\frac{\\frac{1}{2}t^2}{2k^2+\\frac{1}{3}kt}\\right)\\]\nChoose $t=2k\\log(k)$, we have\n\\[\\Pr\\left[V_i>3k\\log(k)\\right]\\leq \\exp(-3\\log(k))=k^{-3}\\]\n\\end{proof}\n\nBy the union bound, $\\Pr[\\max_{i\\in[k]}V_i>3\\cdot k\\log(k)]\\leq k^{-2}$. Therefore, ${\\mathbb{E}}[\\max_{i\\in[k]} V_i]\\leq 3 k\\log k +k^2\\cdot k^{-2}\\leq 4 k\\log k$.\n\\end{proof}\n\n\\notshow{In the analysis of the paper by Chawla and Miller~\\cite{ChawlaM16}, they rely on the following lemma in single buyer auction.\n\\begin{lemma}\\label{lem:shuchi}\n~\\cite{ChawlaM16}~\\cite{FeldmanSZ16}Suppose the buyer is additive within a matroid constraint $\\mathcal{F}$ and let $P_{\\mathcal{F}}=conv(\\{1^S | S\\in \\mathcal{F}\\})$ be the feasibility polytope. For any constant $b\\in (0,1)$ and ex-ante vector $q\\in bP_{\\mathcal{F}}$, let $\\beta$ be the corresponding ex-ante prices. In other words, $\\beta_j$ is chosen such that the probability that the value for item $j$\nexceeds this price is precisely $q_j$. Then the value $\\beta\\cdot q$ can be bounded within some constant factor by the revenue of a posted price mechanism with a more strict constraint, which guarantees that the ex-ante probability of the buyer getting item $j$ is at most $q_j$.\n\\end{lemma}\n\nThe result can be generated for $\\mathcal{F}$ beyond a matroid~\\cite{FeldmanSZ16}. However, Lemma~\\ref{lem:shuchi} does not hold for general downward-close $\\mathcal{F}$. In this section we provide a counterexample with some general downward-close constraint $\\mathcal{F}$, such that the term $\\beta\\cdot q$ cannot be upper bounded by any single buyer mechanism, within in a constant factor.\n\n\\begin{lemma}\nConsider the following single buyer auction with $m=k^2$ i.i.d. items. Items are divided into $k$ disjoint sets $A_1,...,A_k$, each with size $k$. The value distribution $F$ for a single item is defined as the equal-revenue distribution truncated at value $k$, i.e.,\n\\[F(x)=\n\\begin{cases}\n1-\\frac{1}{x},&\\text{if}~~x0$, there exists $q\\in bP_{\\mathcal{F}}$ such that for sufficiently large $k$,\n\\begin{equation}\\label{equ:q times beta}\n\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=\\Theta(\\frac{k}{log(k)})\\cdot SW\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\nFor any $b>0$, consider the following feasible allocation rule. With probability $(1-b)$, don't allocate anything. With probability $b$, give the buyer one of the sets $A_p$ uniformly at random. The corresponding ex-ante probability vector $q$ satisfies $q_j=\\frac{b}{k}, \\forall j\\in [m]$. Thus $q\\in bP_{\\mathcal{F}}$.\n\nSince $q_j<\\frac{1}{k}$, $F^{-1}(1-q_j)=k$ for all $j\\in [m]$. We have\n\\begin{equation}\n\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=k^2\\cdot \\frac{b}{k}\\cdot k=b\\cdot k^2\n\\end{equation}\n\nConsider the optimal social welfare. For every bundle $p$, denote $V_p$ the random variable of the buyer's value for bundle $p$. Notice $A_p\\in \\mathcal{F}$, $V_p$ is the sum of $k$ independent random variables with cdf $F$. With Bernstein Inequality, $V_p=O(klog(k))$ with high probability.\n\\notshow{\\begin{lemma}\\label{lem:bernstein}\n(Bernstein Inequality)~\\cite{bernstein1924modification}: Suppose $X_1,...,X_n$ are independent random variables with zero mean, and $|X_i|\\leq M$ almost surely for all $i$. Then for any $t>0$,\n\\[\\Pr\\left[\\sum_{i=1}X_i>t\\right]\\leq exp\\left(-\\frac{\\frac{1}{2}t^2}{\\sum_{i=1}^nE[X_i^2]+\\frac{1}{3}Mt}\\right)\\]\n\\end{lemma}}\n\\begin{lemma}\nFor any $p\\in [k]$,\n\\[\\Pr\\left[V_p>3\\cdot k\\log(k)\\right]\\leq k^{-3}\\]\n\\end{lemma}\n\\begin{proof}\nLet $X$ be random variable with cdf $F$. Notice $E[X]=\\log(k)$, $E[X^2]=k-1$, and $|X|\\leq k$.\nFor every $p$, by the Bernstein concentration inequality, for any $t>0$,\n\\[\\Pr\\left[V_p-k\\log(k)>t\\right]\\leq exp\\left(-\\frac{\\frac{1}{2}t^2}{k^2+\\frac{1}{3}kt}\\right)\\]\nChoose $t=2k\\log(k)$, we have\n\\[\\Pr\\left[V_p>3k\\log(k)\\right]\\leq exp(-3\\log(k))=k^{-3}\\]\n\\end{proof}\n\nWith union bound,\n\\[\\Pr[\\max_{p\\in [k]}V_p\\geq 3k\\log(k)]\\leq \\sum_{p\\in [k]}\\Pr\\left[V_p>3k\\log(k)\\right]\\leq k^{-2}\\]\n\nNotice the social welfare for the mechanism is at most \\\\\n\\noindent$\\max_{p\\in [k]}V_p$ due to the feasible constraint $\\mathcal{F}$. Also notice that $\\max_{p\\in [k]}V_p\\leq k^2$, we have\n\\begin{align*}\nSW&\\leq E[\\max_{p\\in [k]}V_p]\\leq \\bigg(3k\\log(k)\\cdot \\Pr[\\max_{p\\in [k]}V_p\\leq 3k\\log(k)]\\\\\n&+k^2\\cdot\\Pr[\\max_{p\\in [k]}V_p> 3k\\log(k)]\\bigg)\\\\\n&\\leq 3k\\log(k)+k^2\\cdot k^{-2}=O(k\\log(k))\n\\end{align*}\n\n\\noindent When $k$ is sufficiently large, Equation~\\ref{equ:q times beta} holds.\n\n\\begin{comment}\nNow consider the optimal revenue in this auction. Let $REV_k$ be the optimal revenue selling the items in set $A_k$ to a single buyer. \\ref{HartN12} has shown that $REV_k=\\Theta(n\\log(n))$ for the equal-revenue distribution $F(x)$. Since people are only interested one set of items,\n\\begin{equation}\nREV\\leq \\sum_{k\\in [n]}REV_k=\\Theta(n^2\\log(n))\n\\end{equation}\n\\end{comment}\n\n\\end{proof}}\n\n\n\n\\section{Introduction}\nIn Mechanism Design, we aim to design a mechanism\/system such that a group of strategic participants, who are only interested in optimizing their own utilities, are incentivized to choose actions that also help achieve the designer's objective. Clearly, the quality of the solution with respect to the designer's objective is crucial. However, perhaps one should also pay equal attention to another criterion of a mechanism, that is, its simplicity. When facing a complicated mechanism, participants may be confused by the rules and thus unable to optimize their actions and react in unpredictable ways instead. This may lead to undesirable outcomes and poor performance of the mechanism. An ideal mechanism would be optimal and simple. However, such cases of simple mechanisms being optimal only exist in single-item auctions, with the seminal examples of auctions by Vickrey~\\cite{Vickrey61} and Myerson~\\cite{Myerson81}, while none has been discovered in broader settings. Indeed, we now know that even in fairly simple settings the optimal mechanisms suffer many undesirable properties including randomization, non-monotonicity, and others~\\cite{RochetC98, Tha04, Pavlov11a, HartN13, HartR12, BriestCKW10, DaskalakisDT13, DaskalakisDT14}.\nTo move forward, one has to compromise -- either settle with optimal but somewhat complex mechanisms or turn to simple but approximately optimal solutions.\n\nRecently, there has been extensive research effort focusing on the latter approach, that is, studying the performance of simple mechanisms through the lens of approximation. In particular, a central problem on this front is how to design simple and approximately revenue-optimal mechanisms in multi-item settings. For instance, when bidders have unit-demand valuations, we know sequential posted price mechanisms approximates the optimal revenue due to a line of work initiated by Chawla et al.~\\cite{ChawlaHK07, ChawlaHMS10, ChawlaMS15, CaiDW16}. When buyers have additive valuations, we know that either selling the items separately or running a VCG mechanism with per bidder entry fee approximates the optimal revenue due to a series of work initiated by Hart and Nisan~\\cite{HartN12, CaiH13, LiY13, BabaioffILW14, Yao15, CaiDW16}. Recently, Chawla and Miller~\\cite{ChawlaM16} generalized the two lines of work described above to matroid rank functions\\footnote{{Here is a hierarchy of the valuation functions. additive \\& unit-demand $\\subseteq$ matroid rank $\\subseteq$ constrained additive \\& submodular\n $\\subseteq$ XOS $\\subseteq$ subadditive. A function is constrained additive if it is additive up to some downward closed feasibility constraints. The class of submodular functions is neither a superset nor a subset of the class of constrained additive functions.} See Definition~\\ref{def:valuation classes} for the formal definition. }. They show that a simple mechanism, the sequential two-part tariff mechanism, suffices to extract a constant fraction of the optimal revenue. For subadditive valuations beyond matroid rank functions, we only know how to handle a single buyer~\\cite{RubinsteinW15}\\footnote{All results mentioned above assume that the buyers' valuation distributions are over independent items. For additive and unit-demand valuations, this means a bidder's values for the items are independent. The definition is generalized to subadditive valuations by Rubinstein and Weinberg~\\cite{RubinsteinW15}. See Definition~\\ref{def:subadditive independent}.}. It is a major open problem to extend this result to multiple subadditive buyers.\n\nIn this paper, we unify and strengthen all the results mentioned above via an extension of the duality framework proposed by Cai et al.~\\cite{CaiDW16}. Moreover, we show that even when there are multiple buyers with XOS valuation functions, there exists a simple, deterministic and Dominant Strategy Incentive Compatible (DSIC) mechanism that achieves a constant fraction of the optimal Bayesian Incentive Compatible (BIC) revenue\\footnote{A mechanism is Bayesian Incentive Compatible (BIC) if it is in every bidder's interest to tell the truth, assuming that all other bidders' reported their values. A mechanism is Dominant Strategy Incentive Compatible (DSIC) if it is in every bidder's interest to tell the truth no matter what reports the other bidders make.}. For subadditive valuations, our approximation ratio degrades to $O(\\log m)$.\n\n\\begin{informaltheorem}\n\tThere exists a simple, deterministic and DSIC mechanism that achieves a constant fraction of the optimal BIC revenue in multi-item settings, when the buyers' valuation distributions are XOS over independent items. When the buyers' valuation distributions are subadditive over independent items, our mechanism achieves at least $\\Omega(\\frac{1}{\\log m})$ of the optimal BIC revenue, where $m$ is the number of items.\n\\end{informaltheorem}\n\nThe original paper by Cai et al.~\\cite{CaiDW16} provided a unified treatment for additive and unit-demand valuations. However, it is inadequate to provide an analyzable benchmark for even a single subadditive bidder. In this paper, we show how to extend their duality framework to accommodate general subadditive valuations. Using this extended framework, we substantially improve the approximation ratios for many of the settings discussed above, and in the meantime generalize the results to broader cases. See Table~\\ref{table:comp} for the comparison between the best ratios reported in the literature and the new ratios obtained in this work.\n\n\\begin{table*}\n\n\\centering\n\\begin{tabular}{|c|l|p{2.1cm}|c|p{2cm}|c|c|}\n\\hline\n\t& &\\centering Additive or Unit-demand& \\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{2.2cm}{\\centering Matroid-Rank}}}& \\centering Constrained Additive&\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1cm}{\\centering XOS}}}&\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{2cm}{\\centering Subadditive}}} \\\\\n\\hline\n\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1.3cm}{\\centering Single Buyer}}} & Previous& 6~\\cite{BabaioffILW14} or 4~\\cite{ChawlaMS15}& 31.1* &\\centering 31.1~\\cite{ChawlaM16} & 338* & 338~\\cite{RubinsteinW15} \\\\\\cline{2-7}\n\t\t&This Paper & \\centering - &{11*} &\\centering{11}&{40*}& {40} \\\\\\cline{2-7}\n\n\\hline\\hline\n\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1.3cm}{\\centering Multiple Buyer}}} & Previous& 8~\\cite{CaiDW16} or 24~\\cite{CaiDW16}& 133~\\cite{ChawlaM16}&\\centering ? & ? &? \\\\ \\cline{2-7}\n\t\t& This Paper& \\centering - & 70* &\\centering 70 & 268 &$O(\\log m)$ \\\\\n\\hline\n\\end{tabular}\\\\\n* The result is implied by another result for a more general setting.\n\\caption{Comparison of approximation ratios between previous and current work.}\n\t\t \\label{table:comp}\n\\end{table*}\n\n\\notshow{\n\\begin{table}\n\t\\begin{tabular}{|l|c|c|c|c|c|}\n\t\\hline\n\t\t & Additive or Unit-demand& Matroid-Rank& Constrained Additive&XOS&Subadditive \\\\\n\t\t\\hline\n\t\tSingle Buyer& 6~\\cite{BabaioffILW14} or 4~\\cite{ChawlaMS15}& 33.1~\\cite{ChawlaM16} &$\\rightarrow$ &$\\rightarrow$ & 338~\\cite{RubinsteinW15} \\\\\\hline\n\t\tMultiple Buyers& 8~\\cite{CaiDW16} or 24~\\cite{CaiDW16}& 133~\\cite{ChawlaM16}& ? & ? &? \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t \\caption{The best approximation ratios known prior to this work.}\n\t\t \\label{table:old}\n\\end{table}\n\n\\begin{table}\\centering\n\t\\begin{tabular}{|l|c|c|c|c|}\n\t\\hline\n\t\t &Matroid-Rank& Constrained Additive&XOS&Subadditive \\\\\n\t\t\\hline\n\t\tSingle Buyer& $\\rightarrow$ &14& $\\rightarrow$& 48 \\\\\\hline\n\t\tMultiple Buyers& $\\rightarrow$ & 70 & 268 &$O(\\log m)$ \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t \\caption{New approximation ratios obtained in this work.}\n\t\t \\label{table:new}\n\\end{table}\n\n}\nOur mechanism is either a \\emph{rationed sequential posted price mechanism} (\\textbf{RSPM}) or an \\emph{anonymous sequential posted price with entry fee mechanism} (\\textbf{ASPE}). In an RSPM, there is a price $p_{ij}$ for buyer $i$ if she wants to buy item $j$, and she is allowed to purchase at most one item. We visit the buyers in some arbitrary order and the buyer takes her favorite item among the available items given the item prices for her. Here we allow personalized prices, that is, $p_{ij}$ could be different from $p_{kj}$ if $i\\neq k$. In an ASPE, every buyer faces the same collection of item prices $\\{p_j\\}_{j\\in[m]}$. Again, we visit the buyers in some arbitrary order. For each buyer, we show her the available items and the associated price for each item. Then we ask her to pay the entry fee to enter the mechanism, which may depend on what items are still available and the identity of the buyer. If the buyer accepts the entry fee, she can proceed to purchase any item at the given prices; if she rejects the entry fee, then she will leave the mechanism without receiving anything. Given the entry fee and item prices, the decision making for the buyer is straightforward, as she only accepts the entry fee when the surplus for winning her favorite bundle is larger than the entry fee. Therefore, both RSPM and ASPE are DSIC and ex-post Individually Rational (ex-post IR).\n\n\n\\subsection{Our Contributions}\nTo obtain the new generalizations, we provide important extensions to the duality framework in~\\cite{CaiDW16}, as well as novel analytic techniques and new simple mechanisms.\n\n\\vspace{.05in}\n\\noindent \\textbf{1. Accommodating subadditive valuations:} the original duality framework in~\\cite{CaiDW16} already unified the additive case and unit-demand case by providing an approximately tight upper bound for the optimal revenue using a single dual solution. A trivial upper bound for the revenue is the social welfare, which may be arbitrarily bad in the worst case. The duality based upper bound in~\\cite{CaiDW16} improves this trivial upper bound, the social welfare, by substituting the value of each buyer's favorite item with the corresponding Myerson's virtual value. However, the substitution is viable only when the following condition holds -- the buyer's marginal gain for adding an item solely depends on her value for that item (assuming it's feasible to add that item\\footnote{WLOG, we can reduce any constrained additive valuation to an additive valuation with a feasibility constraint (see Definition~\\ref{def:valuation classes})}), but not the set of items she has already received. This applies to valuations that are additive, unit-demand and more generally constrained additive, but breaks under more general valuation functions, e.g., submodular, XOS or subadditive valuations. As a consequence, the original dual solution from~\\cite{CaiDW16} fails to provide a nice upper bound for more general valuations. To overcome this difficulty, we take a different approach. Instead of directly studying the dual of the original problem, we first relax the valuations and argue that the optimal revenue of the relaxed valuation is comparable to the original one. Then, since we choose the relaxation in a particular way, by applying a dual solution similar to the one in~\\cite{CaiDW16} to the relaxed valuation, we recover an upper bound of the optimal revenue for the relaxed valuation resembling the appealing format of the one in~\\cite{CaiDW16}. Combining these two steps, we obtain an upper bound for subadditive valuations that is easy to analyze. Indeed, we use our new upper bound to improve the approximation ratio for a single subadditive buyer from $338$~\\cite{RubinsteinW15} to $40$. See Section~\\ref{sec:valuation relaxation} for more details.\n\n\\vspace{.05in}\n\\noindent\\textbf{2. An adaptive dual:} our second major change to the framework is that we choose the dual in an adaptive manner. In~\\cite{CaiDW16}, a dual solution $\\lambda$ is chosen up front inducing a virtual value function $\\Phi(\\cdot)$, then the corresponding optimal virtual welfare is used as a benchmark for the optimal revenue. Finally, it is shown that the revenue of some simple mechanism is within a constant factor of the optimal virtual welfare. Unfortunately, when the valuations are beyond additive and unit-demand, the optimal virtual welfare for this particular choice of virtual value function becomes extremely complex and hard to analyze. Indeed, it is already challenging to bound when the buyers' valuations are $k$-demand. In this paper, we take a more flexible approach. For any particular allocation rule $\\sigma$, we tailor a special dual $\\lambda^{(\\sigma)}$ based on $\\sigma$ in a fashion that is inspired by Chawla and Miller's ex-ante relaxation~\\cite{ChawlaM16}. Therefore, the induced virtual valuation $\\Phi^{(\\sigma)}$ also depends on $\\sigma$. By duality, we can show that the optimal revenue obtainable by $\\sigma$ is still upper bounded by the virtual welfare with respect to $\\Phi^{(\\sigma)}$ under allocation rule $\\sigma$. Since the virtual valuation is designed specifically for allocation $\\sigma$, the induced virtual welfare is much easier to analyze. Indeed, we manage to prove that for any allocation $\\sigma$ the induced virtual welfare is within a constant factor of the revenue of some simple mechanism, when bidders have XOS valuations. See Section~\\ref{sec:virtual for relaxed} and~\\ref{sec:choice of beta} for more details.\n\n\n\\vspace{.05in}\n\n\\noindent\\textbf{3. A novel analysis and new mechanism:} with the two contributions above, we manage to derive an upper bound of the optimal revenue similar to the one in \\cite{CaiDW16} but for subadditive bidders. The third major contribution of this paper is a novel approach to analyzing this upper bound. The analysis in~\\cite{CaiDW16} essentially breaks the upper bound into three different terms-- \\textsc{Single}, \\textsc{Tail}~ and \\textsc{Core}, and bound them separately. All three terms are relatively simple to bound for additive and unit-demand buyers, but for more general settings the $\\textsc{Core}$ becomes much more challenging to handle. Indeed, the analysis in~\\cite{CaiDW16} was insufficient to tackle the $\\textsc{Core}$ even when the buyers have $k$-demand valuations\\footnote{The class of $k$-demand valuations is a generalization of unit-demand valuations, where the buyer's value is additive up to $k$ items.}-- a very special case of matroid rank valuations, which itself is a special case of XOS or subadditive valuations. Rubinstein and Weinberg~\\cite{RubinsteinW15} showed how to approximate the $\\textsc{Core}$ for a single subadditive bidder using grand bundling, but their approach does not apply to multiple bidders. Yao~\\cite{Yao15} showed how to approximate the $\\textsc{Core}$ for multiple additive bidders using a VCG with per bidder entry fee mechanism, but again it is unclear how his approach can be extended to multiple k-demand bidders. A recent paper by Chawla and Miller~\\cite{ChawlaM16} finally broke the barrier of analyzing the $\\textsc{Core}$ for multiple $k$-demand buyers. They showed how to bound the $\\textsc{Core}$ for matroid rank valuations using a sequential posted price mechanism by applying the \\emph{online contention resolution scheme (OCRS)} developed by Feldman et al.~\\cite{FeldmanSZ16}. The connection with OCRS is an elegant observation, and one might hope the same technique applies to more general valuations. Unfortunately, OCRS is only known to exist for special cases of downward closed constraints, and as we show in Section~\\ref{sec:core comparison}, the approach by Chawla and Miller cannot yield any constant factor approximation for general constrained additive valuations.\n\nWe take an entirely different approach to bound the $\\textsc{Core}$. Here we provide some intuition behind our mechanism and analysis. The $\\textsc{Core}$ is essentially the optimal social welfare induced by some truncated valuation $v'$, and our goal is to design a mechanism that extracts a constant fraction of the welfare as revenue. Let $M$ be any sequential posted price mechanism. A key observation is that when bidder $i$'s valuation is subadditive over independent items, her utility in $M$, which is the largest surplus she can achieve from the unsold items, is also subadditive over independent items. If we can argue that her utility function is $a$-Lipschitz (Definition~\\ref{def:Lipschitz}) with some small $a$, Talagrand's concentration inequality~\\cite{Talagrand1995concentration,Schechtman2003concentration} allows us to set an entry fee for the bidder so that we can extract a constant fraction of her utility just through the entry fee.\nIf we modify $M$ by introducing an entry fee for every bidder, according to Talagrand's concentration inequality, the new mechanism $M'$ should intuitively have revenue that is a constant fraction of the social welfare obtained by $M$~\\footnote{$M$'s welfare is simply its revenue plus the sum of utilities of the bidders, and $M'$ can extract some extra revenue from the entry fee, which is a constant fraction of the total utility from the bidders.}. Therefore, if there exists a sequential posted price mechanism $M$ that achieves a constant fraction of the optimal social welfare under the truncated valuation $v'$, the modified mechanism $M'$ can obtain a constant fraction of $\\textsc{Core}$ as revenue. Surprisingly, when the bidders have XOS valuations, Feldman et al.~\\cite{FeldmanGL15} showed that there exists an anonymous sequential posted price mechanism that always obtains at least half of the optimal social welfare. Hence, an anonymous sequential posted price with per bidder entry fee mechanism should approximate the $\\textsc{Core}$ well, and this is exactly the intuition behind our ASPE mechanism.\n\n To turn the intuition into a theorem, there are two technical difficulties that we need to address: (i) the Lipschitz constants of the bidders' utility functions turn out to be too large (ii) we deliberately neglected the difference in bidders' behavior under $M$ and $M'$ in hope to keep our discussion in the previous paragraph intuitive. However, due to the entry fee, bidders may end up purchasing completely different items under $M$ and $M'$, so it is not straightforward to see how one can relate the revenue of $M'$ to the welfare obtained by $M$.\n See Section~\\ref{sec:core comparison} for a more detailed discussion on how we overcome these two difficulties.\n \n\n\n\n\n\n\n\n\n\n\\subsection{Related Work}\nIn recent years, we have witnessed several breakthroughs in designing (approximately) optimal mechanisms in multi-dimensional settings. The black-box reduction by Cai et al.~\\cite{CaiDW12a,CaiDW12b,CaiDW13a,CaiDW13b} shows that we can reduce any Bayesian mechanism design problem to a similar algorithm design problem via convex optimization. Through their reduction, it is proved that all optimal mechanisms can be characterized as a distribution of virtual welfare maximizers, where the virtual valuations are computed by an LP. Although this characterization provides important insights about the structure of the optimal mechanism, the optimal allocation rule is unavoidably randomized and might still be complex as the virtual valuations are only a solution of an LP.\n\nAnother line of work considers the ``Simple vs. Optimal'' auction design problem. For instance, a sequence of results~\\cite{ChawlaHK07,ChawlaHMS10,ChawlaMS10,ChawlaMS15} show that sequential posted price mechanism can achieve $\\frac{1}{33.75}$ of the optimal revenue, whenever the buyers have unit-demand valuations over independent items. Another series of results~\\cite{HartN12,CaiH13,LiY13,BabaioffILW14,Yao15} show that the better of selling the items separately and running the VCG mechanism with per bidder entry fee achieves $\\frac{1}{69}$ of the optimal revenue, whenever the buyers' valuations are additive over independent items. Cai et al.~\\cite{CaiDW16} unified the two lines of results and improved the approximation ratios to $\\frac{1}{8}$ for the additive case and $\\frac{1}{24}$ for the unit-demand case using their duality framework.\n\nSome recent works have shown that simple mechanisms can approximate the optimal revenue even when buyers have more sophisticated valuations. For instance, Chawla and Miller~\\cite{ChawlaM16} showed that the sequential two-part tariff mechanism can approximate the optimal revenue when buyers have matroid rank valuation functions over independent items. Their mechanism requires every buyer to pay an entry fee up front, and then run a sequential posted price mechanism on buyers who have accepted the entry fee. Our ASPE is similar to their mechanism, but with two major differences: (i) since buyers are asked to pay the entry fee before the seller visits them, the buyers have to make their decisions based on the expected utility (assuming every other buyer behaves truthfully) they can receive. Hence, the mechanism is only guaranteed to be BIC and interim IR. While in our mechanism, the buyers can see what items are still available before paying the entry fee, therefore the decision making is straightforward and the ASPE is DSIC and ex-post IR; (ii) the item prices in the ASPE are anonymous, while in the sequential two-part tariff mechanism, personalized prices are allowed. For valuations beyond matroid rank functions, Rubinstein and Weinberg~\\cite{RubinsteinW15} showed that for a single buyer whose valuation is subadditive over independent items, either grand bundling or selling the items separately achieves at least $\\frac{1}{338}$ of the optimal revenue.\n\nThe Cai-Devanur-Weinberg duality framework~\\cite{CaiDW16} has been applied to other intriguing Mechanism Design problems. For example, Eden et al. showed that the better of selling separately and bundling together gets an $O(d)$-approximation for a single bidder with ``complementarity-$d$ valuations over independent items''~\\cite{EdenFFTW16a}. The same authors also proved a Bulow-Klemperer result for regular i.i.d. and constrained additive bidders~\\cite{EdenFFTW16b}. Liu and Psomas provided a Bulow-Klemperer result for {dynamic auctions}~\\cite{LiuP16}. Finally, Brustle et al.~\\cite{BrustleCWZ17} extended the duality framework to two-sided markets and used it to design simple mechanisms for approximating the Gains from Trade.\n\nStrong duality frameworks have recently been developed for one additive buyer~\\cite{DaskalakisDT13,DaskalakisDT15,Giannakopoulos14a,GiannakopoulosK14,GiannakopoulosK15}. These frameworks show that the dual problem of revenue maximization can be viewed as an optimal transport\/bipartite matching problem. Hartline and Haghpanah provided an alternative duality framework in~\\cite{HartlineH15}. They showed that if certain paths exist, these paths provide a witness of the optimality of a certain Myerson-type mechanism, but these paths are not guaranteed to exist in general. Similar to the Cai-Devanur-Weinberg framework, Carroll~\\cite{Carroll15} independently made use of a partial Lagrangian over incentive constraints. These duality frameworks have been successfully provide conditions under which a certain type of mechanism is optimal when there is a single unit-demand or additive bidder. However, none of these frameworks succeeds in yielding any approximately optimal results in multi-buyer settings.\n\n\\input{prelim}\n\\input{roadmap}\n\\input{duality}\n\\input{single_subadditive}\n\\input{multi_XOS}\n\\input{example_ocrs}\n\\newpage\n\n\\section{Multiple Bidders}\\label{sec:multi}\n\nIn this section, we prove our main result -- simple mechanisms can approximate the optimal BIC revenue even when there are multiple XOS\/subadditive bidders.\nFirst, we need the definition of supporting prices.\n\\begin{definition}[Supporting Prices~\\cite{DobzinskiNS05}]\\label{def:supporting price}\nFor any $\\alpha\\geq 1$, a type $t$ and a subset $S\\subseteq[m]$, prices $\\{p_j\\}_{j\\in S}$\nare $\\alpha$-supporting prices for $v(t,S)$ if \\textbf{(i)}\t$v(t,S') \\geq \\sum_{j\\in S'} p_j$ for all $S'\\subseteq S$ and \\textbf{(ii)} $\\sum_{j\\in S}p_j\\geq \\frac{v(t,S)}{\\alpha}$.\n\\end{definition}\n\n\n\\begin{theorem}\\label{thm:multi}\nIf for any buyer $i$, any type $t_i\\in T_i$ and any bundle $S\\in [m]$, $v_i(t_i,S)$ has a set of $\\alpha$-supporting prices $\\{\\theta_j^{S}(t_i)\\}_{j\\in S}$, then for any BIC mechanism $M$ and any constant $b\\in (0, 1)$,\n\\begin{align*}\n\\textsc{Rev}(M,v,D)\\leq 32\\alpha \\cdot \\textsc{APostEnRev}\n+\\left(12+\\frac{8}{1-b}+\\alpha\\cdot \\left(\\frac{16}{b(1-b)}+\\frac{96}{1-b}\\right)\\right)\\cdot \\textsc{PostRev}\n\\end{align*}\n\n\\vspace{0.05in}\nIf $v_i(t_i,\\cdot)$ is an XOS valuation for all $i$ and $t_i\\in T_i$, then $\\alpha=1$. By setting $b$ to $\\frac{1}{4}$, we have $$\\textsc{Rev}(M,v,D)\\leq 236\\cdot\\textsc{PostRev}+32\\cdot\\textsc{APostEnRev}.$$ For general subadditive valuations, $\\alpha=O(\\log(m))$ by~\\cite{BhawalkarR11}, hence $$\\textsc{Rev}(M,v,D)\\leq O(\\log(m))\\cdot \\max\\{\\textsc{PostRev},\\textsc{APostEnRev}\\}.$$\n\\end{theorem}\nHere is a sketch of the proof for Theorem~\\ref{thm:multi}. We show how to upper bound $\\textsc{Single}(M,\\beta)$ in Lemma~\\ref{lem:multi_single}. Then, we decompose $\\textsc{Non-Favorite}(M,\\beta)$ into $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ in Lemma~\\ref{lem:multi decomposition}. We show how to construct a simple mechanism to approximate $\\textsc{Tail}(M,\\beta)$ in Section~\\ref{subsection:tail} and how to approximate $\\textsc{Core}(M,\\beta)$ in Section~\\ref{subsection:core}.\n\n\\vspace{.1in}\n \\noindent\\textbf{Analysis of $\\textsc{Single}(M,\\beta)$:} \n\n\n\\begin{lemma}\\label{lem:multi_single}\nFor any mechanism $M$, $$\\textsc{Single}(M, \\beta)\\leq \\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{proof}\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to buyer $i$ and $t_i\\in R_j^{(\\beta)}$, $M'$ serves the agent $(i,j)$. Since there is at most one $R_j^{(\\beta)}$ that $t_i$ belongs to, $M'$ serves at most one agent $(i,j)$ for each of buyer $i$. Hence, $M'$ is feasible in the copies setting, and $\\textsc{Single}(M,\\beta)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare in the same setting. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(M,\\beta)$.\n\nAs showed in~\\cite{ChawlaHMS10, KleinbergW12}, a simple posted-price mechanism with the constraint that every buyer can only purchase one item, i.e., an RSPM, achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/6$ in the original setting. Hence, $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}$.\n\\end{proof}\n\n\n\\vspace{.05in}\n \\noindent\\textbf{Core-Tail Decomposition of $\\textsc{Non-Favorite}(M,\\beta)$:} we decompose $\\textsc{Non-Favorite}(M, \\beta)$ into two terms $\\textsc{Tail}(M, \\beta)$ and $\\textsc{Core}(M, \\beta)$\\footnote{In~\\cite{CaiDW16}, $\\textsc{Non-Favorite}$ is decomposed into four different terms $\\textsc{Under}$, $\\textsc{Over}$, $\\textsc{Core}$ and $\\textsc{Tail}$. We essentially merge the first three terms into $\\textsc{Core}(M, \\beta)$ in our decomposition.}. First, we need the following definition.\n\\begin{definition}\\label{def:c_i}\nFor every buyer $i$, let $c_i :=\\inf\\big\\{x\\geq 0:\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+x\\right]\\leq \\frac{1}{2}\\big\\}.$ For every $t_i \\in T_i$, let $\\mathcal{T}_i(t_i)=\\{j\\ |\\ V_i(t_{ij})\\geq \\beta_{ij}+c_i\\}$ and $\\mathcal{C}_i(t_i) = [m]\\backslash\\mathcal{T}_i(t_i)$.\n\\end{definition}\n Since $v_i(t_i,\\cdot)$ is subadditive for all $i$ and $t_i\\in T_i$, we have $v_i(t_i,S)\\leq v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. The term $\\textsc{Non-Favorite}(M,\\beta)$ can be decomposed into $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ based on the inequality above. The complete proof of Lemma~\\ref{lem:multi decomposition} can be found in Appendix~\\ref{appx:multi}.\n\n \\begin{lemma}~\\label{lem:multi decomposition}\n\t\t\\begin{align*} &\\textsc{Non-Favorite}(M,\\beta)\\\\\n\t\t\t\\leq& \\sum_i\\sum_{t_i}f_i(t_i) \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))~~~~~~~~~~(\\textsc{Core}(M,\\beta))\\\\\n\t\t\t+&\\sum_i\\sum_j \\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot\\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]~~~~~~~(\\textsc{Tail}(M,\\beta))\n\t\t\\end{align*}\n\t\t\\end{lemma}\n\n\\subsection{Analyzing $\\textsc{Tail}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\n\nIn this section we show how to bound $\\textsc{Tail}(M,\\beta)$ with the revenue of an RSPM.\n\\begin{lemma}\\label{lem:multi-tail}\n\tFor any BIC mechanism $M$, $\\textsc{Tail}(M, \\beta)\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{lemma}\n\nWe first fix a few notations. Let $$P_{ij}\\in\\argmax_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],$$\n\\begin{align*}\nr_{ij}&=(P_{ij}+\\beta_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]=\\max_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],\n\\end{align*}\n$r_i=\\sum_j r_{ij}$, and $r=\\sum_i r_i$. We show in the following Lemma that $r$ is an upper bound of $\\textsc{Tail}(M,\\beta)$.\n\\begin{lemma}\\label{lem:tail and r}\nFor any BIC mechanism $M$, $\\textsc{Tail}(M,\\beta)\\leq r.$\n\\end{lemma}\n\n\\begin{proof}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(M,\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]\\leq1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{proof}\n\nNext, we argue that $r$ can be approximated by an RSPM. Indeed, we prove a stronger lemma, which is also useful for analyzing $\\textsc{Core}(M,\\beta)$.\n\n\\begin{lemma}\\label{lem:neprev}\nLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be a collection of non-negative numbers, such that for any buyer $i$\n$$\\sum_{j\\in [m]} \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq 1\/2,$$ then\n\\begin{equation*}\n\\sum_i\\sum_j (x_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nConsider a RSPM that sells item $j$ to buyer $i$ at price $\\xi_{ij}=x_{ij}+\\beta_{ij}$. The mechanism\nvisits the buyers in some arbitrary order. Notice that when it is buyer $i$'s turn, she purchases exactly item $j$ and pays $x_{ij}+\\beta_{ij}$ if all of the following three conditions hold: (i) $j$ is still available, (ii) $V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}$ and (iii) $\\forall k\\neq j, V_i(t_{ik})< x_{ik}+\\beta_{ik}$. The second condition means buyer $i$ can afford item $j$. The third condition means she cannot afford any other item $k\\neq j$. Therefore, buyer $i$'s purchases exactly item $j$.\n\nNow let us compute the probability that all three conditions hold. Since every buyer's valuation is subadditive over the items, item $j$ is purchased by someone else only if there exists a buyer $k\\neq i$ who has $V_k(t_{kj})\\geq \\xi_{kj}$. Because $x_{kj}\\geq 0$ for all $k$, by the union bound, the event described above happens with probability at most $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]$, which is less than $b$ by property (i) of Lemma~\\ref{lem:requirement for beta}. Therefore, condition (i) holds with probability at least $(1-b)$. Clearly, condition (ii) holds with probability $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. Finally, condition (iii) holds with at least probability $1\/2$, because according to our assumption of the $x_{ij}$s, the probability that there exists any item $k\\neq j$ such that $V_i(t_{ik})\\geq x_{ik}+\\beta_{ik}$ is no more than $1\/2$. Since the three conditions are independent, buyer $i$ purchases exactly item $j$ with probability at least $\\frac{(1-b)}{2}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. So the expected revenue of this mechanism is at least $\\frac{(1-b)}{2}\\cdot \\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$.\n\\end{proof}\n\n\\notshow{\n\\begin{corollary}\\label{cor:bound tail }\n\\begin{equation}\n\\textsc{Tail}(M,\\beta)\\leq r\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nSince $P_{ij}\\geq c_i$, it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$\n. Therefore,\n$$r= \\sum_{i,j}(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\nOur statement follows from the above inequality and Lemma~\\ref{lem:tail and r}.\\end{proof}\n}\n\n\\begin{prevproof}{Lemma}{lem:multi-tail}\nSince $P_{ij}\\geq c_i$, it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$\n. Therefore,\n\\begin{equation}\\label{r and prev}\nr= \\sum_{i,j}(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\nOur statement follows from the above inequality and Lemma~\\ref{lem:tail and r}.\\end{prevproof}\n\n\n\nWe have done the analysis for $\\textsc{Tail}{(M,\\beta)}$. Before starting the analysis for $\\textsc{Core}{(M,\\beta)}$, we show that $r_i$ is within a constant factor of $c_i$. This Lemma is useful for bounding $\\textsc{Core}{(M,\\beta)}$.\n\n\\begin{lemma}\\label{lem:c_i}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot c_i$ and $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot\\textsc{PostRev}$.\n\\end{lemma}\n\\begin{proof}\nBy the definition of $P_{ij}$,\n\\begin{align*}\nr_i&= \\sum_j (\\beta_{ij}+P_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]\n\\geq \\sum_j (\\beta_{ij}+c_i)\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\\\\n&\\geq\\sum_j c_i\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\geq\\frac{1}{2}\\cdot c_i\n\\end{align*}\nThe last inequality is because when $c_i>0$,\n$\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+c_i\\right]$ is at least $\\frac{1}{2}$. As $\\sum_i c_i\/2 \\leq r$, by Inequality~(\\ref{r and prev}), \n$\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\n\n\\subsection{Analyzing $\\textsc{Core}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\n\nIn this section we upper bound $\\textsc{Core}(M,\\beta)$. Recall that\n$$\\textsc{Core}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$$\nWe can view it as the welfare of another valuation function $v'$ under allocation $\\sigma^{(\\beta)}$ where $v'_i(t_i, S) = v_i(t_i,S\\cap\\mathcal{C}_i(t_i))$. In other words, we ``truncate'' the function at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold. The new function still satisfies monotonicity, subadditivity and no externalities.\n\nWe first compare existing methods for analyzing the $\\textsc{Core}$ with our approach before jumping into the proofs.\n\n\\subsubsection{Comparison between the Existing Methods and Our Approach}\\label{sec:core comparison}\nAs all results in the literature~\\cite{ChawlaHMS10, Yao15, CaiDW16,ChawlaM16} only study special cases of constrained additive valuations, we restrict our attention to constrained additive valuations in the comparison, but our approach also applies to XOS and subadditive valuations.\n\nWe compare our approach to the state of the art result by Chawla and Miller~\\cite{ChawlaM16}. They separate $\\textsc{Core}(M,\\beta) $ into two parts: (i) the welfare obtained from values below $\\beta$, and (ii) the welfare obtained from values between $\\beta$ and $\\beta+c$\\footnote{In particular, if bidder $i$ is awarded a bundle $S$ that is feasible for her, the contribution for the first part is $\\sum_{j\\in S} \\min\\left\\{\\beta_{ij},t_{ij}\\right\\}\\cdot \\mathds{1}\\left[t_{ij}< \\beta_{ij}+c_i \\right]$ and the contribution to the second part is $\\sum_{j\\in S} \\left(t_{ij}-\\beta_{ij}\\right)^+\\cdot \\mathds{1}\\left[t_{ij}< \\beta_{ij}+c_i \\right]$ }.\n It is not hard to show that the latter can be upper bounded by the revenue of a sequential posted price with per bidder entry fee mechanism.\n Due to their choice of $\\beta$ (similar to the second property of Lemma~\\ref{lem:requirement for beta}), the former is upper bounded by $\\sum_{i,j} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[t_{ij}\\geq \\beta_{ij}\\right]$.\n It turns out when every bidder's feasibility constraint is a matroid, one can use the OCRS from~\\cite{FeldmanSZ16} to design a sequential posted price mechanism to approximate this expression.\n However, as we show in Example~\\ref{ex:counterexample ocrs}, $\\sum_{i,j} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[t_{ij}\\geq \\beta_{ij}\\right]$ could be $\\Omega\\left(\\frac{\\sqrt{m}}{\\log m}\\right)$ times larger than the optimal social welfare when the bidders have general downward closed feasibility constraints.\n Hence, such approach cannot yield any constant factor approximation for general constrained additive valuations.\n\nAs explained in the intro, we take an entirely different approach. We first construct the posted prices $\\{Q_j\\}_{j\\in[m]}$ for our ASPE (Definition~\\ref{def:posted prices}), Feldman et al.~\\cite{FeldmanGL15} showed that the anonymous posted price mechanism with these prices achieves welfare $\\Omega\\left(\\textsc{Core}(M,\\beta)\\right)$. If all bidders have valuations that are subadditive over independent items, for any bidder $i$ and any set of available items $S$, $i$'s surplus for $S$ under valuation $v'_i(t_i, \\cdot)$ ($max_{S'\\subseteq S}~v'_i(t_i,S') -\\sum_{j\\in S'} Q_j$) is also subadditive over independent items. According to Talagrand's concentration inequality, the surplus concentrates and its expectation is upper bounded by its median and its Lipschitz constant $a$. One can extract at least half of the median by setting the median of the surplus as the entry fee. How about the Lipschitz constant $a$? Unfortunately, $a$ could be as large as $\\frac{1}{2}\\max_{j\\in[m]}\\{\\beta_{ij}+c_i\\}$, which is too large to be bounded.\n\nHere is how we overcome this difficulty. Instead of considering $v'$, we construct a new valuation $\\hat{v}$ that is always dominated by the true valuation $v$. We consider the social welfare induced by $\\sigma^{(\\beta)}$ under $\\hat{v}$ and define it as $\\widehat{\\textsc{Core}}(M,\\beta)$. In Section~\\ref{sec:proxy core}, we show that $\\widehat{\\textsc{Core}}(M,\\beta)$ is not too far away from $\\textsc{Core}(M,\\beta)$, so it suffices to approximate $\\widehat{\\textsc{Core}}(M,\\beta)$ (Lemma~\\ref{lem:hat Q}). But why is $\\widehat{\\textsc{Core}}(M,\\beta)$ easier to approximate? The reason is two-fold. \\textbf{(i)} For any bidder $i$ and any set of available items $S$, bidder $i$'s surplus for $S$ under $\\hat{v}_i(t_i,\\cdot)$ (defined as $\\mu_i(t_i,S)$ in Definition~\\ref{def:entry fee}, which is $max_{S'\\subseteq S}~ \\hat{v}_i(t_i,S') -\\sum_{j\\in S'} Q_j$), is not only subadditive over independent items, but also has a small Lipschitz constant $\\tau_i$ (Lemma~\\ref{lem:property of mu}). Indeed, these Lipschitz constants are so small that $\\sum_i \\tau_i$ and can be upper bounded by $\\textsc{PostRev}$ (Lemma~\\ref{lem:tau_i}). \\textbf{(ii)} If we set the entry fee of our ASPE to be the median of $\\mu_i(t_i,S)$ when $t_i$ is drawn from $D_i$, using a proof inspired by Feldman et al.~\\cite{FeldmanGL15}, we can show that our ASPE's revenue collected from the posted prices plus the expected surplus of the bidders (over the randomness of all bidders' types) approximates $\\widehat{\\textsc{Core}}(M,\\beta)$ (implied by Lemma~\\ref{lem:lower bounding mu}). Again by Talagrand's concentration inequality, we can bound bidder $i$'s expected surplus by our entry fee and $\\tau_i$ (Lemma~\\ref{lem:concentration entry fee}). As $\\hat{v}$ is always smaller than the true valuation $v$, thus for any type $t_i$ of bidder $i$ and any available items $S$, the surplus for $S$ under $v_i(t_i,\\cdot)$ must be larger than $\\mu_i(t_i,S)$, and the entry fee is accepted with probability at least $1\/2$. Putting everything together, we demonstrate that we can approximate $\\textsc{Core}(M,\\beta)$ with an ASPE or an RSPM (Lemma~\\ref{lem:upper bounding Q}).\n\\subsubsection{Construction of $\\widehat{\\textsc{Core}}(M,\\beta)$}\\label{sec:proxy core}\n\nWe first show that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$ and any set $S$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$ for $v_i(t_i,\\cdot)$, then for any $t_i$ {and $S$} there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. Moreover, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$ for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\n{The last equality is because $\\gamma_j^S(t_i)=0$ for $j\\in S\\backslash\\mathcal{C}_i(t_i)$. }Also, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$ for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})\\geq\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})<\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{proof}\n\n\n\nNext, we define the prices of our ASPE.\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n{ $\\textsc{Core}(M,\\beta)$ can be upper bounded by $\\sum_{j\\in [m]}Q_j$. The proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).}\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(M,\\beta)$.\n\\end{lemma}\n\\begin{proof}\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(M,\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)\\\\\n&=\\alpha\\cdot \\sum_{j\\in[m]}\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\vspace{0.05in}\nIn the following definitions, we define $\\widehat{\\textsc{Core}}(M,\\beta)$ which is the welfare of another function $\\hat{v}$ under the same allocation $\\sigma^{(\\beta)}$.\n\n \n\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\}.$$\n\\end{definition}\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $Y_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$, $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap Y_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$. Moreover, let $$\\widehat{\\textsc{Core}}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{v}_i(t_i,S).$$\n\\end{definition}\n\nIn the next two Lemmas, we prove some useful properties of $\\tau_i$. In particular, we argue that $\\sum_{i\\in[n]} \\tau_i$ can be upper bounded by $\\frac{4}{1-b}\\cdot \\textsc{PostRev}$ (Lemma~\\ref{lem:tau_i}).\n \\begin{lemma}\\label{lem:beta_ij}\n\\begin{align*}\n\\sum_i\\sum_j \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\right]\n \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nAccording to the definition of $\\tau_i$, for every buyer $i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$,\n and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. Our statement follows directly from Lemma~\\ref{lem:neprev}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n$$\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$$.\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative, \\begin{align*}\n \\sum_i\\sum_j \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\n \\geq \\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right].\n \\end{align*}\nAccording to the definition of $\\tau_i$, when $\\tau_i>0$, $$\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}.$\nTherefore, $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$ due to Lemma~\\ref{lem:beta_ij}.\n\\end{proof}\n\n\nIn the following two Lemmas, we compare $\\widehat{\\textsc{Core}}(M,\\beta)$ with $\\textsc{Core}(M,\\beta)$. The proof of Lemma~\\ref{lem:hat gamma} is postponed to Appendix~\\ref{appx:multi}.\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:hat Q}\n\tLet $$\\hat{Q}_j = \\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{\\gamma}_j^{S}(t_i).$$ Then,\n\t$$\\sum_{j\\in[m]} \\hat{Q}_j\\leq \\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{proof}\nFrom the definition of $\\hat{Q}_j$, it is easy to see that $Q_j\\geq \\hat{Q}_j$ for every $j$. So we only need to argue that $\\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}$.\n\\begin{equation}\\label{eq:first}\n\t\\begin{aligned}\n\t&\\sum_{j} \\left(Q_j- \\hat{Q}_j\\right) = \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left(\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\\\\\n\t= & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\tThis first inequality is because $\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)$ is non-zero only when $V_i(t_{ij})\\geq Q_j+\\tau_i$, and the difference is upper bounded by $\\beta_{ij}$ when $V_i(t_{ij})\\leq \\beta_{ij}$ and upper bounded by $\\beta_{ij}+c_i$ when $V_i(t_{ij})> \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}. The second inequality is due to property (ii) in Lemma~\\ref{lem:requirement for beta}. The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\\end{proof}\n\n\n\nBy Lemma~\\ref{lem:hat gamma}, $\\sum_{j\\in[m]}\\hat{Q}_j\\leq \\widehat{\\textsc{Core}}(M,\\beta)\/2$. By Lemma~\\ref{lem:core and q_j}, $\\sum_{j\\in[m]}{Q}_j\\leq {\\textsc{Core}}(M,\\beta)\/2\\alpha$. Hence, Lemma~\\ref{lem:hat Q} shows that to approximate $\\textsc{Core}(M,\\beta)$, it suffices to approximate $\\widehat{\\textsc{Core}}(M,\\beta)$. Indeed, we will use $\\sum_{j\\in[m]} \\hat{Q}_j$ as an proxy for $\\textsc{Core}(M,\\beta)$ in our analysis of the ASPE.\n\\subsubsection{Design and Analysis of Our ASPE}\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$ \\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\big(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\big).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{0$, then $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+c_i\\right]\\geq \\frac{1}{2}$. As $\\sum_i c_i\/2 \\leq r$, by Corollary~\\ref{cor:bound tail }, $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\subsection{Analyzing $\\textsc{Core}(\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\nIn this section, we bound $\\textsc{Core}(\\beta)$ using the sum of the revenue of a few simple mechanisms. First, we show that if we ``truncate'' the function $v(\\cdot,\\cdot)$ at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold, the new function still satisfies monotonicity, subadditivity and no externalities.\n\\begin{lemma}\\label{lem:valuation v_i'}\n\tLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be a set of nonnegative numbers. For any buyer $i$, any type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij})< x_{ij}\\}$, and let $$\\bar{v}_i(t_i, S) = v_i(t_i,S\\cap X_i(t_i)),$$ for any set $S\\subseteq[m]$. Then for any bidder $i$, any type $t_i\\in T_i$, $\\bar{v}_i(t_i,\\cdot)$, satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{lemma}\n\t\t\\begin{prevproof}{Lemma}{lem:valuation v_i'}\n\t\t We will argue these three properties one by one.\n\t\\begin{itemize}\n\t\t\\item \\emph{Monotonicity:} For all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, since $v_i(t_i,\\cdot)$ is monotone, $$\\bar{v}_i(t_i,U)=v_i(t_i,U\\cap X_i(t_i))\\leq v_i(t_i,V\\cap X_i(t_i))=\\bar{v}(t_i,V)$$ Thus $\\bar{v}_i(t_i,\\cdot)$ is monotone.\n\t\t\\item \\emph{Subadditivity:} For all $t_i\\in T_i$ and $U,V\\subseteq [m]$. Hence, $(U\\cup V)\\cap X_i(t_i)=(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i))$.\\mingfeinote{Since $v_i(t_i,\\cdot)$ is subadditive}, we have\n\\begin{align*}\n&\\bar{v}_i(t_i,U\\cup V)=v_i(t_i,(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i)))\\\\\n &~~~~~~~~~~~~~\\leq v_i(t_i,U\\cap X_i(t_i))+v_i(t_i,V\\cap X_i(t_i))= \\bar{v}_i(t_i,U)+\\bar{v}_i(t_i,V).\n\\end{align*}\n\\item \\emph{No externalities:} For any $t_i\\in T_i$, $S\\subseteq [m]$, and any $t_i'\\in T_i$ such that $t_{ij}=t_{ij}'$ for all $j\\in S$, to prove $\\bar{v}_i(t_i,S)=\\bar{v}_i(t_i',S)$, it suffices to show $S\\cap X_i(t_i)=S\\cap X_i(t_i')$. Since $V_i(t_{ij})=V_i(t_{ij}')$ for any item $j\\in S$, $j\\in S\\cap X_i(t_i)$ if and only if $j\\in S\\cap X_i(t_i')$.\n\t\\end{itemize}\n\t\\end{prevproof}\n\t\t\n\t\\begin{corollary}~\\label{cor:v_i'}\n Let $${v}'_i(t_i, S) = v_i(t_i,S\\cap \\mathcal{C}_i(t_i)),$$ then or any bidder $i$, any type $t_i\\in T_i$, ${v}'_i(t_i, \\cdot)$ satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tSimply set $x_{ij}$ to be $\\beta_{ij}+c_i$ in Lemma~\\ref{lem:valuation v_i'}.\n\t\\end{proof}\n\t\nNext, we argue that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$ for $v_i(t_i,\\cdot)$ and any set $S$, then for any $t_i$ there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$ and any set $S$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. In particular, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})\\leq \\beta_{ij}+c_i]$ for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:supporting prices for v'}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\nAlso, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})\\leq \\beta_{ij}+c_i]$ for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})>\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})\\leq\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{prevproof}\n\n\\notshow{\\begin{lemma}\n\tIf for any buyer $i$, type $t_i$, $v_i(t_i,\\cdot)$ is an XOS valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $1$-supporting prices for $v'(t_i,\\cdot)$ and $S$. If $v_i(t_i,\\cdot)$ is an subadditive valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $\\log m$-supporting prices $v'(t_i,\\cdot)$ and $S$.\n\\end{lemma}\n\\begin{proof}\n\t\\yangnote{Fill in the proof. Argue $v'$ remains to be XOS is $v$ is XOS. And by the previous Lemma, we know $v'$ is subadditive so we already have $\\log m$ supporting prices.}\n\\end{proof}}\n\nNext, we rewrite $\\textsc{Core}(\\beta)$ using $v'(\\cdot,\\cdot)$,\n\n$$\\textsc{Core}(\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v'_i(t_i,S).$$\n\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(\\beta).$$\n\\end{lemma}\n\\begin{proof}\n\tThe proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)=\\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_j \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\},$$ and define $A_i$ to be $\\{j\\ |\\ \\beta_{ij}\\leq Q_j+\\tau_i\\}$.\n\\end{definition}\n We have the following Lemma:\n\n \\begin{lemma}\\label{lem:beta_ij}\n$$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nBy the definition of $\\tau_i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$\\footnote{This clearly holds if $V_i(t_{ij})$ is drawn from a continuous distribution. When $V_i(t_{ij})$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.} for every buyer $i$ and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. By Lemma~\\ref{lem:neprev}, we have $$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n\\begin{equation*}\n\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative, $\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]$ is clearly no smaller than $\\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]$. According to the definition of $\\tau_i$, when $\\tau_i>0$, $\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}$\\footnote{See the proof of Lemma~\\ref{lem:requirement for beta}.}. Therefore, we have $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\notshow{\n\\begin{lemma}\\label{lem:beta_ij}\n\\begin{equation}\n\\sum_i\\sum_{j\\not\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tThe proof is similar to the proof of Lemma~\\ref{lem:tau_i}. Again, we let $x_{ij} = \\max \\{\\beta_{ij},Q_j+\\tau_i\\}-\\beta_{ij}$. Clearly, we also have $$\\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}+x_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}.$$ Note that for any $j\\in A_i$, $x_{ij} = 0$, so the inequality above directly implies our claim.\n\\end{proof}\n}\n\n\n\\notshow{\\begin{definition}\\label{def:v hat}\nWe construct a new subadditive valuation $\\hat{v}_i(t_i,\\cdot)$ for every buyer $i$ and type $t_i\\in T_i$ such that $$\n\\hat{v}_i(t_i,S) = \\max_{\\ell} \\sum_{j\\in S} \\min\\{t_{ij}^{(\\ell)}, Q_j+\\tau_i \\}, $$ for every $S\\subseteq[m]$. Similarly, let $$\\hat{\\gamma}^S_j(t_i) = \\min\\{\\gamma_j^S(t_i), Q_j+\\tau_i \\}$$ for every buyer $i$, type $t_i\\in T_i$ and $S\\subseteq[m]$.\n\\end{definition}\n}\n\nHere, we define a new function $\\hat{v}(\\cdot,\\cdot)$, which will be useful in analyzing the revenue of ASPE.\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$, $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap X_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $$\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:hat gamma}\nBy Lemma~\\ref{lem:valuation v_i'} and Definition~\\ref{def:v hat}, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\t$$\\hat{v}_i(t_i,S')= v_i(t_i,\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}).$$\n\tSince $\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}$ is a subset of $S'$, it is also a subset of $S$. Therefore,\n\t$$v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq \\sum_{j: j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\t\n\n\t\\end{prevproof}\n\n\\subsection{Anonymous Sequential Posted Price Mechanism with Entry Fee}\\label{sec:ASPE}\n\t\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$\\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\left(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\right).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{ \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}). The second inequality is due to the choice of $\\beta_{ij}$ (Lemma~\\ref{lem:requirement for beta}). The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\t\\end{prevproof}\n\n\nLet $M_i^{(\\beta)}(t)$ be the set of items allocated to buyer $i$ by mechanism $M^{(\\beta)}$when the reported type profile is $t$. We argue that in expectation over all type profiles, we can provide a lower bound of the sum of $\\mu_i(t_i,S_i(t_{\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all $U\\subseteq V\\subseteq S_i(t_{0$, $\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}$\\footnote{See the proof of Lemma~\\ref{lem:requirement for beta}.}. Therefore, we have $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\notshow{\n\\begin{lemma}\\label{lem:beta_ij}\n\\begin{equation}\n\\sum_i\\sum_{j\\not\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tThe proof is similar to the proof of Lemma~\\ref{lem:tau_i}. Again, we let $x_{ij} = \\max \\{\\beta_{ij},Q_j+\\tau_i\\}-\\beta_{ij}$. Clearly, we also have $$\\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}+x_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}.$$ Note that for any $j\\in A_i$, $x_{ij} = 0$, so the inequality above directly implies our claim.\n\\end{proof}\n}\n\n\n\\notshow{\\begin{definition}\\label{def:v hat}\nWe construct a new subadditive valuation $\\hat{v}_i(t_i,\\cdot)$ for every buyer $i$ and type $t_i\\in T_i$ such that $$\n\\hat{v}_i(t_i,S) = \\max_{\\ell} \\sum_{j\\in S} \\min\\{t_{ij}^{(\\ell)}, Q_j+\\tau_i \\}, $$ for every $S\\subseteq[m]$. Similarly, let $$\\hat{\\gamma}^S_j(t_i) = \\min\\{\\gamma_j^S(t_i), Q_j+\\tau_i \\}$$ for every buyer $i$, type $t_i\\in T_i$ and $S\\subseteq[m]$.\n\\end{definition}\n}\n\nHere, we define a new function $\\hat{v}(\\cdot,\\cdot)$, which will be useful in analyzing the revenue of ASPE.\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $Y_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$, $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap Y_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $$\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:hat gamma}\nBy Lemma~\\ref{lem:valuation v_i'} and Definition~\\ref{def:v hat}, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\n\t$$\\hat{v}_i(t_i,S')= v_i\\left(t_i,S'\\cap Y_i(t_i)\\right)\\geq v_i\\left(t_i, \\left(S'\\cap Y_i(t_i)\\right)\\cap \\mathcal{C}_i(t_i)\\right) =v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right).$$\n\tSince $S'\\cap Y_i(t_i)\\subseteq S$,\n\t$$v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right) \\geq \\sum_{j\\in S'\\cap Y_i(t_i)}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\n\n\t\\end{prevproof}\n\n\\subsection{Anonymous Sequential Posted Price Mechanism with Entry Fee}\\label{sec:ASPE}\n\t\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$\\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\left(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\right).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{ \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}. The second inequality is due to property (ii) in Lemma~\\ref{lem:requirement for beta}. The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\t\\end{prevproof}\n\n\nLet $M_i^{(\\beta)}(t)$ be the set of items allocated to buyer $i$ by mechanism $M^{(\\beta)}$when the reported type profile is $t$. We argue that in expectation over all type profiles, we can provide a lower bound of the sum of $\\mu_i(t_i,S_i(t_{\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all {$U, V\\subseteq [m]$}, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\n\\end{prevproof}\n\nNow, we are ready to prove Lemma~\\ref{lem:concentration entry fee}.\n\n\\begin{prevproof}{Lemma}{lem:concentration entry fee}\nIt directly follows from Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}. For any $i$ and $t_{\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all {$U, V\\subseteq [m]$}, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\\end{prevproof}\n\n\\section{Efficient Approximation for Symmetric Bidders}\\label{sec:symmetric computation}\nIn this section, we sketch how to compute the RSPM and ASPE to approximate the optimal revenue in polynomial time for symmetric bidders\\footnote{Bidders are symmetric if for any two bidders $i$ and $i'$, we have $v_i(\\cdot,\\cdot) = v_{i'}(\\cdot,\\cdot)$ and $D_{ij}=D_{i'j}$ for all $j$.}. For any given BIC mechanism $M$, one can follow our proof to construct in polynomial time an RSPM and an ASPE such that the better of the two achieves a constant fraction of $M$'s revenue. We will describe the construction of the RSPM and the ASPE separately in this section. The difficulty of applying the method described above to construct the desired simple mechanisms is that we need to know an (approximately) revenue-maximizing mechanism $M^*$. We will show how to circumvent this difficulty when the bidders are symmetric.\n\n Indeed, we can directly construct an RSPM that approximates the $\\textsc{PostRev}$. As we have restricted the buyers to purchase at most one item in an RSPM, the $\\textsc{PostRev}$ is upper bounded by the optimal revenue of the unit-demand setting where buyer $i$ has value $V_i(t_{ij})$ for item $j$ when her type is $t_i$. By~\\cite{CaiDW16}, we know that the optimal revenue in this unit-demand setting is upper bounded by $4\\textsc{OPT}^{\\textsc{Copies-UD}}$, so one can simply use the RSPM constructed in~\\cite{ChawlaHMS10} to extract revenue at least $\\frac{\\textsc{PostRev} }{24}$. Note that the construction is independent of $M$.\n\n Unlike the RSPM, our construction for the ASPE heavily relies on $\\beta$ which depends on $M$ (Lemma~\\ref{lem:requirement for beta}). Given $\\beta$, we first compute $c_i$s according to Definition~\\ref{def:c_i}. Next, we compute the $Q_j$s (Definition~\\ref{def:posted prices}). Finally, we compute the $\\tau_i$s (Defintion~\\ref{def:tau}) and use them to compute the entry fee (Definition~\\ref{def:entry fee}). A few steps of the algorithm above requires sampling from the type distributions, but it is not hard to argue that a polynomial number of samples suffices. The main reason that the information about $M$ is necessary is because our construction crucially relies on the choice of $\\beta$. Next, we argue that for symmetric bidders, we can essentially choose a $\\beta$ that satisfies all requirements in Lemma~\\ref{lem:requirement for beta} for all mechanisms.\n\n When bidders are symmetric, the important observation is that the optimal mechanism must also be symmetric, and for any symmetric mechanism we can directly construct a $\\beta$ that satisfies all the requirements in Lemma~\\ref{lem:requirement for beta}. For every $i\\in [n], j\\in [m]$, choose $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Clearly, this choice satisfies property (i) in Lemma~\\ref{lem:requirement for beta}. Furthermore, the ex-ante probability for any bidder $i$ to win item $j$ is the same in any symmetric mechanism, and therefore is no more than $1\/n$. Hence, property (ii) in Lemma~\\ref{lem:requirement for beta} is also satisfied. Given this $\\beta$, we can essentially follow the algorithm mentioned above to construct the ASPE. The only difference is that we no longer know the $\\sigma$, which is required when computing the $Q_j$s. This can be resolved by considering the welfare maximizing mechanism $M'$ with respect to $v'$. We compute the prices $Q_j$ using the allocation rule of $M'$ and construct our ASPE. As $M'$ is also symmetric, our $\\beta$ satisfies all requirements in Lemma~\\ref{lem:requirement for beta} with respect to $M'$. Therefore, Lemma~\\ref{lem:upper bounding Q} implies that either this ASPE or the RSPM constructed above has at least a constant fraction of $\\textsc{Core}(M',\\beta)$ as revenue. Since $M'$ is welfare maximizing, $\\textsc{Core}(M',\\beta)\\geq \\textsc{Core}(M^*,\\beta)$, where $M^*$ is the revenue optimal mechanism. Therefore, we construct in polynomial time a simple mechanism whose revenue is a constant fraction of the optimal BIC revenue.\n\n\n\\section{Analysis for the Single-Bidder Case}\\label{sec:single_appx}\n\n\n\\begin{comment}\n\\begin{prevproof}[lemma]\\ref{lem:single-single}\nRecall that $\\textsc{Single}(M)=\\sum_{t\\in T}f(t)\\cdot$ \\\\\n\\noindent$\\sum_{j\\in[m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\pi^{(\\beta)}_{j}(t)\\cdot {\\tilde{\\varphi}}_{j}(V(t_{j}))$.\n\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to the buyer and $t\\in R_j^{(\\beta)}$, $M'$ serves the agent $j$. $M'$ is feasible in the copies setting as there is at most one agent being served, and $\\textsc{Single}(M)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare in the same setting. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(M)$.\n\nAs shown in~\\cite{ChawlaHMS10}, when there is a single buyer, a simple posted-price mechanism with the constraint that the buyer can only purchase one item achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/2$ in the original setting. Therefore, by the definition of $\\textsc{SRev}$ we have $2\\textsc{SRev}\\geq\\textsc{OPT}^{\\textsc{Copies-UD}}$.\n\\end{prevproof}\n\\end{comment}\n\n\n\\begin{prevproof}{lemma}{lem:single decomposition}\n{Recall that for all $t\\in T$ and $S\\subseteq [m]$, $v(t,S)\\leq v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$.} We replace every $v(t,S)$ in $\\textsc{Non-Favorite}(M)$ with $v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$. Also since $R^{\\beta}_0=\\emptyset$, the corresponding term is simply $0$. First, the contribution from $v\\left(t,S\\cap \\mathcal{C}(t)\\right)$ is upper bounded by the \\textsc{Core}(M).\n\n\\begin{align*}\n& \\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\left(\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot v\\left(t,(S\\backslash\\{j\\})\\cap \\mathcal{C}(t)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot v\\left(t,S\\cap \\mathcal{C}(t)\\right)\\right)\\\\\n\\leq& \\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{S}^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))\\quad(\\textsc{Core}(M))\n\\end{align*}\n\nThe inequality comes from the monotonicity of $v(t,\\cdot)$.\n\nNext, we upper bound the contribution from $\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$ by the $\\textsc{Tail}(M)$.\n\\begin{align*}\n&\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\left(\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}(t)}V(t_k)+\\sum_{S:j\\not\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}(t)}V(t_k)\\right) \\\\\n=&\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_j)\\cdot \\mathds{1}\\left[t\\not\\in R_j^{(\\beta)}\\right]\\cdot \\pi_j^{(\\beta)}(t)~~~~~~~\\text{{(Recall $\\pi_{j}^{(\\beta)}(t)=\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)$)}}\\\\\n\\leq &\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_{j})\\cdot\n{\\mathds{1}\\left[t \\not\\in R_j^{(\\beta)}\\right]}~~~\\text{($\\pi_j^{(\\beta)}(t)\\leq 1$)}\\\\\n\\leq &{\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_{j})\\cdot \\mathds{1}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]~~~\\text{(Definition of $R_j^{(\\beta)}$)}}\\\\\n=&{\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_{j})\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]\\quad(\\textsc{Tail}(M))}\n\\end{align*}\n\\end{prevproof}\n\n\\begin{comment}\n\\subsubsection{Analyzing $\\textsc{Tail}(M)$~in the Single-Bidder Case}\n\\begin{prevproof}[lemma]{lem:single-tail}\nSince $\\textsc{Tail}(M)=\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$, for each type $t_j\\in T_j$ consider the mechanism that posts the same price $V(t_j)$ for each item but only allows the buyer to purchase at most one. Notice if there exists $k\\not= j$ such that $V(t_k)\\geq V(t_j)$, the mechanism is guaranteed to sell one item obtaining revenue $V(t_j)$. Thus, the revenue obtained by this mechanism\nis at least $V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$. By definition, this is no more than $\\textsc{SRev}$.\n\n\\begin{equation}\\label{equ:single-tail}\n\\textsc{Tail}(M)\\leq \\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot \\textsc{SRev}\\notshow{\\leq}{=} 2\\cdot \\textsc{SRev}\n\\end{equation}\n\n{\nThe last equality is because by the definition of $c$, \\\\\n\\noindent$\\sum_j \\Pr_{t_j}[V(t_j)\\geq c]=2$.\\footnote{This clearly holds if $V(t_j)$ is drawn from a continuous distribution. When $V(t_j)$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.}\n}\n\\end{prevproof}\n\\end{comment}\n\n\\begin{prevproof}{Lemma}{lem:single subadditive}\nWe argue the three properties one by one.\n\n\\begin{itemize}[leftmargin=0.7cm]\n\\item \\emph{Monotonicity:} For all $t\\in T$ and $U\\subseteq V\\subseteq [m]$, $U\\cap \\mathcal{C}(t)\\subseteq V\\cap \\mathcal{C}(t)$. Since $v(t,\\cdot)$ is monotone,\n$$v'(t,U)=v\\left(t,U\\cap \\mathcal{C}(t)\\right)\\leq v\\left(t,V\\cap \\mathcal{C}(t)\\right)=v'(t,V).$$ Thus, $v'(t,\\cdot)$ is monotone.\n\\item \\emph{Subadditivity:} For all $t\\in T$ and $U,V\\subseteq [m]$, notice $(U\\cup V)\\cap \\mathcal{C}(t)=\\left(U\\cap\\mathcal{C}(t)\\right)\\cup \\left(V\\cap\\mathcal{C}(t)\\right)$, we have\n$$v'(t,U\\cup V)=v\\left(\\left(t,(U\\cap\\mathcal{C}(t)\\right)\\cup \\left(V\\cap\\mathcal{C}(t)\\right)\\right)\\leq v\\left(t,U\\cap\\mathcal{C}(t)\\right)+v\\left(t,V\\cap\\mathcal{C}(t)\\right)=v'(t,U)+v'(t,V).$$\n\\item \\emph{No externalities:} For any $t\\in T$, $S\\subseteq [m]$, and any $t'\\in T$ such that $t_{j}=t_{j}'$ for all $j\\in S$, to prove $v'(t,S)=v'(t',S)$, it is enough to show $S\\cap \\mathcal{C}(t)=S\\cap \\mathcal{C}(t')$. Since $V(t_j)=V(t_j')$ for any $j\\in S$, $j\\in S\\cap \\mathcal{C}(t)$ if and only if $j\\in S\\cap \\mathcal{C}(t')$.\n\\end{itemize}\n\\end{prevproof}\n\n\\begin{prevproof}{Lemma}{lem:single Lipschitz}\nFor any $t,t'\\in T$, and set $X,Y\\subseteq [m]$, define set $H=\\left\\{j\\in X\\cap Y:t_j=t_j'\\right\\}$. Since $v'(\\cdot,\\cdot)$ has no externalities, $v'(t',H)=v'(t,H)$. Therefore,\n\\begin{align*}\n|v'(t,X)-v'(t',Y)|&=\\max\\left\\{v'(t,X)-v'(t',Y),v'(t',Y)-v'(t,X)\\right\\}\\\\\n&\\leq \\max\\left\\{v'(t,X)-v'(t',H),v'(t',Y)-v'(t,H)\\right\\}\\quad\\text{(Monotonicity)}\\\\\n&\\leq \\max\\left\\{v'(t,X\\backslash H),v'(t',Y\\backslash H)\\right\\}\\quad\\text{(Subadditivity)}\\\\\n& = \\max\\left\\{v\\left(t,(X\\backslash H)\\cap \\mathcal{C}(t)\\right),v\\left(t',(Y\\backslash H)\\cap\\mathcal{C}(t)\\right)\\right\\}\\quad\\text{(Definition of $v'(\\cdot,\\cdot)$)}\\\\\n&\\leq c\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\\\\n&\\leq c\\cdot (|X\\Delta Y|+|X\\cap Y|-|H|)\n\\end{align*}\nThe second last inequality is because both $v(t,\\cdot)$ and $v(t',\\cdot)$ are subadditive and for any item $j\\in \\mathcal{C}(t)$ ($\\mathcal{C}(t')$) the single-item valuation $V(t_j)$ ($V(t'_j)$) is less than $c$.\n\\end{prevproof}\n\n\\section{Proof of Lemma~\\ref{lem:relaxed valuation}}\n\\begin{lemma}\\label{lem:relaxed valuation stronger}\n\tIn a $n$-player $m$-item combinatorial auction, for any absolute constant $\\eta\\in(0,1)$ and $\\epsilon>0$, any two type profile distributions $D, D'$ on type profile set $T$ and $T'$ accordingly($T$ and $T'$ might be different), any two valuation functions $\\{v_i(\\cdot,\\cdot)\\}_{i\\in[n]}$, $\\{v_i'(\\cdot,\\cdot)\\}_{i\\in[n]}$, assume for every $i$ there exists a coupling $\\hat{D_i}$ for $D_i$ and $D_i'$ such that $\\forall t_i\\in T_i,t_i'\\in T_i', \\hat{D_i}(t_i,t_i')>0$, $v_i'(t_i',S)\\geq v_i(t_i,S)$ holds for subset $S$. Here $\\hat{D_i}(t_i,t_i')$ is the coupling probability. Then for any BIC mechanism $M$ for valuation functions $\\{v_i(\\cdot,\\cdot)\\}_{i\\in[n]}$ with respect to $D$, there exists a BIC mechanism $M'$ for valuation functions $\\{v_i'(\\cdot,\\cdot)\\}_{i\\in[n]}$ with respect to distribution $D'$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{t_i'\\in T_i'}f_i'(t_i')\\cdot\\sum_{S: j\\in S}\\sigma'_{iS}(t_i')\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\t\t\\item $\\textsc{Rev}(M, v, D)\\leq$\\\\\n$~~~~\\frac{1}{1-\\eta}\\cdot{\\textsc{Rev}(M',v', D')}+\\frac{1}{\\eta}\\cdot \\sum_i\\sum_{t_i'\\in T_i'}\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]}\\hat{D_i}(t_i,t_i')\\cdot\\sigma'_{iS}(t_i')\\cdot \\left(v_i'(t_i', S)-v_i(t_i, S)\\right)+\\epsilon$.\n\t\\end{enumerate}\n\t$\\textsc{Rev}(M, v, D)$ is the revenue of the mechanism $M$ while the buyers' types are drawn from $D$ and their valuations are $v$ (similarly defined for $\\textsc{Rev}(M', v', D')$). $\\sigma'_{iS}(t_i')$ is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i'$ in mechanism $M'$ with respect to $D'$ and $\\sigma_{iS}(t_i)$ is the probability for the same event in mechanism $M$ with respect to $D$\n\\end{lemma}\n\n\\begin{prevproof}{lemma}{lem:relaxed valuation stronger}\nReaders who are familiar with the $\\epsilon$-BIC to BIC reduction~\\cite{HartlineKM11, BeiH11,DaskalakisW12} might have already realized that the problem here is quite similar. Our proof will follow essentially the same approach.\n\nFirst, we construct mechanism $M'$, which has two phases:\n\\vspace{.1in}\n\n\\noindent{\\bf Phase 1: Surrogate Sale}\n\\begin{enumerate}\n\t\\item For each buyer $i$, create $\\ell-1$ \\emph{replicas} sampled i.i.d. from $D_i'$ and $\\ell$ \\emph{surrogates} sampled i.i.d. from $D_i$. The value of $\\ell$ will be specified later.\n\t\\item Ask each buyer to report her type $t_i'$.\n\t\\item For each buyer $i$, create a weighted bipartite graph with the replicas and the bidder $i$ on the left and the surrogates on the right. The edge weight between a replica (bidder $i$) with type $r_i$ and a surrogate with type $s_i$ is the expected value for a bidder with valuation $v_i'(r_i,\\cdot)$ to receive bidder $i$'s interim allocation in $M$ when she reported $s_i$ as her type subtract the expected payment of bidder $i$ multiplied by $(1-\\eta)$. Formally, the weight is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i'(r_i,S) - (1-\\eta)p_i(s_i).$\n\t\\item Compute the VCG matching and prices on the bipartite graph created for each buyer $i$. If a replica (or bidder $i$) is unmatched in the VCG matching, match her to a random unmatched surrogate. The surrogate selected for buyer $i$ is whoever she is matched to.\n\\end{enumerate}\n\n\\vspace{.1in}\n\\noindent{\\bf Phase 2: Surrogate Competition}\n\\begin{enumerate}\n\t\\item Apply mechanism $M$ on the type profiles of the selected surrogates $\\vec{s}$. Let $M_i(\\vec{s})$ and $P_i(\\vec{s})$ be the corresponding allocated bundle and payment of buyer $i$.\n\t\\item If buyer $i$ is matched to her surrogate in the VCG matching, give her bundle $M_i(\\vec{s})$ and charge her $(1-\\eta)\\cdot P_i(\\vec{s})$ plus the VCG price. If buyer $i$ is not matched in the VCG matching, award them nothing and charge them nothing.\n\t\\end{enumerate}\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:same distribution}\n\tIf all buyers play $M'$ truthfully, then the distribution of types of the surrogate chosen for buyer $i$ is exactly $D_i$.\n\\end{lemma}\n\\begin{proof}\nIn the mechanism, first the buyer $i$'s type and $\\ell-1$ replicas are sampled i.i.d. from the distribution $D_i'$, while $\\ell$ surrogates are sampled i.i.d. from the distribution $D_i$. Now, imagine a different order of sampling. We first sample the $\\ell$ replicas and $\\ell$ surrogates, then we pick one replica to be buyer $i$ uniformly at random. The two different orders above provide exactly the same joint distribution over the replicas, surrogates and buyer $i$. So we only need to argue that in the second order of sampling, the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$. Note that the perfect matching (VCG matching plus the uniform random matching with the leftover replicas\/surrogates) only depends on the types but not the identity of the node (replica or buyer $i$). So we can decide who is buyer $i$ after we have decided the perfect matching. Since buyer $i$ is chosen uniformly at random among the replicas, the chosen surrogate is also uniformly at random. Clearly, the distribution of the types of a surrogate chosen uniformly at random is also $D_i$. The assumption that buyer $i$ is reporting truthfully is crucial, because otherwise the distribution of buyer $i$'s reported type will be different from the type of a replica, and in that case, we cannot use the second sampling order.\n\\end{proof}\n\n\\begin{lemma}\n\t$M'$ is a BIC mechanism with respect to valuation $v'$.\n\\end{lemma}\n\\begin{proof}\n\tWe need to argue that for every buyer $i$ reporting truthfully is a best response, if every other buyer is truthful. In the VCG mechanism, buyer $i$ faces a competition with the replicas to win a surrogate. If buyer $i$ has type $t_i'$, then her value for winning a surrogate with type $s_i$ in the VCG mechanism is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i'(t_i',S) - (1-\\eta)p_i(s_i)$ due to Lemma~\\ref{lem:same distribution}. Clearly, if buyer $i$ reports truthfully, the weights on the edges between her and all the surrogates will be exactly her value for winning those surrogates. Since buyer $i$ is in a VCG mechanism, reporting the true edge weights is a dominant strategy for her, therefore reporting truthfully is also a best response for her assuming the other buyers are truthful.\n\t\\end{proof}\n\t\n\\begin{lemma}\n\tFor any $i$ and $j$, $\\sum_{t_i'\\in T_i'}f_i'(t_i')\\cdot\\sum_{S: j\\in S} \\sigma'_{iS}(t_i')\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe LHS is the ex-ante probability for buyer $i$ to win item $j$ in $M'$, and the RHS is the corresponding probability in $M$. By Lemma~\\ref{lem:same distribution}, we know the surrogate selected by buyer $i$ is participating in $M$ against all other surrogates whose types are drawn from $D_{-i}$. Therefore, the ex-ante probability for the surrogate chosen by buyer $i$ to win item $j$ is the same as RHS. Clearly, this surrogate's ex-ante probability for winning any item should be at least as large as the ex-ante probability for $i$ to win the item in $M'$.\n\t\\end{proof}\n\t\nNext, we want to compare $\\textsc{Rev}(M',v', D')$ with $\\textsc{Rev}(M,v, D)$. The following simple Lemma relates both quantities to the expected prices charged to the surrogates by mechanism $M$. As in the proof of Lemma~\\ref{lem:same distribution}, we change the order of the sampling. We first sample $\\ell$ replicas and $\\ell$ surrogates then select a replica uniformly at random to be buyer $i$.\nLet $s_i^{k}\\in T_i$ and $r_i^{k}\\in T_i'$ be the type of the $k$-th surrogate and replica, $\\bold{s_i}= (s_i^{1},\\ldots, s_i^{\\ell})$, $\\bold{r_i}=(r_i^{1},\\ldots, r_i^{\\ell})$ and $V(\\bold{s_i},\\bold{r_i})$ be the VCG matching between surrogates and replicas with types $\\bold{s_i}$ and $\\bold{r_i}$. \t\n\\begin{lemma}\\label{lem:revenue by surrogates}\nFor every buyer $i$, her expected payments in $M'$ is at least $$(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right],$$ and her expected payments in $M$ is $${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right].$$\n\\end{lemma}\n\\begin{proof}\n\tThe revenue of $M'$ contains two parts -- the prices paid by the chosen surrogates and the revenue of the VCG mechanism. Let's compute the first part. For buyer $i$ and each realization of $\\bold{r_i}$ and $\\bold{s_i}$ only when the buyer $i$'s chosen surrogate is in $ V(\\bold{s_i},\\bold{r_i})$, $i$ pays the surrogate price. Since each surrogate is selected with probability $1\/\\ell$, the expected surrogate price paid by buyer $i$ is exactly $(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]$. Since the VCG payments are nonnegative, we have proved our first statement.\n\t\n\tThe expected payment from buyer $i$ in $M$ is ${\\mathbb{E}}_{t_i\\sim D_i}\\left[p_i(t_i)\\right]$. Since all $s_i^k$ is drawn from $D_i$, this is exactly the same as ${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]$.\n\\end{proof}\n\nIf the VCG matching is always perfect, then Lemma~\\ref{lem:revenue by surrogates} already shows that the revenue of $M'$ is at least $(1-\\eta)$ fraction of the revenue of $M$. But since the VCG matching may not be perfect, we need to show that the total expected price from surrogates who are not in the VCG matching is small. We prove this in two steps. First, we consider a different type of matching $X(\\bold{s_i},\\bold{r_i})$ -- a maximal matching that only matches replicas and surrogates that have the same type, and show that the expected cardinality of $X(\\bold{s_i},\\bold{r_i})$ is close to $\\ell$. Then we argue that for any realization $\\bold{r_i}$ and $\\bold{s_i}$ the total payments from surrogates that are in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$ is small.\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:equal type matching}\nFor every buyer $i$, the expected cardinality of a maximal matching that only matches replicas and surrogates with the same type is at least $\\ell-\\sqrt{|T_i|\\cdot \\ell}$.\n\\end{lemma}\n\n\nThe proof can be found in Hartline et. al.~\\cite{HartlineKM11}.\n\\begin{corollary}\\label{cor:bound revenue by X}\nLet $\\mathcal{R} = \\max_{i,t_i\\in T_i}\\max_{S\\in[m]} v_i(t_i,S)$, then\n$${\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\geq {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]- \\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}.$$\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a IR mechanism when the buyers' valuations are $v$, $\\mathcal{R}\\geq p_i(t_i)$ for any buyer $i$ and any type $t_i$ of $i$. Our claim follows from Lemma~\\ref{lem:equal type matching}.\n\\end{proof}\n\nNow we implement the second step of our argument. The plan is to show the total prices from surrogates that are unmatched by going from $X(\\bold{s_i},\\bold{r_i})$ to $V(\\bold{s_i},\\bold{r_i})$. For any $\\bold{s_i},\\bold{r_i}$, $V(\\bold{s_i},\\bold{r_i})\\cup X(\\bold{s_i},\\bold{r_i})$ can be decompose into a disjoint collection augmenting paths and cycles. If a surrogate is matched in $X(\\bold{s_i},\\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$, then it must be the starting point of an augmenting path. The following Lemma upper bounds the price of this surrogate.\n\\begin{lemma}\\label{lem:bounding the price for each augmenting path}\n\tFor any buyer $i$ and any realization of $\\bold{s_i}$ and $\\bold{r_i}$, let $P$ be an augmenting path that starts with a surrogate that is in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$. It has the form of either (a) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}}\\right)$ when the path ends with a surrogate, or (b)$\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}},r_i^{\\theta(k)}\\right)$ when the path ends with a replica, where $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$ and matched to $s_i^{\\rho(j+1)}$ (whenever $s_i^{\\rho(j+1)}$ exists) for any $j$.\n\t\\begin{align*}&\\sum_{s_i^{\\rho(j)}\\in P\\cap X(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)-\\sum_{s_i^{\\rho(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)\\leq\\\\\n\t &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{\\eta}\\cdot\\sum_{r_i^{\\theta(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'(r_i^{\\theta(j)},S)-v_i(r_i^{\\theta(j)},S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tSince $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$, $r_i^{\\theta(j)}$ must be equal to $s_i^{\\rho(j)}$. $M$ is a BIC mechanism when buyers valuations are $v$, therefore the expected utility for reporting the true type is better than lying. Hence, the following holds for all $j$:\n\t\\begin{equation}\\label{eq:BIC for M}\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\geq \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\t\n\t\\end{equation}\n\nThe VCG matching finds the maximum weight matching, so the total edge weights in path $P$ and $V(\\bold{s_i},\\bold{r_i})$ is at least as large as the total edge weights in path $P$ and $X(\\bold{s_i},\\bold{r_i})$. Mathematically, it is the following inequalities.\n\\begin{itemize}\n\\item If $P$ has format (a): \\begin{align}\\label{eq:VCG great a}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\\item If $P$ has format (b): \\begin{align}\\label{eq:VCG great b}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\n\\end{itemize}\n\nNext, we further relax the RHS of inequality~(\\ref{eq:VCG great a}) using inequality~(\\ref{eq:BIC for M}).\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great a})}\\\\\n\t\\geq& \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M})}\\\\\n\\end{align*}\nWe can obtain the following inequality by combining the relaxation above with the LHS of inequality~(\\ref{eq:VCG great a}) and rearrange the terms.\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq p_i\\left(s_i^{\\rho(1)}\\right)-p_i\\left(s_i^{\\rho(k)}\\right).$$\nThe inequality above is exactly the inequality in the statement of this Lemma when $P$ has format (a).\n\nSimilarly, we have the following relaxation when $P$ has format (b):\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great b})}\\\\\n\t\\geq& \\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M} and $M$ is IR)}\\\\\n\\end{align*}\nAgain, by combining the relaxation with the LHS of inequality~(\\ref{eq:VCG great b}), we can prove our claim when $P$ has format (b).\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq p_i\\left(s_i^{\\rho(1)}\\right).$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem: gap between X and V}\n\t\\begin{align*}\n&{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\leq\\\\\n&~~~~~~~~~~~~~~~{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]+\\frac{1}{\\eta}\\cdot\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tDue to Lemma~\\ref{lem:bounding the price for each augmenting path}, for any buyer $i$ and any realization of $\\bold{r_i}$ and $\\bold{s_i}$, we have\n\t$$\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}-\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\leq \\frac{1}{\\eta\\cdot\\ell}\\cdot\\sum_{s_i^k \\in V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{k}\\right)\\cdot \\left(v_i'(r_i^{\\omega(k)},S)-v_i(r_i^{\\omega(k)},S)\\right),$$ where $r_i^{\\omega(k)}$ is the replica that is matched to $s_i^k$ in $ V(\\bold{s_i},\\bold{r_i})$. If we take expectation over $\\bold{r_i}$ and $\\bold{s_i}$ on the RHS, the expectation means whenever mechanism $M'$ awards buyer $i$ (with type $t_i$) bundle $S$, $\\frac{1}{\\eta}\\cdot\\left(v_i'(t_i, S)-v_i(t_i, S)\\right)$ is contributed to the expectation. Therefore, the expectation of the RHS is the same as $$\\frac{1}{\\eta}\\cdot\\left(\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)\\right).$$ This completes the proof of the Lemma.\n\\end{proof}\n\nNow, we are ready to prove Lemma~\\ref{lem:relaxed valuation stronger}.\n\\begin{align*}\n\t&\\textsc{Rev}(M, v, D)\\\\\n\t=& \\sum_i E_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\\\\\n\t\\leq & \\sum_i\\left({\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right] +\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}\\right)~~\\text{(Corollary~\\ref{cor:bound revenue by X})}\\\\\n\t\\leq &\\sum_i {\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem: gap between X and V})}\\\\\n\t\\leq & \\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M',v',D)\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\n\\end{align*}\n\nSince $|T_i|$ and $\\cal{R}$ are finite numbers, we can take $\\ell$ to be sufficiently large, so that $\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} < \\epsilon\/(1-\\eta)$. Therefore, we finished the proof of Lemma~\\ref{lem:relaxed valuation stronger}.\n\\end{prevproof}\n\n\n\n\n\n\n\\section{Proof of Lemma~\\ref{lem:relaxed valuation}}\\label{sec:proof_relaxed_valuation}\nWe first prove some properties of $v^{(\\beta)}$, which will be useful for proving Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:relaxed larger}\n\tFor any $\\beta_i$, $t_i\\in T_i$ and $S\\in[m]$, $v_i^{(\\beta_i)}(t_i,S)\\geq v_i(t_i,S)$.\n\\end{lemma}\n\\begin{proof}\n\tThis follows from the fact that $v_i(t_i,\\cdot)$ is a subadditive function over bundles of items for all $t_i$.\n\\end{proof}\n\n\\begin{lemma}\n\tFor any $\\beta_i$ and $t_i\\in T_i$, $v_i^{(\\beta_i)}(t_i,\\cdot)$ is a monotone, subadditive function over the items.\n\\end{lemma}\n\\begin{proof}\nMonotonicity follows directly from the monotonicity of $v_i(t_i,\\cdot)$. We only argue subadditivity here. If $t_i$ belongs to $R_0^{(\\beta_i)}$, $v_i^{(\\beta_i)}(t_i,\\cdot)=v_i(t_i,\\cdot)$. So it is clearly a subadditive function. If $t_i$ belongs to $R_j^{(\\beta_i)}$ for some $j>0$ and $j$ is not in either $U$ or $V$, then clearly $v_i^{(\\beta_i)}(t_i,U\\cup V)\\leq v_i^{(\\beta_i)}(t_i,U)+v_i^{(\\beta_i)}(t_i,V)$. If $j$ is in one of the two sets, without loss of generality let's assume it is in $U$. Then $v_i^{(\\beta_i)}(t_i,U)+v_i^{(\\beta_i)}(t_i,V)=v_i(t_i,U\\backslash\\{j\\})+V_i(t_{ij})+v_i(t_i,V)\\geq v_i(t_i,V\\cup (U\\backslash\\{j\\}))+V_i(t_{ij})= v_i^{(\\beta_i)}(t_i,U\\cup V)$.\n\\end{proof}\n\n\nHere we prove a stronger version of Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:relaxed valuation stronger}\nFor any $\\beta$, any absolute constant $\\eta\\in(0,1)$ and any BIC mechanism $M$ for subadditive valuations $\\{v_i(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, there exists a BIC mechanism $M^{(\\beta)}$ for valuations $\\{v_i^{(\\beta_i)}(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\t\t\\item $\\textsc{Rev}(M, v, D)\\leq$\\\\\n$~~~~\\frac{1}{1-\\eta}\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+\\frac{1}{\\eta}\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$.\n\t\\end{enumerate}\n\t$\\textsc{Rev}(M, v, D)$ (or $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$) is the revenue of the mechanism $M$ (or $M^{(\\beta)}$) while the buyers' types are drawn from $D$ and buyer $i$'s valuation is $v_i(t_i,\\cdot)$ (or $v_i^{(\\beta_i)}(t_i,\\cdot)$). $\\sigma_{iS}(t_i)$ (or $\\sigma^{(\\beta)}_{iS}(t_i)$) is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i$ in mechanism $M$ (or $M^{(\\beta)}$).\n\\end{lemma}\n\n\n\\begin{prevproof}{lemma}{lem:relaxed valuation stronger}\nReaders who are familiar with the $\\epsilon$-BIC to BIC reduction~\\cite{HartlineKM11, BeiH11,DaskalakisW12} might have already realized that the problem here is quite similar. Our proof will follow essentially the same approach.\n\nFirst, we construct mechanism $M^{(\\beta)}$, which has two phases:\n\\vspace{.1in}\n\n\\noindent{\\bf Phase 1: Surrogate Sale}\n\\begin{enumerate}\n\t\\item For each buyer $i$, create $\\ell-1$ \\emph{replicas} and $\\ell$ \\emph{surrogates} sampled i.i.d. from $D_i$. The value of $\\ell$ will be specified later.\n\t\\item Ask each buyer to report her type $t_i$.\n\t\\item For each buyer $i$, create a weighted bipartite graph with the replicas and the buyer $i$ on the left and the surrogates on the right. The edge weight between a replica (or buyer $i$) with type $r_i$ and a surrogate with type $s_i$ is the expected value for a bidder with valuation $v_i^{(\\beta_i)}(r_i,\\cdot)$ to receive buyer $i$'s interim allocation in $M$ when she reported $s_i$ as her type subtract the interim payment of buyer $i$ multiplied by $(1-\\eta)$. Formally, the weight is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i^{(\\beta_i)}(r_i,S) - (1-\\eta)p_i(s_i)$, where $p_i(s_i)$ is the interim payment for buyer $i$ if she reported $s_i$.\n\t\\item Compute the VCG matching and prices on the bipartite graph created for each buyer $i$. If a replica (or bidder $i$) is unmatched in the VCG matching, match her to a random unmatched surrogate. The surrogate selected for buyer $i$ is whoever she is matched to.\n\\end{enumerate}\n\n\\vspace{.1in}\n\\noindent{\\bf Phase 2: Surrogate Competition}\n\\begin{enumerate}\n\t\\item Apply mechanism $M$ on the type profiles of the selected surrogates $\\vec{s}$. Let $M_i(\\vec{s})$ and $P_i(\\vec{s})$ be the corresponding allocated bundle and payment of buyer $i$.\n\t\\item If buyer $i$ is matched to her surrogate in the VCG matching, give her bundle $M_i(\\vec{s})$ and charge her $(1-\\eta)\\cdot P_i(\\vec{s})$ plus the VCG price. If buyer $i$ is not matched in the VCG matching, award them nothing and charge them nothing.\n\t\\end{enumerate}\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:same distribution}\n\tIf all buyers play $M^{(\\beta)}$ truthfully, then the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$.\n\\end{lemma}\n\\begin{proof}\nIn the mechanism, first the buyer $i$'s type is sampled from the distribution, then we sampled $\\ell-1$ replicas and $\\ell$ surrogates i.i.d. from the same distribution. Now, imagine a different order of sampling. We first sample the $\\ell$ replicas and $\\ell$ surrogates, then we pick one replica to be buyer $i$ uniformly at random. The two different orders above provide exactly the same joint distribution over the replicas, surrogates and buyer $i$. So we only need to argue that in the second order of sampling, the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$. Note that the perfect matching (VCG matching plus the uniform random matching with the leftover replicas\/surrogates) only depends on the types but not the identity of the node (replica or buyer $i$). So we can decide who is buyer $i$ after we have decided the perfect matching. Since buyer $i$ is chosen uniformly at random among the replicas, the chosen surrogate is also uniformly at random. Clearly, the distribution of the types of a surrogate chosen uniformly at random is also $D_i$. The assumption that buyer $i$ is reporting truthfully is crucial, because otherwise the distribution of buyer $i$'s reported type will be different from the type of a replica, and in that case, we cannot use the second sampling order.\n\\end{proof}\n\n\\begin{lemma}\n\t$M^{(\\beta)}$ is a BIC mechanism with respect to valuation $v^{(\\beta)}$.\n\\end{lemma}\n\\begin{proof}\n\tWe need to argue that for every buyer $i$ reporting truthfully is a best response, if every other buyer is truthful. In the VCG mechanism, buyer $i$ faces a competition with the replicas to win a surrogate. If buyer $i$ has type $t_i$, then her value for winning a surrogate with type $s_i$ in the VCG mechanism is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i^{(\\beta_i)}(t_i,S) - (1-\\eta)p_i(s_i)$ due to Lemma~\\ref{lem:same distribution}. Clearly, if buyer $i$ reports truthfully, the weights on the edges between her and all the surrogates will be exactly her value for winning those surrogates. Since buyer $i$ is in a VCG mechanism, reporting the true edge weights is a dominant strategy for her, therefore reporting truthfully is also a best response for her assuming the other buyers are truthful. It is critical that the other buyers are reporting truthfully, otherwise we cannot invoke Lemma~\\ref{lem:same distribution} and buyer $i$'s value for winning a surrogate with type $s_i$ may be different from the weight on the corresponding edge.\n\t\\end{proof}\n\t\n\\begin{lemma}\n\tFor any $i$ and $j$, $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe LHS is the ex-ante probability for buyer $i$ to win item $j$ in $M^{(\\beta)}$, and the RHS is the corresponding probability in $M$. By Lemma~\\ref{lem:same distribution}, we know the surrogate selected by buyer $i$ is participating in $M$ against all other surrogates whose types are drawn from $D_{-i}$. Therefore, the ex-ante probability for the surrogate chosen by buyer $i$ to win item $j$ is the same as RHS. Clearly, the chosen surrogate's ex-ante probability for winning any item should be at least as large as the ex-ante probability for buyer $i$ to win the item in $M^{(\\beta)}$.\n\t\\end{proof}\n\t\nNext, we want to compare $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$ with $\\textsc{Rev}(M,v, D)$. The following simple Lemma relates both quantities to the expected prices charged to the surrogates by mechanism $M$. As in the proof of Lemma~\\ref{lem:same distribution}, we change the order of the sampling. We first sample $\\ell$ replicas and $\\ell$ surrogates then select a replica uniformly at random to be buyer $i$.\nLet $s_i^{k}$ and $r_i^{k}$ be the type of the $k$-th surrogate and replica, $\\bold{s_i}= (s_i^{1},\\ldots, s_i^{\\ell})$, $\\bold{r_i}=(r_i^{1},\\ldots, r_i^{\\ell})$ and $V(\\bold{s_i},\\bold{r_i})$ be the VCG matching between surrogates and replicas with types $\\bold{s_i}$ and $\\bold{r_i}$. We will slightly abuse notation by using $s_i^k$ (or $r_i^j$) $\\in V(\\bold{s_i},\\bold{r_i})$ to denote that $s_i^k$ (or $r_i^j$) is matched in the VCG matching $V(\\bold{s_i},\\bold{r_i})$.\n\\begin{lemma}\\label{lem:revenue by surrogates}\nFor every buyer $i$, her expected payments in $M^{(\\beta)}$ is at least $$(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right],$$ and her expected payments in $M$ is $${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right].$$\n\\end{lemma}\n\\begin{proof}\n\tThe revenue of $M^{(\\beta)}$ contains two parts -- the prices paid by the chosen surrogates and the revenue of the VCG mechanism. Let's compute the first part. For buyer $i$ and each realization of $\\bold{r_i}$ and $\\bold{s_i}$ only when the buyer $i$'s chosen surrogate is in $ V(\\bold{s_i},\\bold{r_i})$, she pays the surrogate price. Since each surrogate is selected with probability $1\/\\ell$, the expected surrogate price paid by buyer $i$ is exactly $(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]$. Since the VCG payments are nonnegative, we have proved our first statement.\n\t\n\tThe expected payment from buyer $i$ in $M$ is ${\\mathbb{E}}_{t_i\\sim D_i}\\left[p_i(t_i)\\right]$. Since all $s_i^k$ is drawn from $D_i$, this is exactly the same as ${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]$.\n\\end{proof}\n\nIf the VCG matching is always perfect, then Lemma~\\ref{lem:revenue by surrogates} already shows that the revenue of $M^{(\\beta)}$ is at least $(1-\\eta)$ fraction of the revenue of $M$. But since the VCG matching may not be perfect, we need to show that the total expected price from surrogates who are not in the VCG matching is small. We prove this in two steps. First, we consider another matching $X(\\bold{s_i},\\bold{r_i})$ -- a maximal matching that only matches replicas and surrogates that have the same type, and show that the expected cardinality of $X(\\bold{s_i},\\bold{r_i})$ is close to $\\ell$. Then we argue that for any realization $\\bold{r_i}$ and $\\bold{s_i}$ the total payments from surrogates that are in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$ is small.\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:equal type matching}\nFor every buyer $i$, the expected cardinality of a maximal matching that only matches replicas and surrogates with the same type is at least $\\ell-\\sqrt{|T_i|\\cdot \\ell}$.\n\\end{lemma}\n\n\nThe proof can be found in Hartline et al.~\\cite{HartlineKM11}.\n\\begin{corollary}\\label{cor:bound revenue by X}\nLet $\\mathcal{R} = \\max_{i,t_i\\in T_i}\\max_{S\\in[m]} v_i(t_i,S)$, then\n$${\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\geq {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]- \\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}.$$\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a IR mechanism when the buyers' valuations are $v$, $\\mathcal{R}\\geq p_i(t_i)$ for any buyer $i$ and any type $t_i$ of $i$. Our claim follows from Lemma~\\ref{lem:equal type matching}.\n\\end{proof}\n\nNow we implement the second step of our argument. The plan is to show the total prices from surrogates that are unmatched by going from $X(\\bold{s_i},\\bold{r_i})$ to $V(\\bold{s_i},\\bold{r_i})$. For any $\\bold{s_i},\\bold{r_i}$, $V(\\bold{s_i},\\bold{r_i})\\cup X(\\bold{s_i},\\bold{r_i})$ can be decompose into a disjoint collection augmenting paths and cycles. If a surrogate is matched in $X(\\bold{s_i},\\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$, then it must be the starting point of an augmenting path. The following Lemma upper bounds the price of this surrogate.\n\\begin{lemma}[Adapted from~\\cite{DaskalakisW12}]\\label{lem:bounding the price for each augmenting path}\n\tFor any buyer $i$ and any realization of $\\bold{s_i}$ and $\\bold{r_i}$, let $P$ be an augmenting path that starts with a surrogate that is matched in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$. It has the form of either (a) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}}\\right)$ when the path ends with a surrogate, or\\\\ (b) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}},r_i^{\\theta(k)}\\right)$ when the path ends with a replica, where $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$ and matched to $s_i^{\\rho(j+1)}$ in $V(\\bold{s_i},\\bold{r_i})$ (whenever $s_i^{\\rho(j+1)}$ exists) for any $j$.\n\t\\begin{align*}&\\sum_{s_i^{\\rho(j)}\\in P\\cap X(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)-\\sum_{s_i^{\\rho(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)\\leq\\\\\n\t &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S \\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}(r_i^{\\theta(j)},S)-v_i(r_i^{\\theta(j)},S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tSince $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$, $r_i^{\\theta(j)}$ must be equal to $s_i^{\\rho(j)}$. $M$ is a BIC mechanism when buyers valuations are $v$, therefore the expected utility for reporting the true type is better than lying. Hence, the following holds for all $j$:\n\t\\begin{equation}\\label{eq:BIC for M}\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\geq \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\t\n\t\\end{equation}\n\nThe VCG matching finds the maximum weight matching, so the total edge weights in path $P \\cap V(\\bold{s_i},\\bold{r_i})$ is at least as large as the total edge weights in path $P\\cap X(\\bold{s_i},\\bold{r_i})$. Mathematically, it is the following inequalities.\n\\begin{itemize}\n\\item If $P$ has format (a): \\begin{align}\\label{eq:VCG great a}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\\item If $P$ has format (b): \\begin{align}\\label{eq:VCG great b}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\n\\end{itemize}\n\nNext, we further relax the RHS of inequality~(\\ref{eq:VCG great a}) using inequality~(\\ref{eq:BIC for M}).\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great a})}\\\\\n\t\\geq& \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M})}\\\\\n\\end{align*}\nWe can obtain the following inequality by combining the relaxation above with the LHS of inequality~(\\ref{eq:VCG great a}) and rearrange the terms.\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq p_i\\left(s_i^{\\rho(1)}\\right)-p_i\\left(s_i^{\\rho(k)}\\right).$$\nThe inequality above is exactly the inequality in the statement of this Lemma when $P$ has format (a).\n\nSimilarly, we have the following relaxation when $P$ has format (b):\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great b})}\\\\\n\t\\geq& \\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M} and $M$ is IR)}\\\\\n\\end{align*}\nAgain, by combining the relaxation with the LHS of inequality~(\\ref{eq:VCG great b}), we can prove our claim when $P$ has format (b).\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq p_i\\left(s_i^{\\rho(1)}\\right).$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem: gap between X and V}\nFor any $\\beta$,\n\t\\begin{align*}\n&{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\leq\\\\\n&~~~~~~~~~~~~~~~{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]+\\frac{1}{\\eta}\\cdot\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tDue to Lemma~\\ref{lem:bounding the price for each augmenting path}, for any buyer $i$ and any realization of $\\bold{r_i}$ and $\\bold{s_i}$, we have\n\t$$\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}-\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\leq \\frac{1}{\\eta\\cdot\\ell}\\cdot\\sum_{s_i^k \\in V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{k}\\right)\\cdot \\left(v_i^{(\\beta_i)}(r_i^{\\omega(k)},S)-v_i(r_i^{\\omega(k)},S)\\right),$$ where $r_i^{\\omega(k)}$ is the replica that is matched to $s_i^k$ in $ V(\\bold{s_i},\\bold{r_i})$. If we take expectation over $\\bold{r_i}$ and $\\bold{s_i}$ on the RHS, the expectation means whenever mechanism $M^{(\\beta)}$ awards buyer $i$ (with type $t_i$) bundle $S$, $\\frac{1}{\\eta}\\cdot\\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$ is contributed to the expectation. Therefore, the expectation of the RHS is the same as $$\\frac{1}{\\eta}\\cdot\\left(\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\right).$$ This completes the proof of the Lemma.\n\\end{proof}\n\nNow, we are ready to prove Lemma~\\ref{lem:relaxed valuation stronger}.\n\\begin{align*}\n\t&\\textsc{Rev}(M, v, D)\\\\\n\t=& \\sum_i {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\\\\\n\t\\leq & \\sum_i\\left({\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right] +\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}\\right)~~\\text{(Corollary~\\ref{cor:bound revenue by X})}\\\\\n\t\\leq &\\sum_i {\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem: gap between X and V})}\\\\\n\t\\leq & \\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\n\\end{align*}\n\nSince $|T_i|$ and $\\mathcal{R}$ are finite numbers, we can take $\\ell$ to be sufficiently large, so that $\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} < \\epsilon$ for any $\\epsilon$. Let $P^{(\\beta)}$ be the set of all BIC mechanisms that satisfy the first condition in Lemma~\\ref{lem:relaxed valuation stronger}. Clearly, $P^{(\\beta)}$ is a compact set and contains all $M^{(\\beta)}$ we constructed (by choosing different values for $\\ell$). Notice that both $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)$ and $\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$ are linear functions over the allocation\/price rules of mechanism $M^{(\\beta)}$. Therefore, \\begin{align*}\n \t&\\textsc{Rev}(M, v, D)\\\\\n \t\\leq &\\max_{M^{(\\beta)}\\in P^{(\\beta)}} \\left(\\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\right).\n \\end{align*}\nThis completes the proof of Lemma~\\ref{lem:relaxed valuation stronger}.\n\\end{prevproof}\n\n\\section{Paper Organization}\\label{sec:roadmap}\nIn this section, we provide the roadmap to our paper. In Section~\\ref{sec:duality}, we review the Duality framework of~\\cite{CaiDW16}.\n\nIn Section~\\ref{sec:flow}, we derive an upper bound of the optimal revenue for subadditive bidders by combining the duality framework with our new techniques, i.e. valuation relaxation and adaptive dual variables. Our main result in this section, Theorem~\\ref{thm:revenue upperbound for subadditive}, shows that the revenue can be upper bounded by two terms -- $\\textsc{Non-Favorite}$ and $\\textsc{Single}$ defined in Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism}.\n\nIn Section~\\ref{sec:single}, we use the single bidder case to familiarize the readers with some basic ideas and techniques used to bound $\\textsc{Single}$ and $\\textsc{Non-Favorite}$. The main result of this section, Theorem~\\ref{thm:single}, shows that the optimal revenue for a single subadditive bidder is upper bounded by $24\\textsc{SRev}$ and $16\\textsc{BRev}$.\n\nSection~\\ref{sec:multi} contains the main result of this paper. We show how to upper bound the optimal revenue for XOS (or subadditive) bidders with a constant number of (or $O(\\log m)$) $\\textsc{PostRev}$ (the optimal revenue obtainable by an RSPM) and $\\textsc{APostEnRev}$ ((the optimal revenue obtainable by an ASPE). In particular, $\\textsc{Single}$ can be upper bounded by the optimal revenue $\\textsc{OPT}^{\\textsc{Copies-UD}}$ in the copies setting which is again upper bounded by $6\\textsc{PostRev}$. We further decompose $\\textsc{Non-Favorite}$ into two terms $\\textsc{Tail}$ and $\\textsc{Core}$, and show how to bound $\\textsc{Tail}$ in Section~\\ref{subsection:tail} and how to bound $\\textsc{Core}$ in Section~\\ref{subsection:core}.\n\n\n\n\\section{Warm Up: Single Bidder}\\label{sec:single}\nTo warm up, we first study the case where there is a single subadditive buyer and show how to improve the approximation ratio from $338$ to $40$. Since there is only one buyer, we will drop the subscript $i$ in the notations. As specified in Section~\\ref{sec:choice of beta}, we use a $\\beta$ that satisfies both properties in Lemma~\\ref{lem:requirement for beta}. For a single buyer, we can simply set $\\beta_{j}$ to be $0$ for all $j$. We use $\\textsc{Single}(M), \\textsc{Non-Favorite}(M)$ in the following proof to denote the corresponding terms in Theorem~\\ref{thm:revenue upperbound for subadditive} for $\\beta=\\textbf{0}$. Notice $R_0^{(\\textbf{0})}=\\emptyset$. Theorem~\\ref{thm:single} shows that the optimal revenue is within a constant factor of the better of selling separately and grand bundling.\n\n\\begin{theorem}\\label{thm:single}\nFor a single buyer whose valuation distribution is subadditive over independent items, \n\\[\\textsc{Rev}(M,v,D)\\leq 24\\cdot\\textsc{SRev}+16\\cdot\\textsc{BRev}\\]\nfor any BIC mechanism $M$.\n\\end{theorem}\n\nRecall that the revenue for mechanism $M$ is upper bounded by $4\\cdot \\textsc{Non-Favorite}(M)+2\\cdot\\textsc{Single}(M)$ (Theorem~\\ref{thm:revenue upperbound for subadditive}). We first upper bound $\\textsc{Single}(M)$ by $\\textsc{OPT}^{\\textsc{Copies-UD}}$. Since $\\sigma^{(\\beta)}_{S}(t)$ is a feasible allocation in the original setting, $ \\mathds{1}[t\\in R_j^{(\\beta)}]\\cdot\\pi^{(\\beta)}_{j}(t)$ with $\\pi^{(\\beta)}_j(t)=\\sum_{S:j\\in S}\\sigma^{(\\beta)}_{S}(t)$ is a feasible allocation in the copies setting, and therefore $\\textsc{Single}(M)$ is the Myerson Virtual Welfare of a certain allocation in the copies setting, which is upper bounded by $\\textsc{OPT}^{\\textsc{Copies-UD}}$. By~\\cite{ChawlaHMS10}, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is at most $2\\cdot\\textsc{SRev}$.\n\\begin{lemma}\\label{lem:single-single}\nFor any BIC mechanism $M$, $\\textsc{Single} (M)\\leq \\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 2\\cdot\\textsc{SRev}.$\n\\end{lemma}\n\nWe decompose $\\textsc{Non-Favorite}(M)$ into two terms $\\textsc{Core}(M)$ and $\\textsc{Tail}(M)$, and bound them separately. For every $t\\in T$, define $\\mathcal{C}(t)=\\{j:V(t_j)< c\\}$, $\\mathcal{T}(t)=[m]\\backslash \\mathcal{C}(t)$. Here the threshold $c$ is chosen as\n\\begin{equation}\\label{equ:single-def of c}\nc:=\\inf\\left\\{x\\geq 0:\\ \\sum_j \\Pr_{t_j}\\left[V(t_j)\\geq x\\right]\\leq 2\\right\\}.\n\\end{equation}\nSince $v(t,\\cdot)$ is subadditive for all $t\\in T$ , we have for every $S\\subseteq [m]$, $v(t,S)\\leq v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$. {We decompose $\\textsc{Non-Favorite}(M)$ based on the inequality above.} Proof of Lemma~\\ref{lem:single decomposition} can be found in Appendix~\\ref{sec:single_appx}.\n\n\n\n\n\\begin{lemma}\\label{lem:single decomposition}\n\\begin{align*}\t\\textsc{Non-Favorite}(M)\n\\leq &\t\\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{S}^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))~~~~~~~~~\\quad(\\textsc{Core}(M))\\\\\n+&\\sum_j\\sum_{t_{j}:V(t_{j})\\geq c}f_{j}(t_{j})\\cdot V(t_{j})\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]~~~~\\quad(\\textsc{Tail}(M))\n\\end{align*}\n\\end{lemma}\n\n\nUsing the definition of $c$ and $\\textsc{SRev}$, we can upper bound $\\textsc{Tail}(M)$ with a similar argument as in~\\cite{CaiDW16}. \n\\begin{lemma}\\label{lem:single-tail}\nFor any BIC mechanism $M$, $\\textsc{Tail}(M)\\leq 2\\cdot\\textsc{SRev}$.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\textsc{Tail}(M)=\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$, for each type $t_j\\in T_j$ consider the mechanism that posts the same price $V(t_j)$ for each item but only allows the buyer to purchase at most one. Notice if there exists $k\\not= j$ such that $V(t_k)\\geq V(t_j)$, the mechanism is guaranteed to sell one item obtaining revenue $V(t_j)$. Thus, the revenue obtained by this mechanism\nis at least $V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$. By definition, this is no more than $\\textsc{SRev}$.\n\n\\begin{equation}\\label{equ:single-tail}\n\\textsc{Tail}(M)\\leq \\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot \\textsc{SRev}\\notshow{\\leq}{=} 2\\cdot \\textsc{SRev}\n\\end{equation}\n\n{\nThe last equality is because by the definition of $c$,\n\\noindent$\\sum_j \\Pr_{t_j}[V(t_j)\\geq c]=2$.\\footnote{This clearly holds if $V(t_j)$ is drawn from a continuous distribution. When $V(t_j)$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.}\n}\n\\end{proof}\n\n\nThe $\\textsc{Core}(M)$ is upper bounded by ${\\mathbb{E}}_{t}[v'(t,[m])]$ where $v'(t,S)$\n$= v(t,S\\cap \\mathcal{C}(t))$. We argue that $v'(t,\\cdot)$ is drawn from a distribution that is subadditive over independent items and $v'(\\cdot,\\cdot)$ is $c$-Lipschitz (see Definition~\\ref{def:Lipschitz}). Using a concentration bound by Schechtman~\\cite{Schechtman2003concentration}, we show ${\\mathbb{E}}_{t}[v'(t,[m])]$ is upper bounded by the median of random variable $v'(t,[m])$ and $c$, which are upper bounded by $\\textsc{BRev}$ and $\\textsc{SRev}$ respectively.\n\\begin{lemma}\\label{lem:single-core}\nFor any BIC mechanism $M$, $\\textsc{Core}(M) \\leq 3\\cdot\\textsc{SRev}+4\\cdot\\textsc{BRev}$.\n\\end{lemma}\n\nRecall that\n\\begin{equation}\n\\textsc{Core}(M)=\\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq [m]}\\sigma_S^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))\n\\end{equation}\n\nWe will bound $\\textsc{Core}(M)$ with a concentration inequality from~\\cite{Schechtman2003concentration}. It requires the following definition:\n\n\\begin{definition}\\label{def:Lipschitz}\nA function $v(\\cdot,\\cdot)$ is \\textbf{$a$-Lipschitz} if for any type $t,t'\\in T$, and set $X,Y\\subseteq [m]$,\n$$\\left|v(t,X)-v(t',Y)\\right|\\leq a\\cdot \\left(\\left|X\\Delta Y\\right|+\\left|\\{j\\in X\\cap Y:t_j\\not=t_j'\\}\\right|\\right),$$ where $X\\Delta Y=\\left(X\\backslash Y\\right)\\cup \\left(Y\\backslash X\\right)$ is the symmetric difference between $X$ and $Y$.\n\\end{definition}\n\nDefine a new valuation function for the bidder as $v'(t,S)=v(t,S\\cap \\mathcal{C}(t))$, for all $t\\in T$ and $S\\subseteq [m]$. Then $v'(\\cdot,\\cdot)$ is $c-$ Lipschitz, and when $t$ is drawn from the product distribution $D=\\prod_j D_j$, $v'(t,\\cdot)$ remains to be a valuation drawn from a distribution that is subadditive over independent items. See Appendix~\\ref{sec:single_appx} for the proof of Lemma~\\ref{lem:single subadditive} and Lemma~\\ref{lem:single Lipschitz}.\n\n\\begin{lemma}\\label{lem:single subadditive}\nFor all $t\\in T$, $v'(t,\\cdot)$ satisfies monotonicity, subadditivity and no externalities defined in Definition~\\ref{def:subadditive independent}.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:single Lipschitz}\n$v'(\\cdot,\\cdot)$ is $c-$Lipschitz.\n\\end{lemma}\n\nNext, we apply the following concentration inequality to derive Corollary~\\ref{corollary:concentrate}, which is useful to analyze the $\\textsc{Core}(M)$. \n\n\\begin{lemma}~\\cite{Schechtman2003concentration}\\label{lem:schechtman}\nLet $g(t,\\cdot)$ with $t\\sim D=\\prod_j D_j$ be a function drawn from a distribution that is subadditive over independent items of ground set $I$. If $g(\\cdot,\\cdot)$ is $c$-Lipschitz, then for all $a>0, k\\in \\{1,2,...,|I|\\}, q\\in \\mathbb{N}$,\n$$\\Pr_t[g(t,I)\\geq (q+1)a+k\\cdot c]\\leq \\Pr_t[g(t,I)\\leq a]^{-q}q^{-k}.$$\n\\end{lemma}\n\n\\begin{corollary}\\label{corollary:concentrate}\nLet $g(t,\\cdot)$ with $t\\sim D=\\prod_j D_j$ be a function drawn from a distribution that is subadditive over independent items of ground set $I$. If $g(\\cdot,\\cdot)$ is $c$-Lipschitz, then if we let $a$ be the median of the value of the grand bundle $g(t,I)$, i.e. $a=\\inf\\left\\{x\\geq 0: \\Pr_t[g(t,I)\\leq x]\\geq \\frac{1}{2}\\right\\}$,\n$$\\mathds{E}_t[g(t,I)]\\leq 2a+\\frac{5c}{2}.$$\n\\end{corollary}\n\n\\begin{proof}\nLet $Y$ be $g(t,I)$. If we apply Lemma~\\ref{lem:schechtman} to the case where $a$ is the median and $q=2$, we have\n\n\\begin{align*}\n\\Pr_t[Y\\geq 3a]\\cdot{\\mathbb{E}}_{t}[Y|Y\\geq 3a]&= 3a\\cdot \\Pr_t[Y\\geq 3a]+\\int_{y=0}^{\\infty}\\Pr_t[Y\\geq 3a+y]dy\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+c\\cdot\\sum_{k=0}^{|I|} \\Pr_t[Y\\geq 3a+k\\cdot c] \\quad(Y\\leq |I|\\cdot c)\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+c\\cdot \\sum_{k=0}^2 \\Pr_t[Y > a]+ c\\cdot\\sum_{k=3}^{|I|} 4\\cdot 2^{-k}\\quad(\\text{Lemma~\\ref{lem:schechtman}})\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+\\frac{5}{2}c\\\\\n\\end{align*}\n\nWith the inequality above, we can upper bound the expected value of $Y$.\n\\begin{align*}\n{\\mathbb{E}}_{t}[Y]&\\leq a\\cdot \\Pr_t[Y\\leq a]+3a\\cdot \\Pr_{t}[Y\\in (a,3a)]+\\Pr_t[Y\\geq 3a]\\cdot{\\mathbb{E}}_{t}[Y|Y\\geq 3a]\\\\\n&\\leq a\\cdot \\Pr_t[Y\\leq a]+3a\\cdot \\Pr_{t}[Y\\in (a,3a)]+3a\\cdot \\Pr_t[Y\\geq 3a]+\\frac{5}{2}c\\\\\n&= a+2a\\cdot \\Pr_{t}[Y>a]+\\frac{5}{2}c\\\\\n&\\leq 2a+\\frac{5}{2}c\n\\end{align*}\n\\end{proof}\n\n\nNow, we are ready to prove Lemma~\\ref{lem:single-core}.\n\n\\begin{prevproof}{Lemma}{lem:single-core}\nLet $\\delta$ be the median of $v'(t,[m])$ when $t$ is sampled from distribution $D$. Now consider the mechanism that sells the grand bundle with price $\\delta$. Notice that the bidder's valuation for the grand bundle is $v(t,[m])\\geq v'(t,[m])$. Thus with probability at least $\\frac{1}{2}$,\n the bidder purchases the bundle. Thus, $\\textsc{BRev}\\geq \\frac{1}{2}\\delta$.\n\nAccording to Corollary~\\ref{corollary:concentrate},\n\n\\begin{comment}\n\\begin{equation}\\label{equ:single-core}\n\\begin{aligned}\n\\textsc{Core}(M)&\\leq \\mathds{E}_{t\\sim D}[v'(t,[m])]\\leq 2\\delta+\\frac{5c}{2}\\\\\n&< 4\\cdot\\textsc{BRev}+3\\cdot\\textsc{SRev} \\text{(Lemma~\\ref{lem:single-bound for c}, Inequality~\\ref{equ:bound for delta})}\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\n\\begin{equation}\\label{equ:single-core-prev}\n\\textsc{Core}(M)\\leq \\mathds{E}_{t\\sim D}[v'(t,[m])]\\leq 2\\delta+\\frac{5c}{2}\n\\end{equation}\n\nIt remains to argue that the Lipchitz constant $c$ can be upper bounded using $\\textsc{SRev}$. Notice that by AM-GM Inequality,\n\\begin{align*}\n&\\Pr_t\\left[\\exists j\\in [m], V(t_j)\\geq c\\right]= 1-\\prod_{j}\\Pr_{t_j}[V(t_j)< c]\\\\\n\\geq& 1-(\\frac{\\sum_j \\Pr_{t_j}[V(t_j)< c]}{m})^m\n= 1-(1-\\frac{2}{m})^m\n\\geq 1-e^{-2}\n\\end{align*}\n\n\n\nConsider the mechanism that posts price $c$ for each item but only allow the buyer to purchase one item. Then with probability at least $1-e^{-2}$, the mechanism sells one item obtaining expected revenue $(1-e^{-2})\\cdot c$. Thus $c\\leq \\frac{1}{1-e^{-2}}\\cdot\\textsc{SRev}$. Inequality~\\eqref{equ:single-core-prev} becomes\n\n\\begin{equation}\\label{equ:single-core}\n\\textsc{Core}(M)\\leq 2\\delta+\\frac{5c}{2}<4\\cdot\\textsc{BRev}+3\\cdot\\textsc{SRev}\n\\end{equation}\n\n\\end{prevproof}\n\n\\begin{prevproof}{Theorem}{thm:single}\nSince $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 2 \\textsc{SRev}$ (Lemma~\\ref{lem:single-single}) and $\\textsc{Non-Favorite}(M)\\leq 5\\textsc{SRev}+4\\textsc{BRev}$ (Lemma~\\ref{lem:single-tail} and~\\ref{lem:single-core}), $\\textsc{Rev}(M,v,D)\\leq 24\\cdot\\textsc{SRev}+16\\cdot\\textsc{BRev}$ according to Theorem~\\ref{thm:revenue upperbound for subadditive}.\n\\end{prevproof}\n\n\n\n\n\n\n\n\n\n\\section{Sequentially Posted-Price Mechanism with Entry Fee}\\label{sec:spm}\nHere is the formal specification of the Sequential Posted Price with Entry Fee Mechanism.\\\\\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\REQUIRE $\\xi_{ij}$ is the price for bidder $i$ to purchase item $j$ and $\\delta_i(\\cdot)$ is bidder $i$'s entry fee function.\n\\STATE $S\\gets [m]$\n\\FOR{$i \\in [n]$}\n\t\\STATE Show bidder $i$ {the} set of available items $S$, and define entry fee as ${\\delta_i}(S)$.\n \\IF{Bidder $i$ pays the entry fee ${\\delta_i}(S)$}\n \\STATE $i$ receives her favorite bundle $S_i^{*}$, paying $\\sum_{j\\in S_i^{*}}\\xi_{ij}$.\n \\STATE $S\\gets S\\backslash S_i^{*}$.\n \\ELSE\n \\STATE $i$ gets nothing and pays $0$.\n \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\caption{{\\sf Sequential Posted Price with Entry Fee Mechanism}}\n\\label{alg:seq-mech}\n\\end{algorithm}\n\n\\section*{\\homeworkProblemName\n \\enterProblemHeader{\\homeworkProblemName}\n {\\exitProblemHeader{\\homeworkProblemName}\n\n\\newcommand{\\homeworkSectionName}{\n\\newlength{\\homeworkSectionLabelLength}{\n\\newenvironment{homeworkSection}[1\n \n\n \\renewcommand{\\homeworkSectionName}{#1\n \\settowidth{\\homeworkSectionLabelLength}{\\homeworkSectionName\n \\addtolength{\\homeworkSectionLabelLength}{0.25in\n \\changetext{}{-\\homeworkSectionLabelLength}{}{}{\n \\subsection*{\\homeworkSectionName\n \\enterProblemHeader{\\homeworkProblemName\\ [\\homeworkSectionName]}\n {\\enterProblemHeader{\\homeworkProblemName\n\n \n \n \\changetext{}{+\\homeworkSectionLabelLength}{}{}{}\n\n\\newcommand{\\Answer}{\\ \\\\\\textbf{Answer:} }\n\\newcommand{\\Intui}{\\ \\\\\\textbf{Intuition:} }\n\\newcommand{\\Proof}{\\ \\\\\\textbf{Proof:} }\n\\newcommand{\\Acknowledgement}[1]{\\ \\\\{\\bf Acknowledgement:} #1}\n\n\n\n\n\n\\title{\\textmd{\\bf \\Title}\n\\author{\\textbf{\\StudentName}}\\\\{\\large Instructed by \\textit{\\ClassInstructor}}\\\\\\normalsize\\vspace{0.1in}\\small{\\DueDate}}\n\\date{}\n\n\\begin{document}\n\\begin{spacing}{1.1}\n\\maketitle \\thispagestyle{empty}\n\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{corollary}{Corollary}\n\\newtheorem{definition}{Definition}\n\\newtheorem{assignment}{Homework Problem}\n\\newtheorem{notation}{Notation}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{conjecture}{Conjecture}\n\n\\section{Multiple Bidder, Additive With Matroid Constraint}\n\\subsection{Flow}\nFor a feasible $\\pi\\in P(\\mathcal{F},D)$, let M be the mechanism that induces $\\pi(\\cdot)$. For all $i\\in [n], j\\in [m]$, define\n\\begin{equation}\nq_{ij}^{\\pi}=b\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\pi_{ij}(t_i)\n\\end{equation}\nwhere $b\\in (0,1)$ is a constant parameter that we will determine later. Choose $\\beta_{ij}^{\\pi}$ such that $Pr[t_{ij}\\geq \\beta_{ij}^{\\pi}]=q_{ij}^{\\pi}$ (Assume all the distributions are continuous so $q_{ij}^{\\pi}$ exists).\n\nNotice that we have the following for all $j$ since $\\pi$ is feasible, i.e., without over-allocation:\n\\begin{equation}\n\\sum_i q_{ij}=b\\cdot\\sum_i \\sum_{t\\in T}f(t)\\pi_{ij}(t_i)=b\\cdot\\sum_{t\\in T}f(t)\\sum_i\\pi_{ij}(t_i)\\leq b\n\\end{equation}\n\n~\\\\\n\nLet $R_0^{\\pi}=\\{t_i\\in T_i|t_{ik}<\\beta_{ik},\\forall k\\in [m]\\}$, and $R_j^{\\pi}=\\{t_i\\in T_i|(\\forall k\\not=j,t_{ij}-\\beta_{ij}^{\\pi}\\geq t_{ik}-\\beta_{ik}^{\\pi})\\cap (t_{ij}\\geq \\beta_{ij}^{\\pi})\\}$.\n\nWe will define the same flow as before: $\\lambda^{\\pi}(t_i',t_i)>0$ if and only if\n\\begin{itemize}\n\\item $t_i\\in R_j^{\\pi}$\n\\item $t_{ik}'=t_{ik},\\forall k\\not=j$\n\\item $t_{ij}'>t_{ij}$\n\\end{itemize}\n\nThen after ironing, the Language function becomes:\n\\begin{equation}\nL(\\lambda^{\\pi},\\pi,p)=\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\bigg(t_{ij}\\cdot \\mathds{1}\\big[t_i\\not\\in R_j^{\\pi}\\big]+\\tilde{\\phi_{ij}}(t_{ij})\\cdot \\mathds{1}\\big[t_i\\in R_j^{\\pi}\\big]\\bigg)\n\\end{equation}\n\nBy the Strong Duality Theorem,\n\\begin{equation}\nREV=\\min_{\\lambda}\\max_{\\pi\\in P(\\mathcal{F},D),p}L(\\lambda,\\pi,p)\\leq \\max_{\\pi\\in P(\\mathcal{F},D),p}L(\\lambda^{\\pi},\\pi,p)\n\\end{equation}\n\nIt's enough to bound $L(\\lambda^{\\pi},\\pi,p)$ for all feasible $\\pi$ to obtain an upper bound for $REV$.\n\n\\subsection{Sequential Mechanism and c-Selectability}\nWe will use a new Sequential mechanism. The mechanism posts a price $\\theta_{ij}$ of each item $j$ for each bidder $i$ and determine an order $\\sigma$ for bidders. Each bidder comes in order $\\sigma$. When bidder $i$ with type $t_i$ comes, suppose the set of items left is $S_i$. The mechanism will let bidder $i$ know $S_i$ and charge him an entry fee $\\delta_i$. If he chooses to join the auction, he pays the entry fee and then takes his favorite bundle $S_i^{*}=\\arg\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}(t_{ij}-\\theta_{ij})$ and pay $\\sum_{j\\in S_i^{*}}\\theta_{ij}$. Let the optimal revenue that such a mechanism can get is SEQ.\n\nWhen bidder $i$ is facing the remaining item set $S_i$, we have no idea on which item he will take. c-Selectability from [\\ref{feldman}] actually take control of it. The following lemma directly comes from the definition of c-Selectability:\n\\begin{lemma}\\label{feldman2015}\n(Feldman 2015) For a downward close $\\mathcal{F}$, if there exists a $c$-selectable greedy OCRS, then in the Sequential mechanism, when it's bidder $i$'s turn, as long as $j\\in S_i$ and $t_{ij}>\\theta_{ij}$, item $j$ is in $S_i^{*}$ with at least probability $c$. The probability is taken over the randomness of $S_i$.\n\\end{lemma}\n\nIf $\\mathcal{F}$ is a matroid, the paper shows that $c=1-b$.\n\n\n\\subsection{Separating into Pieces}\n\\begin{equation}\n\\begin{aligned}\n&L(\\lambda^{\\pi},\\pi,p) \\\\\n=&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\tilde{\\phi_{ij}}(t_{ij})\\cdot \\mathds{1}\\big[t_i\\in R_j^{\\pi}\\big] \\text{(SINGLE)}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot t_{ij}\\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big]\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot t_{ij}\\cdot \\mathds{1}\\big[t_{ij}< \\beta_{ij}^{\\pi}\\big]\\\\\n\\leq& \\text{ SINGLE}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big] \\text{(SURPLUS)}\\\\\n+&\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\cdot \\mathds{1}\\big[t_{ij}\\geq \\beta_{ij}^{\\pi}\\big]+\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\cdot \\mathds{1}\\big[t_{ij}< \\beta_{ij}^{\\pi}\\big]\\\\\n\\leq& \\text{ SINGLE + SURPLUS }+\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi} \\text{(PROPHET)}\n\\end{aligned}\n\\end{equation}\n\n~\\\\\n\n\\begin{equation}\n\\begin{aligned}\nSURPLUS&=\\sum_i\\sum_{t_i: \\exists k,t_{ik}>\\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big] \\\\\n&+\\sum_i\\sum_{t_i: \\forall k,t_{ik}\\leq \\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi}) \\cdot \\mathds{1}\\big[(\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi})\\cap(t_{ij}\\geq \\beta_{ij}^{\\pi})\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\Pr_{t_{i,-j}\\sim T_{i,-j}}\\big[\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\text{ (TAIL)}\\\\\n&+\\sum_i\\sum_{t_i: \\forall k,t_{ik}\\leq \\beta_{ik}^{\\pi}+\\tau_i^{\\pi}}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\cdot \\mathds{1}\\big[t_{ij}\\geq \\beta_{ij}^{\\pi}\\big]\\text{ (CORE)}\\\\\n\\end{aligned}\n\\end{equation}\n\nHere, $\\tau_i^{\\pi}$ is chosen such that\n\\begin{equation}\n\\sum_j \\Pr_{t_{ij}}[t_{ij}\\geq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}]=\\frac{1}{2}\n\\end{equation}\n\n~\\\\\n\n\\subsection{Bounding SINGLE}\n\\begin{lemma}\nFor any downward close $\\mathcal{F}$, $\\text{SINGLE}\\leq \\text{OPT}^{\\text{COPIES}}$.\n\\end{lemma}\n\\begin{proof}\nWe build a new mechanism $M'$ in the Copies setting based on $M$. Whenever $M$ allocates item $j$ to bidder $i$ and $t_i\\in R_j^{\\pi}$, $M'$ serves the agent $(i,j)$. Then $M'$ is feasible in the Copies setting and SINGLE is the ironed virtual welfare of $M'$ with respect to $\\tilde{\\phi}(\\cdot)$. Since the maximum revenue in the Copies setting is achieved by the maximum virtual welfare, thus $\\text{OPT}^{\\text{COPIES}}$ is no less than SINGLE.\n\\end{proof}\n\nThe $\\text{OPT}^{\\text{COPIES}}$ can be achieved $\\frac{1}{6}$-approximately by a post-price mechanism.\n\n~\\\\\n\n\\subsection{Bounding PROPHET}\n\\begin{equation}\n\\begin{aligned}\n\\text{PROPHET }&=\\sum_i\\sum_{t_i\\in T_i}\\sum_j f_i(t_i)\\cdot \\pi_{ij}(t_i)\\cdot \\beta_{ij}^{\\pi}\\\\\n&=\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}(t_i)\\\\\n&=\\frac{1}{b}\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\\label{prophet}\nFor $\\mathcal{F}$, if there exists a $c$-selectable greedy OCRS,\n\\[\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq \\frac{1}{(1-b)\\cdot c}\\cdot \\text{SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider a Sequential mechanism without entry fee and post price $\\theta_{ij}=\\beta_{ij}^{\\pi}$. Then according to Lemma \\ref{feldman2015},\n\\begin{equation}\n\\begin{aligned}\nSEQ&\\geq \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot \\Pr_{t_{-i}}[j\\in S_i]\\cdot \\Pr_{t_{ij}}[t_{ij}>\\beta_{ij}^{\\pi}]\\cdot c\\\\\n&\\geq \\sum_i\\sum_j \\beta_{ij}\\cdot (\\sum_{l=1}^{i-1}q_{lj}^{\\pi})\\cdot q_{ij}^{\\pi}\\cdot c\\\\\n&\\geq (1-b)\\cdot c\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nSpecifically, if $\\mathcal{F}$ is a matroid,\n\\begin{equation}\n\\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq \\frac{1}{\\cdot (1-b)^2}\\cdot SEQ\n\\end{equation}\n\n~\\\\\n\n\\subsection{Bounding TAIL}\nWe will bound TAIL for any downward close $\\mathcal{F}$. Let $P_{ij}=\\arg\\max_{x\\geq \\tau_i^{\\pi}}x\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq x]$, and $r_{ij}=P_{ij}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq P_{ij}]$, $r_i=\\sum_j r_{ij}$, $r=\\sum_i r_i$. We have the following relationship between $r_i$ and $\\tau_i^{\\pi}$:\n\\begin{lemma}\\label{tail0}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot \\tau_i^{\\pi}$.\n\\end{lemma}\n\\begin{proof}\n\\begin{equation}\n\\begin{aligned}\nr_i&= \\sum_j P_{ij}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq P_{ij}]\\\\\n&\\geq \\sum_j \\tau_i^{\\pi}\\cdot Pr[t_{ij}-\\beta_{ij}^{\\pi}\\geq \\tau_i^{\\pi}]\\\\\n&=\\frac{1}{2}\\cdot \\tau_i^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n~\\\\\n\nWe notice by the definition of $r_{ij}$,\n\\begin{equation}\\label{tail1}\n\\begin{aligned}\n\\text{TAIL }&=\\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\Pr_{t_{i,-j}\\sim T_{i,-j}}\\big[\\exists k\\not=j, t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot (t_{ij}-\\beta_{ij}^{\\pi})\\sum_{k\\not=j}\\Pr_{t_{ik}\\sim T_{ik}}\\big[t_{ik}-\\beta_{ik}^{\\pi}\\geq t_{ij}-\\beta_{ij}^{\\pi}\\big]\\\\\n&\\leq \\sum_i\\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}\\\\\n&\\leq \\sum_i r_i\\cdot \\sum_j\\sum_{t_{ij}>\\beta_{ij}^{\\pi}+\\tau_i^{\\pi}}f_{ij}(t_{ij})\\\\\n&=\\frac{1}{2}\\cdot r\n\\end{aligned}\n\\end{equation}\n\nThe following lemma shows that $r$ can be approximately achieved by the Sequential mechanism.\n\\begin{lemma}\\label{tail2}\n\\[r\\leq \\frac{2}{(1-b)}\\text{ SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider the Sequential mechanism with item price $\\theta_{ij}=P_{ij}+\\beta_{ij}^{\\pi}$, $\\sigma=(1,2,...,n)$ and without entry fee. When bidder $i$ comes, he will definitely take only item $j$ and pay $P_{ij}+\\beta_{ij}^{\\pi}$ if:\n\\begin{itemize}\n\\item $j\\in S_i$\n\\item $t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}$\n\\item $\\forall k\\not=j, t_{ij}\\leq P_{ij}+\\beta_{ij}^{\\pi}$\n\\end{itemize}\n\nNotice that due to the second condition, every bidder will take item $j$ with at most $q_{ij}$ probability. Thus we have\n\\begin{equation}\n\\begin{aligned}\nSEQ&\\geq \\sum_i\\sum_j (P_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{-i}}[j\\in S_i]\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\Pr_{t_{i,-j}}[\\forall k\\not=j, t_{ij}\\leq P_{ij}+\\beta_{ij}^{\\pi}]\\\\\n&\\geq \\sum_i\\sum_j P_{ij}\\cdot \\big(\\sum_{l=1}^{i-1}q_{lj}^{\\pi}\\big)\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\Pr_{t_{i,-j}}[\\forall k\\not=j, t_{ij}\\leq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}]\\\\\n&\\geq (1-b)\\cdot \\sum_i\\sum_j P_{ij}\\cdot \\Pr_{t_{ij}}[t_{ij}>P_{ij}+\\beta_{ij}^{\\pi}]\\cdot \\big(1-\\sum_{k\\not=j}\\Pr_{t_{ik}}[t_{ik}\\leq P_{ik}+\\tau_i^{\\pi}]\\big)\\\\\n&\\geq \\frac{1}{2}(1-b)\\cdot \\sum_i\\sum_j r_{ij}\\\\\n&\\geq \\frac{1}{2}(1-b)\\cdot r\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nCombining Equation (\\ref{tail1}) and Lemma \\ref{tail2}, we have\n\\begin{equation}\nTAIL\\leq \\frac{1}{1-b}\\cdot SEQ\n\\end{equation}\n\n\\subsection{Bounding CORE}\nWe will bound CORE for matroid $\\mathcal{F}$. Define $t_{ij}'=(t_{ij}-\\beta_{ij}^{\\pi})\\cdot \\mathds{1}\\big[\\beta_{ij}^{\\pi}\\leq t_{ij}\\leq \\beta_{ij}^{\\pi}+\\tau_i^{\\pi}\\big]\\in [0,\\tau_i^{\\pi}]$. Then since $\\pi(\\cdot)$ is feasible,\n\\begin{equation}\nCORE=\\sum_i\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]\n\\end{equation}\n\nConsider the Sequential mechanism with item price $\\theta_{ij}=\\beta_{ij}^{\\pi}$ and order $\\sigma=(1,2,...,n)$. When it's bidder $i$'s turn, suppose the set of items left is $S_i$. Define the entry fee $\\delta_i$ for bidder $i$ as:\n\\[\\Pr_{t_i'}[\\max_{S \\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\leq \\delta_i]=\\frac{2}{3}\\]\n\nNotice that if bidder $i$ enters the auction, the profit he gets is\n\\[\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}(t_{ij}-\\theta_{ij})\\geq \\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\]\nThus with probability at least $\\frac{1}{3}$, bidder $i$ will pay the entry fee $\\delta_i$. Besides, with the same argument in Lemma \\ref{prophet},\n\\begin{equation}\n\\text{SEQ}\\geq \\sum_i\\frac{1}{3}\\big(\\delta_i+(1-b)^2\\cdot \\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\big)\n\\end{equation}\n\n~\\\\\n\nThe following result from [Schechtman 1999] can be applied to bound $\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]$ for all $i$:\n\n\\begin{lemma}\\label{core2}\n(Schechtman, 1999)Let $f(t,S)$ be a value function that is additive under a downward close constaint, drawn from a distribution D over support $[0,\\tau]$ for some $\\tau\\geq 0$. Let $\\Delta$ be a value such that $\\Pr_{t\\sim D}\\big[f(t,[m])\\leq \\Delta\\big]=\\frac{2}{3}$. Then for all $k>0$,\n\\begin{equation}\n\\Pr_{t\\sim D}\\big[f(t,[m])\\geq 3\\Delta+k\\cdot \\tau\\big]\\leq \\frac{9}{4}\\cdot 2^{-k}\n\\end{equation}\n\\end{lemma}\n\n\\begin{corollary}\\label{core3}\nFor each $i$,\n\\[\\mathbf{E}_{t_i'}\\big[\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'\\big]\\leq 3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4\\ln(2)}\\]\n\\end{corollary}\n\\begin{proof}\nLet $g(t_i')=\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'$. Directly from Lemma \\ref{core2},\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{E}_{t_i'}\\big[g(t')\\big]&= \\mathbf{E}_{t_i'}\\big[g(t_i');g(t_i')<3\\delta_i\\big]+ \\mathbf{E}_{t_i'}\\big[g(t');g(t_i')\\geq 3\\delta_i\\big]\\\\\n&\\leq 3\\delta_i+\\int_0^{\\infty}Pr[g(t_i')>3\\delta_i+y]dy\\\\\n&\\leq 3\\delta_i+\\int_0^{\\infty}\\frac{9}{4}\\cdot 2^{-y\/\\tau}dy\\\\\n&\\leq 3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4\\ln(2)}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\nNow we can finish bounding the CORE with the following lemma.\n\n\\begin{lemma}\n\\[\\text{CORE }+3(1-b)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\\leq (\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}\\]\n\\end{lemma}\n\\begin{proof}\nConsider the mechanism above. Assume $S^{*}=\\arg\\max_{S\\in \\mathcal{F}_i}\\sum_{j\\in S} t_{ij}'$. Notice for all $i,j$, $\\Pr_{t_{-i}}[j\\in S_i]\\geq 1-\\sum_i q_{ij}=1-b$. With Lemma \\ref{tail0}, \\ref{tail2}, and \\ref{core2}, we have\n\\begin{equation}\n\\begin{aligned}\nCORE&=\\sum_i\\mathbf{E}_{t_i'}\\big[\\sum_{j\\in S^{*}} t_{ij}'\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i\\mathbf{E}_{t'}\\big[\\sum_{j\\in S^{*}} t_{ij}'\\cdot \\mathds{1}[j\\in S_i]\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i\\mathbf{E}_{t'}\\big[\\max_{S\\subseteq S_i, S\\in \\mathcal{F}_i}\\sum_{j\\in S}t_{ij}'\\big]\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\sum_i (3\\delta_i+\\frac{9\\tau_i^{\\pi}}{4ln(2)})\\\\\n&\\leq \\frac{1}{1-b}\\cdot \\bigg(9\\cdot \\text{SEQ }-3(1-b)^2\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}+\\frac{9r}{2\\cdot ln(2)}\\bigg)\\\\\n&\\leq (\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}-3(1-b)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n~\\\\\n\n\\subsection{Optimizing the Constant}\nPut all pieces together:\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda^{\\pi},\\pi,p)&\\leq \\text{SINGLE}+\\text{TAIL}+\\text{PROPHET}+\\text{CORE} \\\\\n&\\leq 6\\cdot \\text{SREV}+\\frac{1}{1-b}\\cdot \\text{SEQ}+\\big(\\frac{1}{b}-3(1-b)\\big)\\cdot \\sum_i\\sum_j \\beta_{ij}^{\\pi}\\cdot q_{ij}^{\\pi}+(\\frac{9}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2})\\cdot \\text{SEQ}\\\\\n&\\leq 6\\cdot \\text{SREV}+\\bigg(\\frac{7}{1-b}+\\frac{9}{ln(2)}\\cdot \\frac{1}{(1-b)^2}+\\frac{1}{b(1-b)^2}\\bigg)\\cdot \\text{SEQ}\n\\end{aligned}\n\\end{equation}\n\nWhen $b=\\frac{1}{7}$. The larger one of SREV and SEQ gets a $1\/42$-approximately mechanism.\n\n~\\\\\n\n\n\\section{Single Bidder, Subadditive Valuation}\n\\subsection{Problem Setting\\textsuperscript{[\\ref{subadditive}]}}\nLet $m$ be the number of items, $I=[m]$ be the set of items. For each item $j$, let $\\Omega_j$ be a compact space. Each type $t_j\\in \\Omega_j$ represent the information for item $j$. For all $S\\subseteq I$, denote $\\Omega_S=\\times_{j\\in S}\\Omega_j$. $\\Omega=\\Omega_{I}$. The bidder's type $t=\\langle t_j \\rangle_{j\\in I}$ is drawn from a distribution $D$ on $\\Omega$. After $t$ is drawn, the bidder's valuation for a set of items S is only determined by the vector $\\langle t_j \\rangle_{j\\in S}$. Formally, denote $\\Omega^{*}=\\bigcup_{S\\subseteq I}\\Omega_S$. There exists a valuation function $V:\\Omega^{*}\\to \\mathcal{R}$ such that $\\forall t\\in \\Omega, S\\subseteq I$,\n\\begin{equation}\nv(t,S)=V(\\langle t_j \\rangle_{j\\in S},S)\n\\end{equation}\n\nIn this problem, we assume that the bidder's valuation is subadditive, i.e., $\\forall t\\in \\Omega$, $\\forall P,Q\\subseteq I$,\n\\begin{equation}\nv(t,P\\cup Q)\\leq v(t,P)+v(t,Q)\n\\end{equation}\n\n~\\\\\n\n\\subsection{Duality}\nWe use the following functions(variables) to describe the mechanism:\n\\begin{itemize}\n\\item $p(t)$, $t\\in \\Omega^{+}$: the prize that the bidder should pay when his type is $t$.\n\\item $\\phi(t,t')$, $t\\in \\Omega, t'\\in \\Omega^{+}$: the expect valuation if the bidder pretends to be $t'$ when his type is $t$.\n\\end{itemize}\n\nwhere $\\Omega^{+}=\\Omega\\cup \\{\\emptyset\\}$ which allows the bidder not to participate in the auction. When $t=\\emptyset$, $p(t)=\\phi(\\cdot,t)=0$.\n\n\\textbf{Remark:} For a mechanism, let $\\pi_S(t)$ be the probability for bidder with type $t$ to obtain a set $S$ of item. Then $\\phi(t,t')$ can be written as\n\\begin{equation}\n\\phi(t,t')=\\sum_{S\\subseteq I}\\pi_S(t)v_{t'}(S)\n\\end{equation}\n\nHere we use $\\phi(t,t')$ to replace the original variables $\\pi_j(t)=\\sum_{S:j\\in S}\\pi_S(t)$. The function in fact includes both the probability and the valuation. Like $\\pi$, all the $\\phi(t,t')$'s should stay in some feasible region to avoid over-allocation. We use $\\phi\\in \\mathcal{O}$ to represent it.\n\n~\\\\\n\nOur primal is:\n\\begin{itemize}\n\\item \\textbf{Variables}: $p(t),\\phi(t,t'),\\quad t\\in \\Omega, t'\\in \\Omega^{+}$\n\\item \\textbf{Constraint}:\n\n$\\quad(1)\\phi(t,t)-p(t)\\geq \\phi(t,t')-p(t'),\\quad \\forall t\\in \\Omega, t'\\in \\Omega^{+}$\n\n$\\quad(2)\\phi\\in \\mathcal{O}$\n\n\\item \\textbf{Objective}: \\text{min} $\\sum_{t\\in \\Omega}f(t)p(t)$\n\\end{itemize}\n\n~\\\\\n\nThe Language dual function $L(\\lambda,\\phi,p)$ is\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda,\\phi,p)&=\\sum_{t\\in T}f(t)p(t)+\\sum_{t\\in \\Omega,t'\\in \\Omega^{+}}\\lambda(t,t')\\bigg((\\phi(t,t)-p(t))-(\\phi(t,t')-p(t'))\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}\\bigg(f(t)+\\sum_{t'\\in \\Omega}\\lambda(t',t)-\\sum_{t'\\in \\Omega^{+}}\\lambda(t,t')\\bigg)+\\sum_{t\\in \\Omega}\\bigg(\\phi(t,t)\\cdot\\sum_{t'\\in \\Omega^{+}}\\lambda(t,t')-\\sum_{t'\\in \\Omega}\\lambda(t',t)\\cdot \\phi(t',t)\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}f(t)\\bigg(\\phi(t,t)-\\frac{1}{f(t)}\\sum_{t'\\in \\Omega}\\lambda(t',t)\\big(\\phi(t',t)-\\phi(t,t)\\big)\\bigg)\\\\\n&=\\sum_{t\\in \\Omega}f(t)\\Phi(t)\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Phi(t)=\\phi(t,t)-\\frac{1}{f(t)}\\sum_{t'\\in \\Omega}\\lambda(t',t)\\big(\\phi(t',t)-\\phi(t,t)\\big)\n\\end{equation}\n\nHere we need $\\forall t\\in \\Omega$,\n\\begin{equation}\nf(t)+\\sum_{t'\\in \\Omega}\\lambda(t',t)-\\sum_{t'\\in T^{+}}\\lambda(t,t')=0\n\\end{equation}\nto avoid the unlimited optimal value.\n\n\\subsection{Change the Valuation}\n\nWe will change the valuation and apply the duality to the new valuation. Let\n\n\\begin{equation}\nR_j=\\bigg\\{t\\in\\Omega: j=\\arg\\max_k V(t_k,\\{k\\})\\bigg\\}\n\\end{equation}\n\nWe define a new valuation $\\hat{v}(\\cdot)$. For a type $t\\in \\Omega$, if $t\\in R_j$, then\n\\begin{equation}\n\\begin{aligned}\n\\hat{v}(t,S)=\n\\begin{cases}\nv(t,S\\backslash\\{j\\})+v(t,\\{j\\}), &j\\in S\\\\\nv(t,S), &j\\not\\in S\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\nSince $v(t,\\cdot)$ is subadditive, $\\hat{v}(t,S)\\geq v(t,S)$ for all subset S.\n\nLet $REV(v)$ and $REV(\\hat{v})$ be the optimal revenue for the two valuations. We would like to see that the two values are not far so that we can bound $REV(v)$ by bounding $REV(\\hat{v})$. The following two lemmas give the proof of this fact:\n\n\\begin{lemma}\n\\textsuperscript{[\\ref{subadditive}]}Consider a distributions D with two different valuations $v$ and $\\hat{v}$. Let $\\delta(t,S)=\\hat{v}(t,S)-v(t,S)$. If $\\delta(t,S)\\geq 0$ for all $t\\in T, S\\subseteq I$, then for any $\\epsilon\\in (0,1)$,\n\\begin{equation}\nREV(\\hat{v})\\geq (1-\\epsilon)(REV(v)-\\frac{\\bar\\delta}{\\epsilon})\n\\end{equation}\nwhere $\\bar\\delta=\\mathbb{E}_{t\\sim D}[\\max_{S\\subseteq I}\\delta(t,S)]$.\n\nIf we choose $\\epsilon=\\frac{1}{2}$,\n\\begin{equation}\nREV(v)\\leq 2REV(\\hat{v})+2\\bar\\delta\n\\end{equation}\n\n\\end{lemma}\n\n~\\\\\n\n\\begin{lemma}\nLet $\\bar\\delta'=\\mathbb{E}_{t\\sim D}\\bigg[v(t,I\\backslash\\{j\\}), j=\\arg\\max_k V(t_k,\\{k\\})\\bigg]$, then\n\\begin{equation}\n\\bar\\delta\\leq \\bar\\delta'\\leq 6BREV+9.2SREV\n\\end{equation}\n\\end{lemma}\n\n~\\\\\n\nFor a fixed type $t$, we make a partition $I=C_t\\cup T_t$ based on some cutoff $r$, where $C_t=\\{j\\in I: v(t,\\{j\\})V(t_j,\\{j\\})$, there is at least one bundle with positive profit. The mechanism will definitely sell some item, obtaining expected revenue at least $V(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]$. Thus,\n\\begin{equation}\nV(t_j,\\{j\\})\\cdot \\Pr_{t_{-j}}[t\\not\\in R_j]\\leq SREV\n\\end{equation}\n\n\\begin{equation}\\label{e2}\n(TAIL)\\leq SREV\\cdot \\sum_j\\sum_{t_j:V(t_j,\\{j\\})\\geq r}f_j(t_j)=SREV\n\\end{equation}\n\n~\\\\\n\nNow we can finish the prove of Lemma 2.\n\n\\begin{equation}\n\\begin{aligned}\nPr_t[\\exists j, V(t_j,\\{j\\})\\geq r]&\\geq 1-\\prod_{j}F_j(r)\\\\\n&\\geq 1-(\\frac{\\sum_j F_j(r)}{n})^n\\\\\n&\\geq 1-(1-1\/n)^n\\\\\n&=1-\\frac{1}{e}\n\\end{aligned}\n\\end{equation}\n\nConsider the auction that sells every item with price $r$. Then with probability at least $1-\\frac{1}{e}$, the mechanism will sell at least one item at price at least $r$, obtaining revenue at least $(1-\\frac{1}{e})r$. Thus,\n\\begin{equation}\\label{e3}\nr\\leq \\frac{1}{1-1\/e}\\cdot SREV\n\\end{equation}\n\nCombining Equation (\\ref{e1})(\\ref{e2})(\\ref{e3}), we have\n\\begin{equation}\n\\bar\\delta\\leq (CORE)+(TAIL)\n\\leq 6BREV+\\frac{4r}{ln(2)}+SREV\n\\leq 6BREV+9.2SREV\n\\end{equation}\nwhich finishes Lemma 2.\n\n~\\\\\n\n\\subsection{Construction of the Flow}\nWe apply duality on the new valuation $\\hat{v}$. For $t,t'\\in \\Omega$, $\\lambda(t',t)>0$ if and only if\n\\begin{itemize}\n\\item $t$ and $t'$ only differ on the $j$-th coordinate.\n\\item $t\\in R_j$.\n\\item $V(t_j',\\{j\\})>V(t_j,\\{j\\})$.\n\\end{itemize}\n\nWe now consider $\\hat{v}(t',S)-\\hat{v}(t,S)$ when $\\lambda(t',t)>0$. There are two conditions for subset S:\n\n(1)$j\\not\\in S$, notice that only the $j-$th coordinate is different,\n\\begin{equation}\n\\hat{v}(t',S)=V'(\\langle t_k' \\rangle_{k\\in S},S)=V'(\\langle t_k \\rangle_{k\\in S},S)=\\hat{v}(t,S)\n\\end{equation}\n\n(2)$j\\in S$, we have\n\\begin{equation}\n\\begin{aligned}\n\\hat{v}(t',S)-\\hat{v}(t,S)&=\\bigg(V'(\\langle t_k' \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})+V'(t_j',\\{j\\})\\bigg)-\n\\bigg(V'(\\langle t_k \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})+V'(t_j,\\{j\\})\\bigg)\\\\\n&=\\bigg(V'(\\langle t_k' \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})-V'(\\langle t_k \\rangle_{k\\in S\\backslash\\{j\\}},S\\backslash\\{j\\})\\bigg)+\\bigg(V'(t_j',\\{j\\})-V'(t_j,\\{j\\})\\bigg)\\\\\n&=V'(t_j',\\{j\\})-V'(t_j,\\{j\\})\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Bound the Language Function}\nFor all type $t$, the valuation when bidder tells the truth is\n\n\\begin{equation}\n\\begin{aligned}\n\\phi'(t,t)&=\\sum_{S\\subseteq I}\\pi_S(t)\\hat{v}(t,S)\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\sum_{S:j\\in S}\\pi_S(t)\\big(\\hat{v}(t,S\\backslash\\{j\\})+\\hat{v}(t,\\{j\\})\\big)+\\sum_{S:j\\not\\in S}\\pi_S(t)\\hat{v}(t,S)\\bigg)\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\pi_j(t)\\hat{v}(t,\\{j\\})+\\sum_{S\\subseteq I\\backslash\\{j\\}}\\pi_S(t)\\hat{v}(t,S)\\bigg)\\\\\n&\\leq \\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\pi_j(t)\\hat{v}(t,\\{j\\})+\\hat{v}(t,I\\backslash\\{j\\})\\bigg)\n\\end{aligned}\n\\end{equation}\n\nThe virtual valuation $\\Phi(t)$:\n\n\\begin{equation}\n\\begin{aligned}\n\\Phi(t)&=\\phi'(t,t)-\\frac{1}{f(t)}\\sum_{S\\subseteq I}\\pi_S(t)\\sum_{t'\\in \\Omega}\\lambda(t',t)(\\hat{v}(t',S)-\\hat{v}(t,S))\\\\\n&=\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\hat{v}(t,I\\backslash\\{j\\})+\\pi_i(t)\\big(\\hat{v}(t,\\{j\\})-\\sum_{t_j':V'(t_j',\\{j\\})>V'(t_j,\\{j\\})}\\frac{f_j(t_j')}{f_j(t_j)}\\big)\\bigg)\\\\\n&\\leq\\sum_{j\\in I}\\mathds{1}_{[t\\in R_j]}\\bigg(\\hat{v}(t,I\\backslash\\{j\\})+\\pi_i(t)\\tilde{\\phi}_j(t_j)\\bigg)\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{\\phi}_j(t_j)$ is the Myerson Virtual Value for item j with the marginal distribution on $\\Omega_j$ and valuation $V'(t_j,\\{j\\})$.\n\n\\begin{equation}\n\\begin{aligned}\nL(\\lambda,\\phi',p)&=\\sum_{t\\in \\Omega}f(t)\\Phi(t)\\\\\n&=\\bar\\delta'+\\sum_t\\sum_jf(t)\\pi_j(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\\\\\n&\\leq \\bar\\delta'+\\sum_t\\sum_jf(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\n\\begin{equation}\n\\sum_t\\sum_jf(t)\\tilde{\\phi}_j(t_j)\\mathds{1}_{[t\\in R_j]}\\leq 6SREV\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nConsider the optimal revenue $OneRev$ when the mechanism is only allowed to sell one item. Assume a mechanism M allocate item j only if $t\\in R_j$. Then LHS is the virtual welfare of this mechanism, which is less than $OneRev$. Besides, this allocation rule is same as assuming bidder's valuation is unit-demand. Thus selling separately can reach a $\\frac{1}{6}$-approximation of this revenue.\n\\end{proof}\n\n~\\\\\n\n\\subsection{Conclusion}\nPut everything together we have\n\\begin{equation}\nREV(v)\\leq 2REV(\\hat{v})+2\\bar\\delta \\leq 2L(\\lambda,\\phi',p)+2\\bar\\delta \\leq 4\\bar\\delta'+12SREV \\leq 24BREV+49SREV\n\\end{equation}\n\n\n\n\n\\section{Missing Details of Section~\\ref{sec:flow}}~\\label{sec:appx_flow}\n\\section{Recap: Flow for Additive Valuations}\\label{sec:flow_additive}\nWhen the valuations are additive, we simply view $t_{ij}$ as bidder $i$'s value for receiving item $j$. Although there are many possible ways to define a flow, we focus on a class of simple ones. Every flow in this class $\\lambda^{(\\beta)}$ is parametrized by a set of parameters $\\beta=\\{\\beta_{ij}\\}_{i\\in[n], j\\in[m]}\\in\\R^{nm}$. Based on $\\beta_i=\\{\\beta_{ij}\\}_{j\\in[m]}$, we first partition the type space $T_i$ for each bidder $i$ into $m+1$ regions:\n\\begin{itemize}[leftmargin=0.7cm]\n\t\\item $R_{0}^{(\\beta_i)}$ contains all types $t_i$ such that $t_{ij}<\\beta_{ij}$ for all $j\\in[m]$.\n\t\\item $R_j^{(\\beta_i)}$ contains all types $t_i$ such that $t_{ij}-\\beta_{ij}\\geq 0$ and $j$ is the smallest index in $\\argmax_k\\{t_{ik}-\\beta_{ik}\\}$.\n\\end{itemize}\n\nWe use essentially the same flow as in~\\cite{CaiDW16}. Here we provide a partial specification and state some desirable properties of the flow. See Figure~\\ref{fig:multiflow} for an example with $2$ items and~\\cite{CaiDW16} for a complete description of the flow.\n\\begin{figure}[ht]\n\\colorbox{MyGray}{\n\\begin{minipage}{\\textwidth}\n{\\bf Partial Specification of the flow $\\lambda^{(\\beta)}$:}\n\\begin{enumerate}[leftmargin=0.7cm]\n \\item For every type $t_{i}$ in region $R^{(\\beta_{i})}_{0}$, the flow goes directly to $\\varnothing$ (the super sink).\n \\item For all $j>0$, any flow entering $R^{(\\beta_{i})}_{j}$ is from $s$ (the super source) and any flow leaving $R^{(\\beta_{i})}_{j}$ is to $\\varnothing$.\n \\item For all $t_{i}$ and $t_{i}'$ in $R^{(\\beta_{i})}_{j}$ ($j>0$), {$\\lambda^{(\\beta)}_{i}(t_{i},t_{i}')>0$} only if $t_{i}$ and $t_{i}'$ only differ in the $j$-th coordinate.\n\\end{enumerate}\n\\end{minipage}}\n\\caption{Partial Specification of the flow $\\lambda^{(\\beta)}$.}\n\\label{fig:flow specification}\n\\end{figure}\n\n\\begin{figure}\n \\centering{\\includegraphics[width=0.5\\linewidth]{multi_flow.png}}\n \\caption{An example of $\\lambda^{(\\beta)}_{i}$ for additive bidders with two items.}\n \\label{fig:multiflow}\n\\end{figure}\n\n\n\\begin{lemma}[\\cite{CaiDW16}\\footnote{Note that this Lemma is a special case of Lemma 3 in~\\cite{CaiDW16} when the valuations are additive. }]\\label{lem:additive flow properties}\n\tFor any $\\beta$, there exists a flow $\\lambda^{(\\beta)}_{i}$ such that the corresponding virtual value function $\\Phi_{i}(t_{i}, \\cdot)$ satisfies the following properties:\n\t\\begin{itemize}[leftmargin=0.5cm]\n\t\t\\item For any $t_{i}\\in R^{(\\beta_i)}_{0}$, $\\Phi_{i}(t_{i},S) = \\sum_{k\\in S} t_{ik}$.\n\t\t\\item For any $j>0$, $t_{i}\\in R^{(\\beta_i)}_{j}$, $$\\Phi_{i}(t_{i},S)\\leq \\sum_{k\\in S \\land k\\neq j} t_{ik}+{\\tilde{\\varphi}}_{ij}(t_{ij})\\cdot\\mathds{1}[j\\in S],$$ where ${\\tilde{\\varphi}}_{ij}(\\cdot)$ is Myerson's ironed virtual value function for $D_{ij}$.\n\t\\end{itemize}\n\\end{lemma}\n\nThe properties above are crucial for showing the approximation results for simple mechanisms in~\\cite{CaiDW16}. One of the key challenges in approximating the optimal revenue is how to provide a tight upper bound. A trivial upper bound is the social welfare, which may be arbitrarily bad in the worst case. By plugging the virtual value functions in Lemma~\\ref{lem:additive flow properties} into the partial Lagrangian, we obtain a new upper bound that replaces the value of the buyer's favorite item with the corresponding Myerson's ironed virtual value. As demonstrated in~\\cite{CaiDW16}, this new upper bound is at most $8$ times larger than the optimal revenue when the buyers are additive, and its appealing structure allows the authors to compare the revenue of simple mechanisms to it. In \\notshow{the next section} {Section~\\ref{sec:flow}}, we identify some difficulties in directly applying this flow to subadditive valuations. Then we show how to overcome these difficulties by relaxing the subadditive valuations and obtain a similar upper bound.\n\n\\section{Proof of Lemma~\\ref{lem:subadditive flow properties}}\\label{sec:proof_virtual_relaxation}\n\\begin{lemma}\\label{lem:separate the favorite out in virtual value}\n\tFor any flow $\\lambda^{(\\beta)}_i$ that respects the partial specification in Figure~\\ref{fig:flow specification}, the corresponding virtual valuation function $\\Phi_i^{(\\beta_i)}$ of $v_i^{(\\beta_i)}$ for any buyer $i$ is:\n\\begin{itemize}[leftmargin=0.7cm]\n\\item $v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\n \\left(V_i(t'_{ij})-V_i(t_{ij})\\right)$, if $t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S$.\n\\item $v_i(t_i,S)$, otherwise.\n\\end{itemize}\n\n\\begin{comment}\n\t \\begin{equation*}\n\\begin{aligned}\n\\Phi_i^{(\\beta_i)}(t_i, S)=\n\\begin{cases}\nv_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right) &\\text{if }t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S\\\\\nv_i(t_i,S) & \\text{o.w.}\n\\end{cases}\n\\end{aligned}\n\\end{equation*}\n\\end{comment}\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:separate the favorite out in virtual value}\n\tThe proof follows the definitions of the virtual valuation function (Definition~\\ref{def:virtual value}) and relaxed valuation (Definition~\\ref{def:relaxed valuation}). We use $t_{i,-j}=\\langle t_{i{j'}}\\rangle_{j'\\not=j}$ to denote bidder $i$'s information for all items except item $j$. If $t_i\\in R_j^{(\\beta_i)}$ and $j\\in S$, $v_i^{(\\beta_i)}(t_i,S) = v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})$. Since $\\lambda(t_i,t_i')>0$ only when $t_{i,-j}=t_{i,-j}'$ and $t_i'\\in R_j^{(\\beta_i)}$, $v_i^{(\\beta_i)}(t'_i,S) = v_i(t'_i, S\\backslash \\{j\\})+V_i(t'_{ij})= v_i(t_i, S\\backslash \\{j\\})+V_i(t'_{ij})$. Therefore,\n\\begin{align*}\n \\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S\\backslash \\{j\\})+V_i(t_{ij})\n -\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)\n\\end{align*}\n\t\n\tIf $t_i\\in R_j^{(\\beta_i)}$ and $j\\notin S$ or $t_i\\in R_0^{(\\beta_i)}$, then $v_i^{(\\beta_i)}(t_i,S) =v_i(t_i, S)$. If $t_i\\in R_0^{(\\beta_i)}$, there is no flow entering $t_i$ except from the source, so clearly $\\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S)$. If $t_i\\in R_j^{(\\beta_i)}$, then for any $t'_i$ that only differs from $t_i$ in the $j$-th coordinate, we have $v_i(t'_i, S)=v_i(t_i,S)$, because {$j\\not\\in S$}. Hence, $\\Phi_i^{(\\beta_i)}(t_i, S)=v_i(t_i, S)$.\n\\end{prevproof}\n\n\n\\begin{prevproof}{Lemma}{lem:subadditive flow properties}\n\nLet $\\Psi_{ij}^{(\\beta_i)}(t_i)=V_i(t_{ij})-\\frac{1}{f_i(t_i)}\\sum_{t'_i\\in T_i}\\lambda(t'_i,t_i)\\cdot\\left(V_i(t'_{ij})-V_i(t_{ij})\\right)$. According to Lemma~\\ref{lem:separate the favorite out in virtual value}, it suffices to prove that for any $j>0$, any $t_{i}\\in R^{(\\beta_i)}_{j}$, $\\Psi_{ij}^{(\\beta_i)}(t_i)\\leq {\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$.\n\n\\begin{claim}\nFor any type $t_{i}\\in R^{(\\beta_i)}_{j}$, if we only allow flow from type $t'_{i}$ to $t_{i}$, where $t_{ik}=t'_{ik}$ for all $k\\neq j$ and $t'_{ij}\\in \\argmin_{s\\in T_{ij} \\land V_i(s)> V_i(t_{ij})} V_i(s)$, and the flow $\\lambda(t_i',t_i)$ equals $\\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}$ fraction of the total in flow to $t_i'$, then there exists a flow $\\lambda$ such that\n\\begin{align*}\n\\Psi_{ij}^{(\\beta_i)}(t_i)=\\varphi_{ij}(V_i(t_{ij}))\n=V_i(t_{ij})-\\frac{\\left(V_i({t'_{ij}})-V_i(t_{ij})\\right)\\cdot\\Pr_{v\\sim F_{ij}}[v>V_i(t_{ij})]}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]},\n\\end{align*} where $\\varphi_{ij}(V_i(t_{ij}))$ is the Myerson virtual value for $V_i(t_{ij})$ with respect to $F_{ij}$. \\end{claim}\n\\begin{proof}\n{As the flow only goes from $t_i'$ and $t_i$, where $t_i'$ and $t_i$ only differs in the $j$-th coordinate, and \\\\\n\\noindent$t_{ij}\\in \\argmax_{s\\in T_{ij} \\land V_i(s)< V_i(t_{ij}')} V_i(s)$. If $t_{ij}$ is a type with the largest $V_i(t_{ij})$ value in $T_{ij}$, then there is no flow coming into it except the one from the source, so $\\Psi_{ij}^{(\\beta_i)}(t_i)=V_i(t_{ij})$. For every other value of $t_{ij}$, the in flow is exactly\n\\begin{align*} \\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}\\prod_{k\\neq j} f_{ik}(t_{ik})\\cdot \\sum_{x\\in T_{ij}:V_i({x})>V_i(t_{ij})} f_{ij}(x) \n=\\prod_{k} f_{ik}(t_{ik})\\cdot \\frac{\\Pr_{v\\sim F_{ij}}[v>V_i(t_{ij})]}{\\Pr_{v\\sim F_{ij}}[v=V_i(t_{ij})]}.\\end{align*}\n {This is because each type of the form $(x,t_{i,-j})$ with $V_i(x) > V_i(t_{ij})$ is also in $R^{(\\beta_i)}_{j}$. So $\\frac{f_{ij}(t_{ij})}{\\Pr_{v\\sim F_{ij}}[v= V_i(t_{ij})]}$ of all flow that enters these types will be passed down to $t_{i}$ (and possibly further, before going to the sink), and the total amount of flow entering all of these types from the source is exactly {$\\prod_{k\\neq j} f_{ik}(t_{ik})\\cdot \\sum_{x\\in T_{ij}:V_i({x})>V_i(t_{ij})} f_{ij}(x) $}.} Therefore, $\\Psi_{ij}^{(\\beta_i)}(t_i)=\\varphi_{ij}(V_i(t_{ij}))$. Whenever there is no more type $t_i\\in R_j^{(\\beta_i)}$ with smaller $V_i(t_{ij})$ value, we push all the flow to the sink.}\n\\end{proof}\n\n\nIf $F_{ij}$ is regular, this completes our proof. When $F_{ij}$ is not regular, we can iron the virtual value function in the same way as in \\cite{CaiDW16}. Basically, for two types $t_i,t_i'\\in R^{(\\beta_i)}_{j}$ that only differ in the $j$-th coordinate, if $\\Psi_{ij}^{(\\beta_i)}(t_i)>\\Psi_{ij}^{(\\beta_i)}(t_i')$ but $V_i(t_{ij})0$ contains all types in $T_i$ whose ``favorite'' item is $j$. We first provide a {\\bf Partial Specification of the flow $\\lambda^{(\\beta)}$:}\n\n\\noindent\\textbf{1.} For every type $t_{i}$ in region $R^{(\\beta_{i})}_{0}$, the flow goes directly to $\\varnothing$ (the super sink).\n\n\\noindent \\textbf{2.} For all $j>0$, any flow entering $R^{(\\beta_{i})}_{j}$ is from $s$ (the super source) and any flow leaving $R^{(\\beta_{i})}_{j}$ is to $\\varnothing$.\n\n\\noindent \\textbf{3.} For all $t_{i}$ and $t_{i}'$ in $R^{(\\beta_{i})}_{j}$ ($j>0$), {$\\lambda^{(\\beta)}_{i}(t_{i},t_{i}')>0$} only if $t_{i}$ and $t_{i}'$ only differ in the $j$-th coordinate.\n\n\\notshow{\n\\begin{figure}\n \\centering{\\includegraphics[width=0.5\\linewidth]{multi_flow.png}}\n \\caption{An example of $\\lambda^{(\\beta)}_{i}$ for additive bidders with two items.}\n \\label{fig:multiflow}\n\\end{figure}\n}\n\nFor additive valuations and any type $t_i \\in R_j^{(\\beta_i)}$ , the contribution to the virtual value function $\\Phi(t_i,S)$ from any type $t_i'\\in R_j^{(\\beta_i)}$ is either $0$ if $j\\notin S$, or {$\\lambda_i^{(\\beta)}(t_i', t_i)(v_i(t_i',S)-v_i(t_i,S))=\\lambda_i^{(\\beta)}(t_i', t_i)(t_{ij}'-t_{ij})$} if $t_i$, $t_i'$ only differs on the $j$-th coordinate and $j\\in S$. In either case, the contribution does not depend on $t_{ik}$ for any $k\\neq j$. This is the key property that allows~\\cite{CaiDW16} to choose a flow such that the value of the favorite item is replaced by the corresponding Myerson's ironed virtual value in the virtual value function $\\Phi_i(t_i,\\cdot)$.\nUnfortunately, this property no longer holds for subadditive valuations. When $j\\in S$ and $\\lambda_i^{(\\beta)}(t_i',t_i)>0$, the contribution {$\\lambda_i^{(\\beta)}(t_i', t_i)(v_i(t_i',S)-v_i(t_i,S))$} heavily depends on $t_{ik}$ of all the other item $k\\in S$. All we can conclude is that the contribution lies in the range {$[-\\lambda_i^{(\\beta)}(t_i', t_i)\\cdot V_{i}(t_{ij}), \\lambda_i^{(\\beta)}(t_i', t_i)\\cdot V_{i}(t_{ij}')]$}\\footnote{$v_i(t,\\cdot)$ is subadditive and monotone for every type $t\\in T_i$, therefore $v_i(t_i,S)\\in[v_i(t_i, S\\backslash\\{j\\}),v_i(t_i, S\\backslash\\{j\\})+V_{i}(t_{ij})]$ and $v_i(t'_i,S)\\in[v_i(t'_i, S\\backslash\\{j\\}),v_i(t'_i, S\\backslash\\{j\\})+V_{i}(t'_{ij})]$.}, but this is not sufficient for us to convert the value of item $j$ into the corresponding Myerson's ironed virtual value.\n\n\\subsection{Valuation Relaxation}\\label{sec:valuation relaxation}\nThis is the first major barrier for extending the duality framework to accommodate subadditive valuations. We overcome it by considering a relaxation of the valuation functions. More specifically, for any $\\beta$, we construct another function $v_i^{(\\beta_i)}(\\cdot,\\cdot): T_i\\times 2^{[m]}\\mapsto {\\mathbb{R}_{\\geq 0}}$ for every buyer $i$ such that: (i) for any $t_i$, $v_i^{(\\beta_i)}(t_i,\\cdot)$ is subadditive and monotone, and for every bundle $S$ the new value $v_i^{(\\beta_i)}(t_i,S)$ is no smaller than the original value $v_i(t_i,S)$; (ii) for any BIC mechanism $M$ with respect to the original valuations, there exists another mechanism $M^{(\\beta)}$ that is BIC with respect to the new valuations and its revenue is comparable to the revenue of $M$; (iii) for the new valuations $v^{(\\beta)}$, there exists a flow whose induced virtual value functions have properties similar to those in the additive case.\nProperty (ii) implies that the optimal revenue with respect to $v^{(\\beta)}$ can serve as a proxy for the original optimal revenue. Moreover, due to Theorem~\\ref{thm:revenue less than virtual welfare}, the optimal revenue for $v^{(\\beta)}$ is upper bounded by the partial Lagrangian dual with respect to $v^{(\\beta)}$, which has an appealing format similar to the additive case by property (iii). Thus, we obtain a benchmark for subadditive bidders that resembles the benchmark for additive bidders in~\\cite{CaiDW16}\n\n\\begin{definition}[Relaxed Valuation]\\label{def:relaxed valuation}\n\tGiven $\\beta$, for any buyer $i$, define $v_i^{(\\beta_i)}(t_i,S)=v_i(t_i,S\\backslash\\{j\\})+V_i(t_{ij})$, if the ``favorite'' item is in $S$, i.e., $t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S$. Otherwise, define $v_i^{(\\beta_i)}(t_i,S)=v_i(t_i,S)$.\n\n\n\\begin{comment}\n\\begin{equation}\n\\begin{aligned}\nv_i^{(\\beta_i)}(t_i,S)=\n\\begin{cases}\nv_i(t_i,S\\backslash\\{j\\})+V_i(t_{ij}) &\\text{if the ``favorite'' item is in $S$, i.e., }t_i\\in R_j^{(\\beta_i)} \\text{ and } j\\in S\\\\\nv_i(t_i,S) & \\text{o.w.}\n\\end{cases}\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\\end{definition}\n\nIn the next Lemma, we show that for any BIC mechanism $M$ for $v$, there exists a BIC mechanism $M^{(\\beta)}$ for $v^{(\\beta)}$ such that its revenue is comparable to the revenue of $M$ (property (ii)). Moreover, the ex-ante probability for any buyer $i$ to receive any item $j$ in $M^{(\\beta)}$ is no greater than in $M$ (property (i)). We will see later that this is an important property for our analysis. The proof of Lemma~\\ref{lem:relaxed valuation} is similar to the $\\epsilon$-BIC to BIC reduction in~\\cite{HartlineKM11, BeiH11,DaskalakisW12} and can be found in Appendix~\\ref{sec:proof_relaxed_valuation}.\n\n\n\\begin{lemma}\\label{lem:relaxed valuation}\n\n\tFor any $\\beta$ and any BIC mechanism $M$ for subadditive valuation $\\{v_i(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, there exists a BIC mechanism $M^{(\\beta)}$ for valuations $\\{v_i^{(\\beta_i)}(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, such that\n\n \\vspace{.1in}\n \\noindent \\emph{\\textbf{(i)}} $\\displaystyle\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\n \\vspace{.1in}\n \\noindent \\emph{\\textbf{(ii)}} $\\displaystyle\\textsc{Rev}(M, v, D)\\leq2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}\\displaystyle+2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right).$\n\n\\vspace{0.05in}\n\\noindent$\\textsc{Rev}(M, v, D)$ (or $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$) is the revenue of the mechanism $M$ (or $M^{(\\beta)}$) while the buyers' types are drawn from $D$ and buyer $i$'s valuation is $v_i(t_i,\\cdot)$ (or $v_i^{(\\beta_i)}(t_i,\\cdot)$). $\\sigma_{iS}(t_i)$ (or $\\sigma^{(\\beta)}_{iS}(t_i)$) is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i$ in mechanism $M$ (or $M^{(\\beta)}$)\n\\end{lemma}\n\\notshow{\nFrom now on, we fix $M^{(\\beta)}$ to be the mechanism that is constructed by setting $\\eta$ to be $1\/2$ and $\\epsilon$ be a extremely tiny positive constant $\\epsilon_o$ in Lemma~\\ref{lem:relaxed valuation}.\n\\begin{corollary}\n\tFor any $\\beta$, there exists a mechanism $M^{(\\beta)}$ such that\n\t$$\\textsc{Rev}(M, v, D)\\leq 2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\epsilon_0.$$\n\\end{corollary}}\n\\subsection{Virtual Valuation for the Relaxed Valuation}\\label{sec:virtual for relaxed}\nFor any $\\beta$, based on the same partition of the type sets as in the beginning of Section~\\ref{sec:flow}, we construct a flow $\\lambda^{(\\beta)}$ that respects the partial specification, such that the corresponding virtual valuation function for $v^{(\\beta)}$ has the same appealing properties as in the additive case.\nFor the relaxed valuation, as {$\\lambda_i^{(\\beta)}(t_i, t_i')$} is only positive for types $t_i$, $t_i'\\in R_j^{(\\beta_i)}$ that only differ in the $j$-th coordinate, the contribution from item $j$ to the virtual valuation solely depends on $t_{ij}$ and $t'_{ij}$ but not $t_{ik}$ for any other item $k\\in S$\n. Notice that this property does not hold for the original valuation, and it is the main reason why we choose the relaxed valuation as in Definition~\\ref{def:relaxed valuation}. Moreover, we can choose $\\lambda_i^{(\\beta)}$ carefully so that the virtual valuation of $v^{(\\beta)}$ has the following format:\n\n\n\n\n\\begin{lemma}\\label{lem:subadditive flow properties}\n\tLet $F_{ij}$ be the distribution of $V_i(t_{ij})$ when $t_{ij}$ is drawn from $D_{ij}$. For any $\\beta$, there exists a flow $\\lambda^{(\\beta)}_i$ such that the corresponding virtual value function $\\Phi^{(\\beta_i)}_{i}(t_{i}, \\cdot)$ of valuation $v_i^{(\\beta_i)}(t_i,\\cdot)$ satisfies the following properties:\n\n\\vspace{.05in}\t\n\\noindent 1. For any $t_{i}\\in R^{(\\beta_i)}_{0}$, $\\Phi^{(\\beta_i)}_{i}(t_{i},S) = v_i(t_i, S)$.\n\n\\vspace{.05in}\n\\noindent 2. For any $j>0$, $t_{i}\\in R^{(\\beta_i)}_{j}$, $\\Phi_{i}^{(\\beta_i)}(t_{i},S)\\leq v_i (t_{i}, S)\\cdot\\mathds{1}[j\\notin S]+\\left(v_i (t_{i}, S\\backslash\\{j\\})+{\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))\\right)\\cdot\\mathds{1}[j\\in S],$ where ${\\tilde{\\varphi}}_{ij}(V_i(t_{ij}))$ is the Myerson's ironed virtual value for $V_i(t_{ij})$ with respect to $F_{ij}$.\n\\end{lemma}\n\nThe proof of Lemma \\ref{lem:subadditive flow properties} is postponed to Appendix~\\ref{sec:proof_virtual_relaxation}.\nNext, we use the virtual welfare of the allocation $\\sigma^{(\\beta)}$ to bound the revenue of $M^{(\\beta)}$.\n\n\\begin{lemma}\\label{lem:upper bound the revenue of the relaxed mechanism}\n\tFor any $\\beta$, \\begin{align*} &\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\leq \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot\\Phi^{(\\beta_i)}_i(t_i,S)\\\\\n \\leq &\t\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}\\left[t_i\\in R_0^{(\\beta_i)}\\right]\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S)\\\\\n &+ \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in[m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot \\left(\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\backslash\\{j\\})+\\sum_{S:j\\notin S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S)\\right)\\\\\n &+ \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{j\\in[m]} \\mathds{1}\\left[t_i\\in R_j^{(\\beta_i)}\\right]\\cdot\\pi^{(\\beta)}_{ij}(t_i)\\cdot {\\tilde{\\varphi}}_{ij}(t_{ij}),\\end{align*}\n where $ \\pi_{ij}^{(\\beta)}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)$. {\\bf \\textsc{Non-Favorite}$(M, \\beta)$} denotes the sum of the first two terms. {\\bf \\textsc{Single}$(M, \\beta)$} denotes the last term. \\end{lemma}\n\n\\begin{proof}\nThe Lemma follows easily from the properties in Lemma~\\ref{lem:subadditive flow properties} and Theorem~\\ref{thm:revenue less than virtual welfare}.\n\\end{proof}\n\nWe obtain Theorem~\\ref{thm:revenue upperbound for subadditive} by combining Lemma~\\ref{lem:relaxed valuation} and~\\ref{lem:upper bound the revenue of the relaxed mechanism}.\n \\begin{theorem}\\label{thm:revenue upperbound for subadditive}\nFor any mechanism $M$ and any $\\beta$,\n$$\\textsc{Rev}{(M,v,D)}\\leq 4\\cdot\\textsc{Non-Favorite}(M, \\beta)+2\\cdot\\textsc{Single}(M,\\beta).$$\n\\end{theorem}\n\n\\begin{prevproof}{Theorem}{thm:revenue upperbound for subadditive}\nFirst, let's look at the value of $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)$. When $t_i\\in R_j^{(\\beta_i)}$ for some $j>0$ and $j\\in S$, $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)= v_i(t_i, S\\backslash\\{j\\})+V_i(t_{ij})-v_i(t_i,S)\\leq v_i(t_i, S\\backslash\\{j\\}),$ because $V_i(t_{ij})\\leq v_i(t_i,S)$. For the other cases, $v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)=0$. Therefore,\n\\begin{align*}\\label{eq:bounding delta}\n\t&\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\nonumber\\\\\n\\leq &\\sum_i\\sum_{t_i} f_i(t_i)\\sum_{j} \\mathds{1}[t_i\\in R_j^{(\\beta_i)}]\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\cdot v_i(t_i, S\\backslash\\{j\\})\\nonumber\\\\\n\\leq &\\textsc{Non-Favorite}(M,\\beta)~~~~~~~~~\\text{(Definition of $\\textsc{Non-Favorite}(M,\\beta)$)}\n\\end{align*}\n\nOur statement follows from combining Lemma~\\ref{lem:relaxed valuation}, Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism} with the inequality above.\n\\end{prevproof}\n\\begin{comment}\nNow we bound $\\textsc{Rev}(M, v, D)$. By Lemma~\\ref{lem:relaxed valuation},\n\t\\begin{align*}&\\textsc{Rev}(M, v, D)\\\\\n\t\\leq& 2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}\\\\\n+&2\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\\\\t\n\\leq &2\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+2\\cdot\\textsc{Non-Favorite}(M,\\beta)~~~~\\text{(Equation~(\\ref{eq:bounding delta}))}\\\\\n\\leq & 4\\cdot\\textsc{Non-Favorite}(M,\\beta)+2\\cdot\\textsc{Single}(M,\\beta)~~~~\\text{(Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism})}.\n\t\\end{align*}\n\\end{comment}\n\n\n\n\\subsection{Upper Bound for the Revenue of Subadditive Buyers}~\\label{sec:choice of beta}\n In Section~\\ref{sec:valuation relaxation}, we have argued that for any $\\beta$, there exists a mechanism $M^{(\\beta)}$ such that its revenue with respect to the relaxed valuation $v^{(\\beta)}$ is comparable to the revenue of $M$ with respect to the original valuation. In Section~\\ref{sec:virtual for relaxed}, we have shown for any $\\beta$ how to choose a flow to obtain an upper bound for $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)$ and also an upper bound for $\\textsc{Rev}(M,v,D)$. Now we specify our choice of $\\beta$.\n\nIn~\\cite{CaiDW16}, the authors fixed a particular $\\beta$, and shown that under any allocation rule, the corresponding benchmark can be bounded by the sum of the revenue of a few simple mechanisms. However, for valuations beyond additive and unit-demand, the benchmark becomes much more challenging to analyze\\footnote{Indeed, the difficulties already arise for valuations as simple as $k$-demand. A bidder's valuation is $k$-demand if her valuation is additive subject to a uniform matroid with rank $k$.}. We adopt an alternative and more flexible approach to obtain a new upper bound. Instead of fixing a single $\\beta$ for all mechanisms, we customize a different $\\beta$ for every different mechanism $M$. Next, we relax the valuation and design the flow based on the chosen $\\beta$ as specified in Section~\\ref{sec:valuation relaxation} and \\ref{sec:virtual for relaxed}.\n Then we upper bound the revenue of $M$ with the benchmark in Theorem~\\ref{thm:revenue upperbound for subadditive} and argue that for any mechanism $M$, the corresponding benchmark can be upper bounded by the sum of the revenue of a few simple mechanisms. As we allow $\\beta$, in other words the flow $\\lambda^{(\\beta)}$, to depend on the mechanism, our new approach may provide a better upper bound. As it turns out, our new upper bound is indeed easier to analyze.\n\n Lemma~\\ref{lem:requirement for beta} specifies the two properties of our $\\beta$ that play the most crucial roles in our analysis. We construct such a $\\beta$ in the proof of Lemma~\\ref{lem:requirement for beta}, however the construction is not necessarily unique and any $\\beta$ satisfying these two properties suffices. Note that our construction heavily relies on property \\textbf{(i)} of Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:requirement for beta}\n\tFor any constant $b\\in (0,1)$ and any mechanism $M$, there exists a $\\beta$ such that: for the mechanism $M^{(\\beta)} $ constructed in Lemma~\\ref{lem:relaxed valuation} according to $\\beta$, any $i\\in[n]$ and $j\\in[m]$,\n\n\\noindent\\emph{\\textbf{(i)}} $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]\\leq b$;\n\n\\noindent\\emph{\\textbf{(ii)}} $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\leq \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\/ b$, where $\\pi_{ij}^{(\\beta)}(t_i) = \\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)$.\n\\end{lemma}\n\nBefore proving Lemma~\\ref{lem:requirement for beta}, we provide some intuition behind the two required properties.\nProperty \\textbf{(i)} is used to guarantee that if item $j$'s price for bidder $i$ is higher than $\\beta_{ij}$ for all $i$ and $j$ in an RSPM, for any item $j'$ and any bidder $i'$, $j'$ is still available with probability at least $(1-b)$ when $i'$ is visited. As for any bidder $k\\neq i'$ to purchase item $j'$, $V_k(t_{kj'})$ must be greater than her price for item $j'$. By the union bound, the probability that there exists such a bidder is upper bounded by the LHS of property (i), and therefore is at most $b$. With this guarantee, we can easily show that the RSPM achieves good revenue (Lemma~\\ref{lem:neprev}). Property \\textbf{(ii)} states that the ex-ante probability for bidder $i$ to receive an item $j$ in $M^{(\\beta)}$ is not much bigger than the probability that bidder $i$'s value is larger than item $j$. This is crucial for proving our key Lemma~\\ref{lem:hat Q}, in which we argue that two different valuations provide comparable welfare under the same allocation rule $\\sigma^{(\\beta)}$. With Lemma~\\ref{lem:hat Q}, we can show that the ASPE obtains good revenue.\n\n\\begin{prevproof}{Lemma}{lem:requirement for beta}\n\tWhen there is only one buyer, we can simply set every $\\beta_j$ to be $0$ and both conditions are satisfied.\n\tWhen there are multiple players, we let $$\\beta_{ij}:=\\inf\\{{x\\geq 0}: \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x\\right] \\leq b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\},$$ where $ \\pi_{ij}(t_i)=\\sum_{S:j\\in S}\\sigma_{iS}(t_i)$. Clearly, when the distribution of $V_i(t_{ij})$ is continuous, then\n\\begin{equation}\\label{equ:beta_second_condition}\n\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i),\n\\end{equation}\nand therefore for any $j$, $$\\sum_i\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=b\\cdot\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\leq b.$$\n\nSo the first condition is satisfied. The second condition holds because by the first property in Lemma~\\ref{lem:relaxed valuation}, $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)$.\n\nWhen the distribution for $V_i(t_{ij})$ is discrete, it is possible that Equation~\\ref{equ:beta_second_condition} does not hold, but this is essentially a tie breaking issue and not hard to fix. Let $\\epsilon>0$ be an extremely small constant that is smaller than $\\left|V_i(t_{ij})-V_i(t'_{ij})\\right|$ for any $t_{ij}, t'_{ij}\\in T_{ij}$, any $i$ and any $j$. Let $\\zeta_{ij}$ be a random variable uniformly distributed on $[0,\\epsilon]$, and think of it as a random rebate that the seller gives to bidder $i$ when she purchases item $j$. Now we modify the definition of $\\beta_{ij}$ as $\\beta_{ij}:=\\inf\\{{x\\geq 0}: \\Pr_{t_{ij},\\zeta_{ij}}[V_i(t_{ij})+\\zeta_{ij}\\geq x] \\leq b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)\\}.$\n\n\\notshow{\\begin{equation}\n\\epsilon_1=\\epsilon\\cdot \\frac{\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]-b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)}{\\Pr_{t_{ij}}\\left[V_i(t_{ij})= \\beta_{ij}\\right]}\n\\end{equation}\n\nBy the definition of $\\beta_{ij}$, $\\epsilon_1\\in [0,\\epsilon)$. Let $\\zeta_{ij}$ be a random variable uniformly distributed on $[\\epsilon_1-\\epsilon,\\epsilon_1]$. It is not hard to verify that $\\Pr_{t_{ij},\\zeta_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+\\zeta_{ij}\\right]=b\\cdot\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\pi_{ij}(t_i)$. For those probabilities related to $\\beta_{ij}$ shown in the proofs below, which is written for simplification, we refer to this definition. With regard to this definition, we make essentially small changes for the mechanism described below. Instead of using fixed price, add a small disturbance $\\zeta_{ij}$ on item $j$'s price for bidder $i$. Since $\\epsilon$ can be chosen as small as possible, the revenue will only be affected by a small constant. All the argument maintain to be true.}\n\nBoth of the two properties in Lemma~\\ref{lem:requirement for beta} hold if we replace $V_i(t_{ij})$ with $V_i(t_{ij})+\\zeta_{ij}$. The only change we need to make in the mechanism is to actually give the bidders $\\zeta_{ij}$ as the corresponding rebate. Since we can choose $\\epsilon$ to be arbitrarily small, the sum of the rebate is also arbitrarily small. For the simplicity of the presentation, we will omit $\\zeta_{ij}$ and $\\epsilon$ in the rest of the paper.\nThe random rebate indeed makes our mechanism randomized(according to the random variable $\\zeta_{ij}\\sim [0,\\epsilon]$). However, the randomized mechanism is a uniform distribution of deterministic DSIC mechanisms (after determining all $\\zeta_{ij}$), and the expected revenue of the randomized mechanism is simply the average revenue of all these deterministic mechanisms. Therefore, there must be one realization of the rebates such that the corresponding deterministic mechanism has revenue above the expectation, i.e., the expected revenue of the randomized one. Thus if the randomized mechanism is proved to achieve some approximation ratio, there must exist a deterministic one that achieves the same ratio. The deterministic mechanism will use a fixed value $z_{ij}\\in [0,\\epsilon]$ as the rebate.\n\nSimilarly, the same issue about discrete distributions arises when we define some other crucial parameters later, e.g., in the Definition of $c$, $c_i$ and $\\tau_i$. We can resolve all of them together using the trick (adding a random rebate) described above, and we will not include a detailed proof for those cases.\n\\end{prevproof}\n\n\n\n\n\\subsubsection{Bad Example for Chawla and Miller's Approach}\\label{sec:cs_ocrs}\nLet bidders be constrained additive and $\\mathcal{F}_i$ be bidder $i$ feasibility constraint. We use $P_{\\mathcal{F}_i}=conv(\\{1^S | S\\in \\mathcal{F}_i\\})$ to denote the feasibility polytope of bidder $i$. Let $\\{q_{ij}\\}_{i\\in[n],j\\in[m]}$ be a collection of probabilities that satisfy $\\sum_i q_{ij}\\leq 1\/2$ for all item $j$ and $\\boldsymbol{q}_i = (q_{i1},\\ldots, q_{im})\\in b\\cdot P_{\\mathcal{F}_i}$. Let $\\beta_{ij}=F_{ij}^{-1}(q_{ij})$. The analysis by Chawla and Miller~\\cite{ChawlaM16} needs to upper bound $\\sum_{i, j}\\beta_{ij}\\cdot q_{ij}$ using the revenue of some BIC mechanism. When $\\mathcal{F}_i$ is a matroid for every bidder $i$, this expression can be upper bounded by the revenue of a sequential posted price mechanism constructed using OCRS from~\\cite{FeldmanSZ16}. Here we show that if the bidders have general downward closed feasibility constraints, this expression is gigantic. More specifically, we prove that even when there is only one bidder, the expression could be $\\Omega\\left(\\frac{\\sqrt{m}}{\\log m}\\right)$ times larger than the optimal social welfare.\n\nConsider the following example.\n\\begin{example}\\label{ex:counterexample ocrs}\n\tThe seller is selling $m=k^2$ items to a single bidder. The bidder's value for each item is drawn i.i.d. from distribution $F$, which is the equal revenue distribution truncated at $k$, i.e.,\n\t\\[F(x)=\n\\begin{cases}\n1-\\frac{1}{x},&\\text{if}~~x0$, there exists $q\\in b\\cdot P_{\\mathcal{F}}$ such that for sufficiently large $k$, $$\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=\\Theta(\\frac{k}{\\log k })\\cdot SW$$.\n\\end{lemma}\n\n\\begin{proof}\n\nFor any $b>0$, consider the following feasible allocation rule: w.p. $(1-b)$, don't allocate anything, and w.p. $b$, give the buyer one of the sets $A_i$ uniformly at random. The corresponding ex-ante probability vector $q$ satisfies $q_j=\\frac{b}{k}, \\forall j\\in [m]$. Thus $q\\in b\\cdot P_{\\mathcal{F}}$. Since $q_j<\\frac{1}{k}$, $F^{-1}(1-q_j)=k$ for all $j\\in [m]$. We have $\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=k^2\\cdot \\frac{b}{k}\\cdot k=b\\cdot k^2$. We use $V_i$ to denote the random variable of the bidder's value for set $A_i$. It is not hard to see that $SW={\\mathbb{E}}[\\max_{i\\in[k]} V_i]$. \n\n\\begin{lemma}\nFor any $i\\in [k]$,\n\\[\\Pr\\left[V_i>3\\cdot k\\log(k)\\right]\\leq k^{-3}\\]\n\\end{lemma}\n\\begin{proof}\nLet $X$ be random variable with cdf $F$. Notice $E[X]=\\log(k)$, $E[X^2]=2k$, and $|X|\\leq k$.\nFor every $i$, by the Bernstein concentration inequality, for any $t>0$,\n\\[\\Pr\\left[V_i-k\\log(k)>t\\right]\\leq \\exp\\left(-\\frac{\\frac{1}{2}t^2}{2k^2+\\frac{1}{3}kt}\\right)\\]\nChoose $t=2k\\log(k)$, we have\n\\[\\Pr\\left[V_i>3k\\log(k)\\right]\\leq \\exp(-3\\log(k))=k^{-3}\\]\n\\end{proof}\n\nBy the union bound, $\\Pr[\\max_{i\\in[k]}V_i>3\\cdot k\\log(k)]\\leq k^{-2}$. Therefore, ${\\mathbb{E}}[\\max_{i\\in[k]} V_i]\\leq 3 k\\log k +k^2\\cdot k^{-2}\\leq 4 k\\log k$.\n\\end{proof}\n\n\\notshow{In the analysis of the paper by Chawla and Miller~\\cite{ChawlaM16}, they rely on the following lemma in single buyer auction.\n\\begin{lemma}\\label{lem:shuchi}\n~\\cite{ChawlaM16}~\\cite{FeldmanSZ16}Suppose the buyer is additive within a matroid constraint $\\mathcal{F}$ and let $P_{\\mathcal{F}}=conv(\\{1^S | S\\in \\mathcal{F}\\})$ be the feasibility polytope. For any constant $b\\in (0,1)$ and ex-ante vector $q\\in bP_{\\mathcal{F}}$, let $\\beta$ be the corresponding ex-ante prices. In other words, $\\beta_j$ is chosen such that the probability that the value for item $j$\nexceeds this price is precisely $q_j$. Then the value $\\beta\\cdot q$ can be bounded within some constant factor by the revenue of a posted price mechanism with a more strict constraint, which guarantees that the ex-ante probability of the buyer getting item $j$ is at most $q_j$.\n\\end{lemma}\n\nThe result can be generated for $\\mathcal{F}$ beyond a matroid~\\cite{FeldmanSZ16}. However, Lemma~\\ref{lem:shuchi} does not hold for general downward-close $\\mathcal{F}$. In this section we provide a counterexample with some general downward-close constraint $\\mathcal{F}$, such that the term $\\beta\\cdot q$ cannot be upper bounded by any single buyer mechanism, within in a constant factor.\n\n\\begin{lemma}\nConsider the following single buyer auction with $m=k^2$ i.i.d. items. Items are divided into $k$ disjoint sets $A_1,...,A_k$, each with size $k$. The value distribution $F$ for a single item is defined as the equal-revenue distribution truncated at value $k$, i.e.,\n\\[F(x)=\n\\begin{cases}\n1-\\frac{1}{x},&\\text{if}~~x0$, there exists $q\\in bP_{\\mathcal{F}}$ such that for sufficiently large $k$,\n\\begin{equation}\\label{equ:q times beta}\n\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=\\Theta(\\frac{k}{log(k)})\\cdot SW\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\nFor any $b>0$, consider the following feasible allocation rule. With probability $(1-b)$, don't allocate anything. With probability $b$, give the buyer one of the sets $A_p$ uniformly at random. The corresponding ex-ante probability vector $q$ satisfies $q_j=\\frac{b}{k}, \\forall j\\in [m]$. Thus $q\\in bP_{\\mathcal{F}}$.\n\nSince $q_j<\\frac{1}{k}$, $F^{-1}(1-q_j)=k$ for all $j\\in [m]$. We have\n\\begin{equation}\n\\sum_{j\\in[m]}q_j\\cdot F^{-1}(1-q_j)=k^2\\cdot \\frac{b}{k}\\cdot k=b\\cdot k^2\n\\end{equation}\n\nConsider the optimal social welfare. For every bundle $p$, denote $V_p$ the random variable of the buyer's value for bundle $p$. Notice $A_p\\in \\mathcal{F}$, $V_p$ is the sum of $k$ independent random variables with cdf $F$. With Bernstein Inequality, $V_p=O(klog(k))$ with high probability.\n\\notshow{\\begin{lemma}\\label{lem:bernstein}\n(Bernstein Inequality)~\\cite{bernstein1924modification}: Suppose $X_1,...,X_n$ are independent random variables with zero mean, and $|X_i|\\leq M$ almost surely for all $i$. Then for any $t>0$,\n\\[\\Pr\\left[\\sum_{i=1}X_i>t\\right]\\leq exp\\left(-\\frac{\\frac{1}{2}t^2}{\\sum_{i=1}^nE[X_i^2]+\\frac{1}{3}Mt}\\right)\\]\n\\end{lemma}}\n\\begin{lemma}\nFor any $p\\in [k]$,\n\\[\\Pr\\left[V_p>3\\cdot k\\log(k)\\right]\\leq k^{-3}\\]\n\\end{lemma}\n\\begin{proof}\nLet $X$ be random variable with cdf $F$. Notice $E[X]=\\log(k)$, $E[X^2]=k-1$, and $|X|\\leq k$.\nFor every $p$, by the Bernstein concentration inequality, for any $t>0$,\n\\[\\Pr\\left[V_p-k\\log(k)>t\\right]\\leq exp\\left(-\\frac{\\frac{1}{2}t^2}{k^2+\\frac{1}{3}kt}\\right)\\]\nChoose $t=2k\\log(k)$, we have\n\\[\\Pr\\left[V_p>3k\\log(k)\\right]\\leq exp(-3\\log(k))=k^{-3}\\]\n\\end{proof}\n\nWith union bound,\n\\[\\Pr[\\max_{p\\in [k]}V_p\\geq 3k\\log(k)]\\leq \\sum_{p\\in [k]}\\Pr\\left[V_p>3k\\log(k)\\right]\\leq k^{-2}\\]\n\nNotice the social welfare for the mechanism is at most \\\\\n\\noindent$\\max_{p\\in [k]}V_p$ due to the feasible constraint $\\mathcal{F}$. Also notice that $\\max_{p\\in [k]}V_p\\leq k^2$, we have\n\\begin{align*}\nSW&\\leq E[\\max_{p\\in [k]}V_p]\\leq \\bigg(3k\\log(k)\\cdot \\Pr[\\max_{p\\in [k]}V_p\\leq 3k\\log(k)]\\\\\n&+k^2\\cdot\\Pr[\\max_{p\\in [k]}V_p> 3k\\log(k)]\\bigg)\\\\\n&\\leq 3k\\log(k)+k^2\\cdot k^{-2}=O(k\\log(k))\n\\end{align*}\n\n\\noindent When $k$ is sufficiently large, Equation~\\ref{equ:q times beta} holds.\n\n\\begin{comment}\nNow consider the optimal revenue in this auction. Let $REV_k$ be the optimal revenue selling the items in set $A_k$ to a single buyer. \\ref{HartN12} has shown that $REV_k=\\Theta(n\\log(n))$ for the equal-revenue distribution $F(x)$. Since people are only interested one set of items,\n\\begin{equation}\nREV\\leq \\sum_{k\\in [n]}REV_k=\\Theta(n^2\\log(n))\n\\end{equation}\n\\end{comment}\n\n\\end{proof}}\n\n\n\n\\section{Introduction}\nIn Mechanism Design, we aim to design a mechanism\/system such that a group of strategic participants, who are only interested in optimizing their own utilities, are incentivized to choose actions that also help achieve the designer's objective. Clearly, the quality of the solution with respect to the designer's objective is crucial. However, perhaps one should also pay equal attention to another criterion of a mechanism, that is, its simplicity. When facing a complicated mechanism, participants may be confused by the rules and thus unable to optimize their actions and react in unpredictable ways instead. This may lead to undesirable outcomes and poor performance of the mechanism. An ideal mechanism would be optimal and simple. However, such cases of simple mechanisms being optimal only exist in single-item auctions, with the seminal examples of auctions by Vickrey~\\cite{Vickrey61} and Myerson~\\cite{Myerson81}, while none has been discovered in broader settings. Indeed, we now know that even in fairly simple settings the optimal mechanisms suffer many undesirable properties including randomization, non-monotonicity, and others~\\cite{RochetC98, Tha04, Pavlov11a, HartN13, HartR12, BriestCKW10, DaskalakisDT13, DaskalakisDT14}.\nTo move forward, one has to compromise -- either settle with optimal but somewhat complex mechanisms or turn to simple but approximately optimal solutions.\n\nRecently, there has been extensive research effort focusing on the latter approach, that is, studying the performance of simple mechanisms through the lens of approximation. In particular, a central problem on this front is how to design simple and approximately revenue-optimal mechanisms in multi-item settings. For instance, when bidders have unit-demand valuations, we know sequential posted price mechanisms approximates the optimal revenue due to a line of work initiated by Chawla et al.~\\cite{ChawlaHK07, ChawlaHMS10, ChawlaMS15, CaiDW16}. When buyers have additive valuations, we know that either selling the items separately or running a VCG mechanism with per bidder entry fee approximates the optimal revenue due to a series of work initiated by Hart and Nisan~\\cite{HartN12, CaiH13, LiY13, BabaioffILW14, Yao15, CaiDW16}. Recently, Chawla and Miller~\\cite{ChawlaM16} generalized the two lines of work described above to matroid rank functions\\footnote{{Here is a hierarchy of the valuation functions. additive \\& unit-demand $\\subseteq$ matroid rank $\\subseteq$ constrained additive \\& submodular\n $\\subseteq$ XOS $\\subseteq$ subadditive. A function is constrained additive if it is additive up to some downward closed feasibility constraints. The class of submodular functions is neither a superset nor a subset of the class of constrained additive functions.} See Definition~\\ref{def:valuation classes} for the formal definition. }. They show that a simple mechanism, the sequential two-part tariff mechanism, suffices to extract a constant fraction of the optimal revenue. For subadditive valuations beyond matroid rank functions, we only know how to handle a single buyer~\\cite{RubinsteinW15}\\footnote{All results mentioned above assume that the buyers' valuation distributions are over independent items. For additive and unit-demand valuations, this means a bidder's values for the items are independent. The definition is generalized to subadditive valuations by Rubinstein and Weinberg~\\cite{RubinsteinW15}. See Definition~\\ref{def:subadditive independent}.}. It is a major open problem to extend this result to multiple subadditive buyers.\n\nIn this paper, we unify and strengthen all the results mentioned above via an extension of the duality framework proposed by Cai et al.~\\cite{CaiDW16}. Moreover, we show that even when there are multiple buyers with XOS valuation functions, there exists a simple, deterministic and Dominant Strategy Incentive Compatible (DSIC) mechanism that achieves a constant fraction of the optimal Bayesian Incentive Compatible (BIC) revenue\\footnote{A mechanism is Bayesian Incentive Compatible (BIC) if it is in every bidder's interest to tell the truth, assuming that all other bidders' reported their values. A mechanism is Dominant Strategy Incentive Compatible (DSIC) if it is in every bidder's interest to tell the truth no matter what reports the other bidders make.}. For subadditive valuations, our approximation ratio degrades to $O(\\log m)$.\n\n\\begin{informaltheorem}\n\tThere exists a simple, deterministic and DSIC mechanism that achieves a constant fraction of the optimal BIC revenue in multi-item settings, when the buyers' valuation distributions are XOS over independent items. When the buyers' valuation distributions are subadditive over independent items, our mechanism achieves at least $\\Omega(\\frac{1}{\\log m})$ of the optimal BIC revenue, where $m$ is the number of items.\n\\end{informaltheorem}\n\nThe original paper by Cai et al.~\\cite{CaiDW16} provided a unified treatment for additive and unit-demand valuations. However, it is inadequate to provide an analyzable benchmark for even a single subadditive bidder. In this paper, we show how to extend their duality framework to accommodate general subadditive valuations. Using this extended framework, we substantially improve the approximation ratios for many of the settings discussed above, and in the meantime generalize the results to broader cases. See Table~\\ref{table:comp} for the comparison between the best ratios reported in the literature and the new ratios obtained in this work.\n\n\\begin{table*}\n\n\\centering\n\\begin{tabular}{|c|l|p{2.1cm}|c|p{2cm}|c|c|}\n\\hline\n\t& &\\centering Additive or Unit-demand& \\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{2.2cm}{\\centering Matroid-Rank}}}& \\centering Constrained Additive&\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1cm}{\\centering XOS}}}&\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{2cm}{\\centering Subadditive}}} \\\\\n\\hline\n\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1.3cm}{\\centering Single Buyer}}} & Previous& 6~\\cite{BabaioffILW14} or 4~\\cite{ChawlaMS15}& 31.1* &\\centering 31.1~\\cite{ChawlaM16} & 338* & 338~\\cite{RubinsteinW15} \\\\\\cline{2-7}\n\t\t&This Paper & \\centering - &{11*} &\\centering{11}&{40*}& {40} \\\\\\cline{2-7}\n\n\\hline\\hline\n\\multirow{2}{*}{\\rotatebox[origin=c]{0}{\\parbox[c]{1.3cm}{\\centering Multiple Buyer}}} & Previous& 8~\\cite{CaiDW16} or 24~\\cite{CaiDW16}& 133~\\cite{ChawlaM16}&\\centering ? & ? &? \\\\ \\cline{2-7}\n\t\t& This Paper& \\centering - & 70* &\\centering 70 & 268 &$O(\\log m)$ \\\\\n\\hline\n\\end{tabular}\\\\\n* The result is implied by another result for a more general setting.\n\\caption{Comparison of approximation ratios between previous and current work.}\n\t\t \\label{table:comp}\n\\end{table*}\n\n\\notshow{\n\\begin{table}\n\t\\begin{tabular}{|l|c|c|c|c|c|}\n\t\\hline\n\t\t & Additive or Unit-demand& Matroid-Rank& Constrained Additive&XOS&Subadditive \\\\\n\t\t\\hline\n\t\tSingle Buyer& 6~\\cite{BabaioffILW14} or 4~\\cite{ChawlaMS15}& 33.1~\\cite{ChawlaM16} &$\\rightarrow$ &$\\rightarrow$ & 338~\\cite{RubinsteinW15} \\\\\\hline\n\t\tMultiple Buyers& 8~\\cite{CaiDW16} or 24~\\cite{CaiDW16}& 133~\\cite{ChawlaM16}& ? & ? &? \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t \\caption{The best approximation ratios known prior to this work.}\n\t\t \\label{table:old}\n\\end{table}\n\n\\begin{table}\\centering\n\t\\begin{tabular}{|l|c|c|c|c|}\n\t\\hline\n\t\t &Matroid-Rank& Constrained Additive&XOS&Subadditive \\\\\n\t\t\\hline\n\t\tSingle Buyer& $\\rightarrow$ &14& $\\rightarrow$& 48 \\\\\\hline\n\t\tMultiple Buyers& $\\rightarrow$ & 70 & 268 &$O(\\log m)$ \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t \\caption{New approximation ratios obtained in this work.}\n\t\t \\label{table:new}\n\\end{table}\n\n}\nOur mechanism is either a \\emph{rationed sequential posted price mechanism} (\\textbf{RSPM}) or an \\emph{anonymous sequential posted price with entry fee mechanism} (\\textbf{ASPE}). In an RSPM, there is a price $p_{ij}$ for buyer $i$ if she wants to buy item $j$, and she is allowed to purchase at most one item. We visit the buyers in some arbitrary order and the buyer takes her favorite item among the available items given the item prices for her. Here we allow personalized prices, that is, $p_{ij}$ could be different from $p_{kj}$ if $i\\neq k$. In an ASPE, every buyer faces the same collection of item prices $\\{p_j\\}_{j\\in[m]}$. Again, we visit the buyers in some arbitrary order. For each buyer, we show her the available items and the associated price for each item. Then we ask her to pay the entry fee to enter the mechanism, which may depend on what items are still available and the identity of the buyer. If the buyer accepts the entry fee, she can proceed to purchase any item at the given prices; if she rejects the entry fee, then she will leave the mechanism without receiving anything. Given the entry fee and item prices, the decision making for the buyer is straightforward, as she only accepts the entry fee when the surplus for winning her favorite bundle is larger than the entry fee. Therefore, both RSPM and ASPE are DSIC and ex-post Individually Rational (ex-post IR).\n\n\n\\subsection{Our Contributions}\nTo obtain the new generalizations, we provide important extensions to the duality framework in~\\cite{CaiDW16}, as well as novel analytic techniques and new simple mechanisms.\n\n\\vspace{.05in}\n\\noindent \\textbf{1. Accommodating subadditive valuations:} the original duality framework in~\\cite{CaiDW16} already unified the additive case and unit-demand case by providing an approximately tight upper bound for the optimal revenue using a single dual solution. A trivial upper bound for the revenue is the social welfare, which may be arbitrarily bad in the worst case. The duality based upper bound in~\\cite{CaiDW16} improves this trivial upper bound, the social welfare, by substituting the value of each buyer's favorite item with the corresponding Myerson's virtual value. However, the substitution is viable only when the following condition holds -- the buyer's marginal gain for adding an item solely depends on her value for that item (assuming it's feasible to add that item\\footnote{WLOG, we can reduce any constrained additive valuation to an additive valuation with a feasibility constraint (see Definition~\\ref{def:valuation classes})}), but not the set of items she has already received. This applies to valuations that are additive, unit-demand and more generally constrained additive, but breaks under more general valuation functions, e.g., submodular, XOS or subadditive valuations. As a consequence, the original dual solution from~\\cite{CaiDW16} fails to provide a nice upper bound for more general valuations. To overcome this difficulty, we take a different approach. Instead of directly studying the dual of the original problem, we first relax the valuations and argue that the optimal revenue of the relaxed valuation is comparable to the original one. Then, since we choose the relaxation in a particular way, by applying a dual solution similar to the one in~\\cite{CaiDW16} to the relaxed valuation, we recover an upper bound of the optimal revenue for the relaxed valuation resembling the appealing format of the one in~\\cite{CaiDW16}. Combining these two steps, we obtain an upper bound for subadditive valuations that is easy to analyze. Indeed, we use our new upper bound to improve the approximation ratio for a single subadditive buyer from $338$~\\cite{RubinsteinW15} to $40$. See Section~\\ref{sec:valuation relaxation} for more details.\n\n\\vspace{.05in}\n\\noindent\\textbf{2. An adaptive dual:} our second major change to the framework is that we choose the dual in an adaptive manner. In~\\cite{CaiDW16}, a dual solution $\\lambda$ is chosen up front inducing a virtual value function $\\Phi(\\cdot)$, then the corresponding optimal virtual welfare is used as a benchmark for the optimal revenue. Finally, it is shown that the revenue of some simple mechanism is within a constant factor of the optimal virtual welfare. Unfortunately, when the valuations are beyond additive and unit-demand, the optimal virtual welfare for this particular choice of virtual value function becomes extremely complex and hard to analyze. Indeed, it is already challenging to bound when the buyers' valuations are $k$-demand. In this paper, we take a more flexible approach. For any particular allocation rule $\\sigma$, we tailor a special dual $\\lambda^{(\\sigma)}$ based on $\\sigma$ in a fashion that is inspired by Chawla and Miller's ex-ante relaxation~\\cite{ChawlaM16}. Therefore, the induced virtual valuation $\\Phi^{(\\sigma)}$ also depends on $\\sigma$. By duality, we can show that the optimal revenue obtainable by $\\sigma$ is still upper bounded by the virtual welfare with respect to $\\Phi^{(\\sigma)}$ under allocation rule $\\sigma$. Since the virtual valuation is designed specifically for allocation $\\sigma$, the induced virtual welfare is much easier to analyze. Indeed, we manage to prove that for any allocation $\\sigma$ the induced virtual welfare is within a constant factor of the revenue of some simple mechanism, when bidders have XOS valuations. See Section~\\ref{sec:virtual for relaxed} and~\\ref{sec:choice of beta} for more details.\n\n\n\\vspace{.05in}\n\n\\noindent\\textbf{3. A novel analysis and new mechanism:} with the two contributions above, we manage to derive an upper bound of the optimal revenue similar to the one in \\cite{CaiDW16} but for subadditive bidders. The third major contribution of this paper is a novel approach to analyzing this upper bound. The analysis in~\\cite{CaiDW16} essentially breaks the upper bound into three different terms-- \\textsc{Single}, \\textsc{Tail}~ and \\textsc{Core}, and bound them separately. All three terms are relatively simple to bound for additive and unit-demand buyers, but for more general settings the $\\textsc{Core}$ becomes much more challenging to handle. Indeed, the analysis in~\\cite{CaiDW16} was insufficient to tackle the $\\textsc{Core}$ even when the buyers have $k$-demand valuations\\footnote{The class of $k$-demand valuations is a generalization of unit-demand valuations, where the buyer's value is additive up to $k$ items.}-- a very special case of matroid rank valuations, which itself is a special case of XOS or subadditive valuations. Rubinstein and Weinberg~\\cite{RubinsteinW15} showed how to approximate the $\\textsc{Core}$ for a single subadditive bidder using grand bundling, but their approach does not apply to multiple bidders. Yao~\\cite{Yao15} showed how to approximate the $\\textsc{Core}$ for multiple additive bidders using a VCG with per bidder entry fee mechanism, but again it is unclear how his approach can be extended to multiple k-demand bidders. A recent paper by Chawla and Miller~\\cite{ChawlaM16} finally broke the barrier of analyzing the $\\textsc{Core}$ for multiple $k$-demand buyers. They showed how to bound the $\\textsc{Core}$ for matroid rank valuations using a sequential posted price mechanism by applying the \\emph{online contention resolution scheme (OCRS)} developed by Feldman et al.~\\cite{FeldmanSZ16}. The connection with OCRS is an elegant observation, and one might hope the same technique applies to more general valuations. Unfortunately, OCRS is only known to exist for special cases of downward closed constraints, and as we show in Section~\\ref{sec:core comparison}, the approach by Chawla and Miller cannot yield any constant factor approximation for general constrained additive valuations.\n\nWe take an entirely different approach to bound the $\\textsc{Core}$. Here we provide some intuition behind our mechanism and analysis. The $\\textsc{Core}$ is essentially the optimal social welfare induced by some truncated valuation $v'$, and our goal is to design a mechanism that extracts a constant fraction of the welfare as revenue. Let $M$ be any sequential posted price mechanism. A key observation is that when bidder $i$'s valuation is subadditive over independent items, her utility in $M$, which is the largest surplus she can achieve from the unsold items, is also subadditive over independent items. If we can argue that her utility function is $a$-Lipschitz (Definition~\\ref{def:Lipschitz}) with some small $a$, Talagrand's concentration inequality~\\cite{Talagrand1995concentration,Schechtman2003concentration} allows us to set an entry fee for the bidder so that we can extract a constant fraction of her utility just through the entry fee.\nIf we modify $M$ by introducing an entry fee for every bidder, according to Talagrand's concentration inequality, the new mechanism $M'$ should intuitively have revenue that is a constant fraction of the social welfare obtained by $M$~\\footnote{$M$'s welfare is simply its revenue plus the sum of utilities of the bidders, and $M'$ can extract some extra revenue from the entry fee, which is a constant fraction of the total utility from the bidders.}. Therefore, if there exists a sequential posted price mechanism $M$ that achieves a constant fraction of the optimal social welfare under the truncated valuation $v'$, the modified mechanism $M'$ can obtain a constant fraction of $\\textsc{Core}$ as revenue. Surprisingly, when the bidders have XOS valuations, Feldman et al.~\\cite{FeldmanGL15} showed that there exists an anonymous sequential posted price mechanism that always obtains at least half of the optimal social welfare. Hence, an anonymous sequential posted price with per bidder entry fee mechanism should approximate the $\\textsc{Core}$ well, and this is exactly the intuition behind our ASPE mechanism.\n\n To turn the intuition into a theorem, there are two technical difficulties that we need to address: (i) the Lipschitz constants of the bidders' utility functions turn out to be too large (ii) we deliberately neglected the difference in bidders' behavior under $M$ and $M'$ in hope to keep our discussion in the previous paragraph intuitive. However, due to the entry fee, bidders may end up purchasing completely different items under $M$ and $M'$, so it is not straightforward to see how one can relate the revenue of $M'$ to the welfare obtained by $M$.\n See Section~\\ref{sec:core comparison} for a more detailed discussion on how we overcome these two difficulties.\n \n\n\n\n\n\n\n\n\n\n\\subsection{Related Work}\nIn recent years, we have witnessed several breakthroughs in designing (approximately) optimal mechanisms in multi-dimensional settings. The black-box reduction by Cai et al.~\\cite{CaiDW12a,CaiDW12b,CaiDW13a,CaiDW13b} shows that we can reduce any Bayesian mechanism design problem to a similar algorithm design problem via convex optimization. Through their reduction, it is proved that all optimal mechanisms can be characterized as a distribution of virtual welfare maximizers, where the virtual valuations are computed by an LP. Although this characterization provides important insights about the structure of the optimal mechanism, the optimal allocation rule is unavoidably randomized and might still be complex as the virtual valuations are only a solution of an LP.\n\nAnother line of work considers the ``Simple vs. Optimal'' auction design problem. For instance, a sequence of results~\\cite{ChawlaHK07,ChawlaHMS10,ChawlaMS10,ChawlaMS15} show that sequential posted price mechanism can achieve $\\frac{1}{33.75}$ of the optimal revenue, whenever the buyers have unit-demand valuations over independent items. Another series of results~\\cite{HartN12,CaiH13,LiY13,BabaioffILW14,Yao15} show that the better of selling the items separately and running the VCG mechanism with per bidder entry fee achieves $\\frac{1}{69}$ of the optimal revenue, whenever the buyers' valuations are additive over independent items. Cai et al.~\\cite{CaiDW16} unified the two lines of results and improved the approximation ratios to $\\frac{1}{8}$ for the additive case and $\\frac{1}{24}$ for the unit-demand case using their duality framework.\n\nSome recent works have shown that simple mechanisms can approximate the optimal revenue even when buyers have more sophisticated valuations. For instance, Chawla and Miller~\\cite{ChawlaM16} showed that the sequential two-part tariff mechanism can approximate the optimal revenue when buyers have matroid rank valuation functions over independent items. Their mechanism requires every buyer to pay an entry fee up front, and then run a sequential posted price mechanism on buyers who have accepted the entry fee. Our ASPE is similar to their mechanism, but with two major differences: (i) since buyers are asked to pay the entry fee before the seller visits them, the buyers have to make their decisions based on the expected utility (assuming every other buyer behaves truthfully) they can receive. Hence, the mechanism is only guaranteed to be BIC and interim IR. While in our mechanism, the buyers can see what items are still available before paying the entry fee, therefore the decision making is straightforward and the ASPE is DSIC and ex-post IR; (ii) the item prices in the ASPE are anonymous, while in the sequential two-part tariff mechanism, personalized prices are allowed. For valuations beyond matroid rank functions, Rubinstein and Weinberg~\\cite{RubinsteinW15} showed that for a single buyer whose valuation is subadditive over independent items, either grand bundling or selling the items separately achieves at least $\\frac{1}{338}$ of the optimal revenue.\n\nThe Cai-Devanur-Weinberg duality framework~\\cite{CaiDW16} has been applied to other intriguing Mechanism Design problems. For example, Eden et al. showed that the better of selling separately and bundling together gets an $O(d)$-approximation for a single bidder with ``complementarity-$d$ valuations over independent items''~\\cite{EdenFFTW16a}. The same authors also proved a Bulow-Klemperer result for regular i.i.d. and constrained additive bidders~\\cite{EdenFFTW16b}. Liu and Psomas provided a Bulow-Klemperer result for {dynamic auctions}~\\cite{LiuP16}. Finally, Brustle et al.~\\cite{BrustleCWZ17} extended the duality framework to two-sided markets and used it to design simple mechanisms for approximating the Gains from Trade.\n\nStrong duality frameworks have recently been developed for one additive buyer~\\cite{DaskalakisDT13,DaskalakisDT15,Giannakopoulos14a,GiannakopoulosK14,GiannakopoulosK15}. These frameworks show that the dual problem of revenue maximization can be viewed as an optimal transport\/bipartite matching problem. Hartline and Haghpanah provided an alternative duality framework in~\\cite{HartlineH15}. They showed that if certain paths exist, these paths provide a witness of the optimality of a certain Myerson-type mechanism, but these paths are not guaranteed to exist in general. Similar to the Cai-Devanur-Weinberg framework, Carroll~\\cite{Carroll15} independently made use of a partial Lagrangian over incentive constraints. These duality frameworks have been successfully provide conditions under which a certain type of mechanism is optimal when there is a single unit-demand or additive bidder. However, none of these frameworks succeeds in yielding any approximately optimal results in multi-buyer settings.\n\n\\input{prelim}\n\\input{roadmap}\n\\input{duality}\n\\input{single_subadditive}\n\\input{multi_XOS}\n\\input{example_ocrs}\n\\newpage\n\n\\section{Multiple Bidders}\\label{sec:multi}\n\nIn this section, we prove our main result -- simple mechanisms can approximate the optimal BIC revenue even when there are multiple XOS\/subadditive bidders.\nFirst, we need the definition of supporting prices.\n\\begin{definition}[Supporting Prices~\\cite{DobzinskiNS05}]\\label{def:supporting price}\nFor any $\\alpha\\geq 1$, a type $t$ and a subset $S\\subseteq[m]$, prices $\\{p_j\\}_{j\\in S}$\nare $\\alpha$-supporting prices for $v(t,S)$ if \\textbf{(i)}\t$v(t,S') \\geq \\sum_{j\\in S'} p_j$ for all $S'\\subseteq S$ and \\textbf{(ii)} $\\sum_{j\\in S}p_j\\geq \\frac{v(t,S)}{\\alpha}$.\n\\end{definition}\n\n\n\\begin{theorem}\\label{thm:multi}\nIf for any buyer $i$, any type $t_i\\in T_i$ and any bundle $S\\in [m]$, $v_i(t_i,S)$ has a set of $\\alpha$-supporting prices $\\{\\theta_j^{S}(t_i)\\}_{j\\in S}$, then for any BIC mechanism $M$ and any constant $b\\in (0, 1)$,\n\\begin{align*}\n\\textsc{Rev}(M,v,D)\\leq 32\\alpha \\cdot \\textsc{APostEnRev}\n+\\left(12+\\frac{8}{1-b}+\\alpha\\cdot \\left(\\frac{16}{b(1-b)}+\\frac{96}{1-b}\\right)\\right)\\cdot \\textsc{PostRev}\n\\end{align*}\n\n\\vspace{0.05in}\nIf $v_i(t_i,\\cdot)$ is an XOS valuation for all $i$ and $t_i\\in T_i$, then $\\alpha=1$. By setting $b$ to $\\frac{1}{4}$, we have $$\\textsc{Rev}(M,v,D)\\leq 236\\cdot\\textsc{PostRev}+32\\cdot\\textsc{APostEnRev}.$$ For general subadditive valuations, $\\alpha=O(\\log(m))$ by~\\cite{BhawalkarR11}, hence $$\\textsc{Rev}(M,v,D)\\leq O(\\log(m))\\cdot \\max\\{\\textsc{PostRev},\\textsc{APostEnRev}\\}.$$\n\\end{theorem}\nHere is a sketch of the proof for Theorem~\\ref{thm:multi}. We show how to upper bound $\\textsc{Single}(M,\\beta)$ in Lemma~\\ref{lem:multi_single}. Then, we decompose $\\textsc{Non-Favorite}(M,\\beta)$ into $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ in Lemma~\\ref{lem:multi decomposition}. We show how to construct a simple mechanism to approximate $\\textsc{Tail}(M,\\beta)$ in Section~\\ref{subsection:tail} and how to approximate $\\textsc{Core}(M,\\beta)$ in Section~\\ref{subsection:core}.\n\n\\vspace{.1in}\n \\noindent\\textbf{Analysis of $\\textsc{Single}(M,\\beta)$:} \n\n\n\\begin{lemma}\\label{lem:multi_single}\nFor any mechanism $M$, $$\\textsc{Single}(M, \\beta)\\leq \\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{proof}\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to buyer $i$ and $t_i\\in R_j^{(\\beta)}$, $M'$ serves the agent $(i,j)$. Since there is at most one $R_j^{(\\beta)}$ that $t_i$ belongs to, $M'$ serves at most one agent $(i,j)$ for each of buyer $i$. Hence, $M'$ is feasible in the copies setting, and $\\textsc{Single}(M,\\beta)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare in the same setting. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(M,\\beta)$.\n\nAs showed in~\\cite{ChawlaHMS10, KleinbergW12}, a simple posted-price mechanism with the constraint that every buyer can only purchase one item, i.e., an RSPM, achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/6$ in the original setting. Hence, $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 6\\cdot\\textsc{PostRev}$.\n\\end{proof}\n\n\n\\vspace{.05in}\n \\noindent\\textbf{Core-Tail Decomposition of $\\textsc{Non-Favorite}(M,\\beta)$:} we decompose $\\textsc{Non-Favorite}(M, \\beta)$ into two terms $\\textsc{Tail}(M, \\beta)$ and $\\textsc{Core}(M, \\beta)$\\footnote{In~\\cite{CaiDW16}, $\\textsc{Non-Favorite}$ is decomposed into four different terms $\\textsc{Under}$, $\\textsc{Over}$, $\\textsc{Core}$ and $\\textsc{Tail}$. We essentially merge the first three terms into $\\textsc{Core}(M, \\beta)$ in our decomposition.}. First, we need the following definition.\n\\begin{definition}\\label{def:c_i}\nFor every buyer $i$, let $c_i :=\\inf\\big\\{x\\geq 0:\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+x\\right]\\leq \\frac{1}{2}\\big\\}.$ For every $t_i \\in T_i$, let $\\mathcal{T}_i(t_i)=\\{j\\ |\\ V_i(t_{ij})\\geq \\beta_{ij}+c_i\\}$ and $\\mathcal{C}_i(t_i) = [m]\\backslash\\mathcal{T}_i(t_i)$.\n\\end{definition}\n Since $v_i(t_i,\\cdot)$ is subadditive for all $i$ and $t_i\\in T_i$, we have $v_i(t_i,S)\\leq v_i\\left(t_i,S\\cap \\mathcal{C}_i(t_i)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}_i(t_i)}V_i(t_{ij})$. The term $\\textsc{Non-Favorite}(M,\\beta)$ can be decomposed into $\\textsc{Tail}(M,\\beta)$ and $\\textsc{Core}(M,\\beta)$ based on the inequality above. The complete proof of Lemma~\\ref{lem:multi decomposition} can be found in Appendix~\\ref{appx:multi}.\n\n \\begin{lemma}~\\label{lem:multi decomposition}\n\t\t\\begin{align*} &\\textsc{Non-Favorite}(M,\\beta)\\\\\n\t\t\t\\leq& \\sum_i\\sum_{t_i}f_i(t_i) \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))~~~~~~~~~~(\\textsc{Core}(M,\\beta))\\\\\n\t\t\t+&\\sum_i\\sum_j \\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot V_i(t_{ij})\\cdot\\sum_{k\\neq j} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]~~~~~~~(\\textsc{Tail}(M,\\beta))\n\t\t\\end{align*}\n\t\t\\end{lemma}\n\n\\subsection{Analyzing $\\textsc{Tail}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:tail}\n\nIn this section we show how to bound $\\textsc{Tail}(M,\\beta)$ with the revenue of an RSPM.\n\\begin{lemma}\\label{lem:multi-tail}\n\tFor any BIC mechanism $M$, $\\textsc{Tail}(M, \\beta)\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{lemma}\n\nWe first fix a few notations. Let $$P_{ij}\\in\\argmax_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],$$\n\\begin{align*}\nr_{ij}&=(P_{ij}+\\beta_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]=\\max_{x\\geq c_i}(x+\\beta_{ij})\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})-\\beta_{ij}\\geq x],\n\\end{align*}\n$r_i=\\sum_j r_{ij}$, and $r=\\sum_i r_i$. We show in the following Lemma that $r$ is an upper bound of $\\textsc{Tail}(M,\\beta)$.\n\\begin{lemma}\\label{lem:tail and r}\nFor any BIC mechanism $M$, $\\textsc{Tail}(M,\\beta)\\leq r.$\n\\end{lemma}\n\n\\begin{proof}\n\\begin{equation*}\\label{equ:tail1}\n\\begin{aligned}\n\\textsc{Tail}(M,\\beta)\\leq&\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\\\\n\\leq&\\frac{1}{2}\\cdot\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot(\\beta_{ij}+c_i)~~\\text{(Definition of $c_i$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$)}\\\\\n&+\\sum_i\\sum_j\\sum_{t_{ij}:V_i(t_{ij})\\geq\\beta_{ij}+c_i}f_{ij}(t_{ij})\\cdot \\sum_{k\\not=j}r_{ik}~~(\\text{Definition of $r_{ik}$ and $V_i(t_{ij})\\geq\\beta_{ij}+c_i$})\\\\\n\\leq& \\frac{1}{2}\\cdot\\sum_i\\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\cdot(\\beta_{ij}+c_i)+\\sum_i r_i \\cdot \\sum_j\\Pr_{t_{ij}}[V_i(t_{ij})\\geq\\beta_{ij}+c_i]\\\\\n\\leq &\\frac{1}{2}\\cdot\\sum_i\\sum_j r_{ij}+ \\frac{1}{2}\\cdot\\sum_i r_i~~\\text{(Definition of $r_{ij}$ and $c_i$)}\\\\\n =& r\n\\end{aligned}\n\\end{equation*}\nIn the second inequality, the first term is because $V_{i}(t_{ij})-\\beta_{ij}\\geq c_i$, so $$\\sum_{k\\not=j}\\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\sum_{k} \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq c_i\\right]\\leq1\/2.$$ The second term is because for any $t_{ij}$ such that $V_i(t_{ij})\\geq \\beta_{ij}+c_i$, $$\\left(V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq \\left(\\beta_{ik}+V_i(t_{ij})-\\beta_{ij}\\right)\\cdot \\Pr_{t_{ik}}\\left[ V_i(t_{ik})-\\beta_{ik}\\geq V_i(t_{ij})-\\beta_{ij}\\right]\\leq r_{ik}.$$\n\\end{proof}\n\nNext, we argue that $r$ can be approximated by an RSPM. Indeed, we prove a stronger lemma, which is also useful for analyzing $\\textsc{Core}(M,\\beta)$.\n\n\\begin{lemma}\\label{lem:neprev}\nLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be a collection of non-negative numbers, such that for any buyer $i$\n$$\\sum_{j\\in [m]} \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq 1\/2,$$ then\n\\begin{equation*}\n\\sum_i\\sum_j (x_{ij}+\\beta_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nConsider a RSPM that sells item $j$ to buyer $i$ at price $\\xi_{ij}=x_{ij}+\\beta_{ij}$. The mechanism\nvisits the buyers in some arbitrary order. Notice that when it is buyer $i$'s turn, she purchases exactly item $j$ and pays $x_{ij}+\\beta_{ij}$ if all of the following three conditions hold: (i) $j$ is still available, (ii) $V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}$ and (iii) $\\forall k\\neq j, V_i(t_{ik})< x_{ik}+\\beta_{ik}$. The second condition means buyer $i$ can afford item $j$. The third condition means she cannot afford any other item $k\\neq j$. Therefore, buyer $i$'s purchases exactly item $j$.\n\nNow let us compute the probability that all three conditions hold. Since every buyer's valuation is subadditive over the items, item $j$ is purchased by someone else only if there exists a buyer $k\\neq i$ who has $V_k(t_{kj})\\geq \\xi_{kj}$. Because $x_{kj}\\geq 0$ for all $k$, by the union bound, the event described above happens with probability at most $\\sum_{k\\neq i} \\Pr_{t_{kj}}\\left[V_k(t_{kj})\\geq \\beta_{kj}\\right]$, which is less than $b$ by property (i) of Lemma~\\ref{lem:requirement for beta}. Therefore, condition (i) holds with probability at least $(1-b)$. Clearly, condition (ii) holds with probability $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. Finally, condition (iii) holds with at least probability $1\/2$, because according to our assumption of the $x_{ij}$s, the probability that there exists any item $k\\neq j$ such that $V_i(t_{ik})\\geq x_{ik}+\\beta_{ik}$ is no more than $1\/2$. Since the three conditions are independent, buyer $i$ purchases exactly item $j$ with probability at least $\\frac{(1-b)}{2}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$. So the expected revenue of this mechanism is at least $\\frac{(1-b)}{2}\\cdot \\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq x_{ij}+\\beta_{ij}\\right]$.\n\\end{proof}\n\n\\notshow{\n\\begin{corollary}\\label{cor:bound tail }\n\\begin{equation}\n\\textsc{Tail}(M,\\beta)\\leq r\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nSince $P_{ij}\\geq c_i$, it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$\n. Therefore,\n$$r= \\sum_{i,j}(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\nOur statement follows from the above inequality and Lemma~\\ref{lem:tail and r}.\\end{proof}\n}\n\n\\begin{prevproof}{Lemma}{lem:multi-tail}\nSince $P_{ij}\\geq c_i$, it satisfies the assumption in Lemma~\\ref{lem:neprev} due to the choice of $c_i$\n. Therefore,\n\\begin{equation}\\label{r and prev}\nr= \\sum_{i,j}(\\beta_{ij}+P_{ij})\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq P_{ij}+\\beta_{ij}\\right] \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.\n\\end{equation}\nOur statement follows from the above inequality and Lemma~\\ref{lem:tail and r}.\\end{prevproof}\n\n\n\nWe have done the analysis for $\\textsc{Tail}{(M,\\beta)}$. Before starting the analysis for $\\textsc{Core}{(M,\\beta)}$, we show that $r_i$ is within a constant factor of $c_i$. This Lemma is useful for bounding $\\textsc{Core}{(M,\\beta)}$.\n\n\\begin{lemma}\\label{lem:c_i}\nFor all $i\\in [n]$, $r_i\\geq \\frac{1}{2}\\cdot c_i$ and $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot\\textsc{PostRev}$.\n\\end{lemma}\n\\begin{proof}\nBy the definition of $P_{ij}$,\n\\begin{align*}\nr_i&= \\sum_j (\\beta_{ij}+P_{ij})\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq P_{ij}]\n\\geq \\sum_j (\\beta_{ij}+c_i)\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\\\\n&\\geq\\sum_j c_i\\cdot \\Pr[V_i(t_{ij})-\\beta_{ij}\\geq c_i]\\geq\\frac{1}{2}\\cdot c_i\n\\end{align*}\nThe last inequality is because when $c_i>0$,\n$\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+c_i\\right]$ is at least $\\frac{1}{2}$. As $\\sum_i c_i\/2 \\leq r$, by Inequality~(\\ref{r and prev}), \n$\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\n\n\\subsection{Analyzing $\\textsc{Core}(M,\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\n\nIn this section we upper bound $\\textsc{Core}(M,\\beta)$. Recall that\n$$\\textsc{Core}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i(t_i,S\\cap \\mathcal{C}_i(t_i))$$\nWe can view it as the welfare of another valuation function $v'$ under allocation $\\sigma^{(\\beta)}$ where $v'_i(t_i, S) = v_i(t_i,S\\cap\\mathcal{C}_i(t_i))$. In other words, we ``truncate'' the function at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold. The new function still satisfies monotonicity, subadditivity and no externalities.\n\nWe first compare existing methods for analyzing the $\\textsc{Core}$ with our approach before jumping into the proofs.\n\n\\subsubsection{Comparison between the Existing Methods and Our Approach}\\label{sec:core comparison}\nAs all results in the literature~\\cite{ChawlaHMS10, Yao15, CaiDW16,ChawlaM16} only study special cases of constrained additive valuations, we restrict our attention to constrained additive valuations in the comparison, but our approach also applies to XOS and subadditive valuations.\n\nWe compare our approach to the state of the art result by Chawla and Miller~\\cite{ChawlaM16}. They separate $\\textsc{Core}(M,\\beta) $ into two parts: (i) the welfare obtained from values below $\\beta$, and (ii) the welfare obtained from values between $\\beta$ and $\\beta+c$\\footnote{In particular, if bidder $i$ is awarded a bundle $S$ that is feasible for her, the contribution for the first part is $\\sum_{j\\in S} \\min\\left\\{\\beta_{ij},t_{ij}\\right\\}\\cdot \\mathds{1}\\left[t_{ij}< \\beta_{ij}+c_i \\right]$ and the contribution to the second part is $\\sum_{j\\in S} \\left(t_{ij}-\\beta_{ij}\\right)^+\\cdot \\mathds{1}\\left[t_{ij}< \\beta_{ij}+c_i \\right]$ }.\n It is not hard to show that the latter can be upper bounded by the revenue of a sequential posted price with per bidder entry fee mechanism.\n Due to their choice of $\\beta$ (similar to the second property of Lemma~\\ref{lem:requirement for beta}), the former is upper bounded by $\\sum_{i,j} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[t_{ij}\\geq \\beta_{ij}\\right]$.\n It turns out when every bidder's feasibility constraint is a matroid, one can use the OCRS from~\\cite{FeldmanSZ16} to design a sequential posted price mechanism to approximate this expression.\n However, as we show in Example~\\ref{ex:counterexample ocrs}, $\\sum_{i,j} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[t_{ij}\\geq \\beta_{ij}\\right]$ could be $\\Omega\\left(\\frac{\\sqrt{m}}{\\log m}\\right)$ times larger than the optimal social welfare when the bidders have general downward closed feasibility constraints.\n Hence, such approach cannot yield any constant factor approximation for general constrained additive valuations.\n\nAs explained in the intro, we take an entirely different approach. We first construct the posted prices $\\{Q_j\\}_{j\\in[m]}$ for our ASPE (Definition~\\ref{def:posted prices}), Feldman et al.~\\cite{FeldmanGL15} showed that the anonymous posted price mechanism with these prices achieves welfare $\\Omega\\left(\\textsc{Core}(M,\\beta)\\right)$. If all bidders have valuations that are subadditive over independent items, for any bidder $i$ and any set of available items $S$, $i$'s surplus for $S$ under valuation $v'_i(t_i, \\cdot)$ ($max_{S'\\subseteq S}~v'_i(t_i,S') -\\sum_{j\\in S'} Q_j$) is also subadditive over independent items. According to Talagrand's concentration inequality, the surplus concentrates and its expectation is upper bounded by its median and its Lipschitz constant $a$. One can extract at least half of the median by setting the median of the surplus as the entry fee. How about the Lipschitz constant $a$? Unfortunately, $a$ could be as large as $\\frac{1}{2}\\max_{j\\in[m]}\\{\\beta_{ij}+c_i\\}$, which is too large to be bounded.\n\nHere is how we overcome this difficulty. Instead of considering $v'$, we construct a new valuation $\\hat{v}$ that is always dominated by the true valuation $v$. We consider the social welfare induced by $\\sigma^{(\\beta)}$ under $\\hat{v}$ and define it as $\\widehat{\\textsc{Core}}(M,\\beta)$. In Section~\\ref{sec:proxy core}, we show that $\\widehat{\\textsc{Core}}(M,\\beta)$ is not too far away from $\\textsc{Core}(M,\\beta)$, so it suffices to approximate $\\widehat{\\textsc{Core}}(M,\\beta)$ (Lemma~\\ref{lem:hat Q}). But why is $\\widehat{\\textsc{Core}}(M,\\beta)$ easier to approximate? The reason is two-fold. \\textbf{(i)} For any bidder $i$ and any set of available items $S$, bidder $i$'s surplus for $S$ under $\\hat{v}_i(t_i,\\cdot)$ (defined as $\\mu_i(t_i,S)$ in Definition~\\ref{def:entry fee}, which is $max_{S'\\subseteq S}~ \\hat{v}_i(t_i,S') -\\sum_{j\\in S'} Q_j$), is not only subadditive over independent items, but also has a small Lipschitz constant $\\tau_i$ (Lemma~\\ref{lem:property of mu}). Indeed, these Lipschitz constants are so small that $\\sum_i \\tau_i$ and can be upper bounded by $\\textsc{PostRev}$ (Lemma~\\ref{lem:tau_i}). \\textbf{(ii)} If we set the entry fee of our ASPE to be the median of $\\mu_i(t_i,S)$ when $t_i$ is drawn from $D_i$, using a proof inspired by Feldman et al.~\\cite{FeldmanGL15}, we can show that our ASPE's revenue collected from the posted prices plus the expected surplus of the bidders (over the randomness of all bidders' types) approximates $\\widehat{\\textsc{Core}}(M,\\beta)$ (implied by Lemma~\\ref{lem:lower bounding mu}). Again by Talagrand's concentration inequality, we can bound bidder $i$'s expected surplus by our entry fee and $\\tau_i$ (Lemma~\\ref{lem:concentration entry fee}). As $\\hat{v}$ is always smaller than the true valuation $v$, thus for any type $t_i$ of bidder $i$ and any available items $S$, the surplus for $S$ under $v_i(t_i,\\cdot)$ must be larger than $\\mu_i(t_i,S)$, and the entry fee is accepted with probability at least $1\/2$. Putting everything together, we demonstrate that we can approximate $\\textsc{Core}(M,\\beta)$ with an ASPE or an RSPM (Lemma~\\ref{lem:upper bounding Q}).\n\\subsubsection{Construction of $\\widehat{\\textsc{Core}}(M,\\beta)$}\\label{sec:proxy core}\n\nWe first show that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$ and any set $S$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$ for $v_i(t_i,\\cdot)$, then for any $t_i$ {and $S$} there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. Moreover, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$ for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\n{The last equality is because $\\gamma_j^S(t_i)=0$ for $j\\in S\\backslash\\mathcal{C}_i(t_i)$. }Also, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})< \\beta_{ij}+c_i]$ for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})\\geq\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})<\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{proof}\n\n\n\nNext, we define the prices of our ASPE.\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n{ $\\textsc{Core}(M,\\beta)$ can be upper bounded by $\\sum_{j\\in [m]}Q_j$. The proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).}\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(M,\\beta)$.\n\\end{lemma}\n\\begin{proof}\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(M,\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)\\\\\n&=\\alpha\\cdot \\sum_{j\\in[m]}\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\vspace{0.05in}\nIn the following definitions, we define $\\widehat{\\textsc{Core}}(M,\\beta)$ which is the welfare of another function $\\hat{v}$ under the same allocation $\\sigma^{(\\beta)}$.\n\n \n\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\}.$$\n\\end{definition}\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $Y_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$, $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap Y_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$. Moreover, let $$\\widehat{\\textsc{Core}}(M,\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{v}_i(t_i,S).$$\n\\end{definition}\n\nIn the next two Lemmas, we prove some useful properties of $\\tau_i$. In particular, we argue that $\\sum_{i\\in[n]} \\tau_i$ can be upper bounded by $\\frac{4}{1-b}\\cdot \\textsc{PostRev}$ (Lemma~\\ref{lem:tau_i}).\n \\begin{lemma}\\label{lem:beta_ij}\n\\begin{align*}\n\\sum_i\\sum_j \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\right]\n \\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nAccording to the definition of $\\tau_i$, for every buyer $i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$,\n and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. Our statement follows directly from Lemma~\\ref{lem:neprev}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n$$\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$$.\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative, \\begin{align*}\n \\sum_i\\sum_j \\max \\left\\{\\beta_{ij},Q_j+\\tau_i\\right\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\n \\geq \\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right].\n \\end{align*}\nAccording to the definition of $\\tau_i$, when $\\tau_i>0$, $$\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}.$\nTherefore, $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$ due to Lemma~\\ref{lem:beta_ij}.\n\\end{proof}\n\n\nIn the following two Lemmas, we compare $\\widehat{\\textsc{Core}}(M,\\beta)$ with $\\textsc{Core}(M,\\beta)$. The proof of Lemma~\\ref{lem:hat gamma} is postponed to Appendix~\\ref{appx:multi}.\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:hat Q}\n\tLet $$\\hat{Q}_j = \\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\hat{\\gamma}_j^{S}(t_i).$$ Then,\n\t$$\\sum_{j\\in[m]} \\hat{Q}_j\\leq \\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\n\\begin{proof}\nFrom the definition of $\\hat{Q}_j$, it is easy to see that $Q_j\\geq \\hat{Q}_j$ for every $j$. So we only need to argue that $\\sum_{j\\in[m]}Q_j\\leq \\sum_{j\\in[m]}\\hat{Q}_j+\\frac{(b+1)}{b\\cdot(1-b)}\\cdot \\textsc{PostRev}$.\n\\begin{equation}\\label{eq:first}\n\t\\begin{aligned}\n\t&\\sum_{j} \\left(Q_j- \\hat{Q}_j\\right) = \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left(\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)\\right)\\\\\n\t\\leq & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S: j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\\\\\n\t= & \\frac{1}{2}\\cdot \\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\left (\\beta_{ij}\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq Q_j+\\tau_i\\right]+c_i\\cdot \\mathds{1}\\left[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}\\right]\\right)\n\t\\end{aligned}\n\\end{equation}\n\n\tThis first inequality is because $\\gamma_j^S(t_i)- \\hat{\\gamma}_j^{S}(t_i)$ is non-zero only when $V_i(t_{ij})\\geq Q_j+\\tau_i$, and the difference is upper bounded by $\\beta_{ij}$ when $V_i(t_{ij})\\leq \\beta_{ij}$ and upper bounded by $\\beta_{ij}+c_i$ when $V_i(t_{ij})> \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}. The second inequality is due to property (ii) in Lemma~\\ref{lem:requirement for beta}. The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\\end{proof}\n\n\n\nBy Lemma~\\ref{lem:hat gamma}, $\\sum_{j\\in[m]}\\hat{Q}_j\\leq \\widehat{\\textsc{Core}}(M,\\beta)\/2$. By Lemma~\\ref{lem:core and q_j}, $\\sum_{j\\in[m]}{Q}_j\\leq {\\textsc{Core}}(M,\\beta)\/2\\alpha$. Hence, Lemma~\\ref{lem:hat Q} shows that to approximate $\\textsc{Core}(M,\\beta)$, it suffices to approximate $\\widehat{\\textsc{Core}}(M,\\beta)$. Indeed, we will use $\\sum_{j\\in[m]} \\hat{Q}_j$ as an proxy for $\\textsc{Core}(M,\\beta)$ in our analysis of the ASPE.\n\\subsubsection{Design and Analysis of Our ASPE}\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$ \\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\big(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\big).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{0$, then $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}+c_i\\right]\\geq \\frac{1}{2}$. As $\\sum_i c_i\/2 \\leq r$, by Corollary~\\ref{cor:bound tail }, $\\sum_i c_i\/2\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\subsection{Analyzing $\\textsc{Core}(\\beta)$ in the Multi-Bidder Case}\\label{subsection:core}\nIn this section, we bound $\\textsc{Core}(\\beta)$ using the sum of the revenue of a few simple mechanisms. First, we show that if we ``truncate'' the function $v(\\cdot,\\cdot)$ at some threshold, i.e., only evaluate the items whose value on its own is less than that threshold, the new function still satisfies monotonicity, subadditivity and no externalities.\n\\begin{lemma}\\label{lem:valuation v_i'}\n\tLet $\\{x_{ij}\\}_{i\\in[n], j\\in[m]}$ be a set of nonnegative numbers. For any buyer $i$, any type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij})< x_{ij}\\}$, and let $$\\bar{v}_i(t_i, S) = v_i(t_i,S\\cap X_i(t_i)),$$ for any set $S\\subseteq[m]$. Then for any bidder $i$, any type $t_i\\in T_i$, $\\bar{v}_i(t_i,\\cdot)$, satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{lemma}\n\t\t\\begin{prevproof}{Lemma}{lem:valuation v_i'}\n\t\t We will argue these three properties one by one.\n\t\\begin{itemize}\n\t\t\\item \\emph{Monotonicity:} For all $t_i\\in T_i$ and $U\\subseteq V\\subseteq [m]$, since $v_i(t_i,\\cdot)$ is monotone, $$\\bar{v}_i(t_i,U)=v_i(t_i,U\\cap X_i(t_i))\\leq v_i(t_i,V\\cap X_i(t_i))=\\bar{v}(t_i,V)$$ Thus $\\bar{v}_i(t_i,\\cdot)$ is monotone.\n\t\t\\item \\emph{Subadditivity:} For all $t_i\\in T_i$ and $U,V\\subseteq [m]$. Hence, $(U\\cup V)\\cap X_i(t_i)=(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i))$.\\mingfeinote{Since $v_i(t_i,\\cdot)$ is subadditive}, we have\n\\begin{align*}\n&\\bar{v}_i(t_i,U\\cup V)=v_i(t_i,(U\\cap X_i(t_i))\\cup (V\\cap X_i(t_i)))\\\\\n &~~~~~~~~~~~~~\\leq v_i(t_i,U\\cap X_i(t_i))+v_i(t_i,V\\cap X_i(t_i))= \\bar{v}_i(t_i,U)+\\bar{v}_i(t_i,V).\n\\end{align*}\n\\item \\emph{No externalities:} For any $t_i\\in T_i$, $S\\subseteq [m]$, and any $t_i'\\in T_i$ such that $t_{ij}=t_{ij}'$ for all $j\\in S$, to prove $\\bar{v}_i(t_i,S)=\\bar{v}_i(t_i',S)$, it suffices to show $S\\cap X_i(t_i)=S\\cap X_i(t_i')$. Since $V_i(t_{ij})=V_i(t_{ij}')$ for any item $j\\in S$, $j\\in S\\cap X_i(t_i)$ if and only if $j\\in S\\cap X_i(t_i')$.\n\t\\end{itemize}\n\t\\end{prevproof}\n\t\t\n\t\\begin{corollary}~\\label{cor:v_i'}\n Let $${v}'_i(t_i, S) = v_i(t_i,S\\cap \\mathcal{C}_i(t_i)),$$ then or any bidder $i$, any type $t_i\\in T_i$, ${v}'_i(t_i, \\cdot)$ satisfies monotonicity, subadditivity and no externalities.\t\n\t\\end{corollary}\n\t\\begin{proof}\n\t\tSimply set $x_{ij}$ to be $\\beta_{ij}+c_i$ in Lemma~\\ref{lem:valuation v_i'}.\n\t\\end{proof}\n\t\nNext, we argue that if for any $i$ and $t_i\\in T_i$ there is a set of $\\alpha$-supporting prices for $v_i(t_i,\\cdot)$, then there is a set of $\\alpha$-supporting prices for $v'_i(t_i,\\cdot)$.\n\\begin{lemma}\\label{lem:supporting prices for v'}\n\tIf for any type $t_i$, there exists a set of $\\alpha$-supporting prices $\\{\\theta_j^S(t_i)\\}_{j\\in S}$ for $v_i(t_i,\\cdot)$ and any set $S$, then for any $t_i$ there also exists a set of $\\alpha$-supporting prices $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ for $v'_i(t_i,\\cdot)$ and any set $S$. In particular, $\\gamma_j^S(t_i)=\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)$ if $j\\in S\\cap \\mathcal{C}_i(t_i)$ and $\\gamma_j^S(t_i)=0$ otherwise. In particular, $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})\\leq \\beta_{ij}+c_i]$ for all $i$, $t_i$, $j$ and $S$.\n\\end{lemma}\n\n\\begin{prevproof}{Lemma}{lem:supporting prices for v'}\nIt suffices to verify that $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ satisfies the two properties of $\\alpha$-supporting prices.\nFor any $S'\\subseteq S$, $S'\\cap \\mathcal{C}_i(t_i)\\subseteq S\\cap \\mathcal{C}_i(t_i)$. Therefore,\n\\begin{equation*}\nv_i'(t_i,S')=v_i(t_i,S'\\cap \\mathcal{C}_i(t_i))\\geq \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)= \\sum_{j\\in S'\\cap \\mathcal{C}_i(t_i)}\\gamma_j^S(t_i) = \\sum_{j\\in S'}\\gamma_j^S(t_i)\n\\end{equation*}\n\nAlso, we have\n\\begin{equation*}\n\\sum_{j\\in S}\\gamma_j^S(t_i)=\\sum_{j\\in S\\cap \\mathcal{C}_i(t_i)}\\theta^{S\\cap \\mathcal{C}_i(t_i)}_j(t_i)\\geq\\frac{v_i(t_i,S\\cap \\mathcal{C}_i(t_i))}{\\alpha}=\\frac{v_i'(t_i,S)}{\\alpha}\n\\end{equation*}\n\nThus, $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ defined above is a set of $\\alpha$-supporting prices for $v_i'(t_i,\\cdot)$. Next, we argue that $\\gamma_j^S(t_i)\\leq V_i(t_{ij})\\cdot\\mathds{1}[V_i(t_{ij})\\leq \\beta_{ij}+c_i]$ for all $i$, $t_i$, $j \\in S$. If $V_i(t_{ij})>\\beta_{ij}+c_i$, $j\\not\\in \\mathcal{C}_i(t_i)$, by definition $\\gamma_j^S(t_i)=0$. Otherwise if $V_i(t_{ij})\\leq\\beta_{ij}+c_i$, then $\\{j\\}\\subseteq S\\cap \\mathcal{C}_i(t_i)$, by the first property of $\\alpha$-supporting prices, $\\gamma_j^S(t_i)\\leq v'_i(t_i,\\{j\\})=V_i(t_{ij})$.\n\\end{prevproof}\n\n\\notshow{\\begin{lemma}\n\tIf for any buyer $i$, type $t_i$, $v_i(t_i,\\cdot)$ is an XOS valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $1$-supporting prices for $v'(t_i,\\cdot)$ and $S$. If $v_i(t_i,\\cdot)$ is an subadditive valuation function, then there exists $\\{\\gamma_j^{S}(t_i)\\}_{j\\in S}$ to be a $\\log m$-supporting prices $v'(t_i,\\cdot)$ and $S$.\n\\end{lemma}\n\\begin{proof}\n\t\\yangnote{Fill in the proof. Argue $v'$ remains to be XOS is $v$ is XOS. And by the previous Lemma, we know $v'$ is subadditive so we already have $\\log m$ supporting prices.}\n\\end{proof}}\n\nNext, we rewrite $\\textsc{Core}(\\beta)$ using $v'(\\cdot,\\cdot)$,\n\n$$\\textsc{Core}(\\beta)=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v'_i(t_i,S).$$\n\n\n\\begin{definition}\\label{def:posted prices}\nWe define a price $Q_j$ for each item $j$ as follows,\n\t\\begin{equation*}\nQ_j=\\frac{1}{2}\\cdot \\sum_i \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i),\n\\end{equation*}\nwhere $\\{\\gamma_j^S(t_i)\\}_{j\\in S}$ are the $\\alpha$-supporting prices of $v'_i(t_i,\\cdot)$ and set $S$ for any bidder $i$ and type $t_i \\in T_i$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:core and q_j}\n\t$$2\\alpha\\cdot\\sum_{j\\in [m]}Q_j\\geq \\textsc{Core}(\\beta).$$\n\\end{lemma}\n\\begin{proof}\n\tThe proof follows from the definition of $\\alpha$-supporting prices (Definition~\\ref{def:supporting price}) and the definition of $Q_j$ (Definition~\\ref{def:posted prices}).\n\t\\begin{equation*}\\label{equ:core and q_j}\n\\begin{aligned}\n\\textsc{Core}(\\beta)&=\\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot v_i'(t_i,S)\\\\\n&\\leq \\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\sum_{j\\in S}\\gamma_j^{S}(t_i)=\\alpha\\cdot \\sum_i\\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\sum_j \\sum_{S:j\\in S}\\sigma_{iS}^{(\\beta)}(t_i)\\cdot \\gamma_j^{S}(t_i)\\\\\n&=2\\alpha\\cdot \\sum_{j\\in [m]}Q_j\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\begin{definition}\\label{def:tau}\nLet $$\\tau_i := \\inf\\{x\\geq 0: \\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max\\{\\beta_{ij},Q_j+x\\}\\right]\\leq \\frac{1}{2}\\},$$ and define $A_i$ to be $\\{j\\ |\\ \\beta_{ij}\\leq Q_j+\\tau_i\\}$.\n\\end{definition}\n We have the following Lemma:\n\n \\begin{lemma}\\label{lem:beta_ij}\n$$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{lemma}\n\\begin{proof}\nBy the definition of $\\tau_i$, $\\sum_j \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]=\\frac{1}{2}$\\footnote{This clearly holds if $V_i(t_{ij})$ is drawn from a continuous distribution. When $V_i(t_{ij})$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.} for every buyer $i$ and $\\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\geq \\beta_{ij}$. By Lemma~\\ref{lem:neprev}, we have $$\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\leq \\frac{2}{1-b}\\cdot \\textsc{PostRev}.$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem:tau_i}\n\\begin{equation*}\n\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $Q_j$ is nonnegative, $\\sum_i\\sum_j \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]$ is clearly no smaller than $\\sum_i \\tau_i\\cdot \\sum_j \\Pr\\left[V_i(t_{ij})\\geq{\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]$. According to the definition of $\\tau_i$, when $\\tau_i>0$, $\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}$\\footnote{See the proof of Lemma~\\ref{lem:requirement for beta}.}. Therefore, we have $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\notshow{\n\\begin{lemma}\\label{lem:beta_ij}\n\\begin{equation}\n\\sum_i\\sum_{j\\not\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tThe proof is similar to the proof of Lemma~\\ref{lem:tau_i}. Again, we let $x_{ij} = \\max \\{\\beta_{ij},Q_j+\\tau_i\\}-\\beta_{ij}$. Clearly, we also have $$\\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}+x_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}.$$ Note that for any $j\\in A_i$, $x_{ij} = 0$, so the inequality above directly implies our claim.\n\\end{proof}\n}\n\n\n\\notshow{\\begin{definition}\\label{def:v hat}\nWe construct a new subadditive valuation $\\hat{v}_i(t_i,\\cdot)$ for every buyer $i$ and type $t_i\\in T_i$ such that $$\n\\hat{v}_i(t_i,S) = \\max_{\\ell} \\sum_{j\\in S} \\min\\{t_{ij}^{(\\ell)}, Q_j+\\tau_i \\}, $$ for every $S\\subseteq[m]$. Similarly, let $$\\hat{\\gamma}^S_j(t_i) = \\min\\{\\gamma_j^S(t_i), Q_j+\\tau_i \\}$$ for every buyer $i$, type $t_i\\in T_i$ and $S\\subseteq[m]$.\n\\end{definition}\n}\n\nHere, we define a new function $\\hat{v}(\\cdot,\\cdot)$, which will be useful in analyzing the revenue of ASPE.\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $X_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$, $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap X_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $$\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:hat gamma}\nBy Lemma~\\ref{lem:valuation v_i'} and Definition~\\ref{def:v hat}, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\t$$\\hat{v}_i(t_i,S')= v_i(t_i,\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}).$$\n\tSince $\\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}$ is a subset of $S'$, it is also a subset of $S$. Therefore,\n\t$$v'_i(t_i, \\{j\\ |\\ j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i\\}) \\geq \\sum_{j: j\\in S' \\land V_i(t_{ij})< Q_j+\\tau_i}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\t\n\n\t\\end{prevproof}\n\n\\subsection{Anonymous Sequential Posted Price Mechanism with Entry Fee}\\label{sec:ASPE}\n\t\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$\\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\left(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\right).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{ \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}). The second inequality is due to the choice of $\\beta_{ij}$ (Lemma~\\ref{lem:requirement for beta}). The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\t\\end{prevproof}\n\n\nLet $M_i^{(\\beta)}(t)$ be the set of items allocated to buyer $i$ by mechanism $M^{(\\beta)}$when the reported type profile is $t$. We argue that in expectation over all type profiles, we can provide a lower bound of the sum of $\\mu_i(t_i,S_i(t_{\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all $U\\subseteq V\\subseteq S_i(t_{0$, $\\sum_j \\Pr\\left[V_i(t_{ij})\\geq {\\max \\{\\beta_{ij},Q_j+\\tau_i\\}}\\right]= \\frac{1}{2}$\\footnote{See the proof of Lemma~\\ref{lem:requirement for beta}.}. Therefore, we have $\\sum_{i\\in[n]} \\tau_i\\leq \\frac{4}{1-b}\\cdot \\textsc{PostRev}$.\n\\end{proof}\n\n\\notshow{\n\\begin{lemma}\\label{lem:beta_ij}\n\\begin{equation}\n\\sum_i\\sum_{j\\not\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tThe proof is similar to the proof of Lemma~\\ref{lem:tau_i}. Again, we let $x_{ij} = \\max \\{\\beta_{ij},Q_j+\\tau_i\\}-\\beta_{ij}$. Clearly, we also have $$\\sum_i\\sum_j (\\beta_{ij}+x_{ij})\\cdot \\Pr\\left[V_i(t_{ij})\\geq \\beta_{ij}+x_{ij}\\right]\\leq \\frac{2}{(1-b)}\\cdot \\textsc{PostRev}.$$ Note that for any $j\\in A_i$, $x_{ij} = 0$, so the inequality above directly implies our claim.\n\\end{proof}\n}\n\n\n\\notshow{\\begin{definition}\\label{def:v hat}\nWe construct a new subadditive valuation $\\hat{v}_i(t_i,\\cdot)$ for every buyer $i$ and type $t_i\\in T_i$ such that $$\n\\hat{v}_i(t_i,S) = \\max_{\\ell} \\sum_{j\\in S} \\min\\{t_{ij}^{(\\ell)}, Q_j+\\tau_i \\}, $$ for every $S\\subseteq[m]$. Similarly, let $$\\hat{\\gamma}^S_j(t_i) = \\min\\{\\gamma_j^S(t_i), Q_j+\\tau_i \\}$$ for every buyer $i$, type $t_i\\in T_i$ and $S\\subseteq[m]$.\n\\end{definition}\n}\n\nHere, we define a new function $\\hat{v}(\\cdot,\\cdot)$, which will be useful in analyzing the revenue of ASPE.\n\n\\begin{definition}\\label{def:v hat}\nFor every buyer $i$ and type $t_i\\in T_i$, let $Y_i(t_i)=\\{j\\ |\\ V_i(t_{ij}) < Q_j + \\tau_i\\}$, $$ \\hat{v}_i(t_i,S) =v_i\\left(t_i,S\\cap Y_i(t_i)\\right)$$\nand\n$$\\hat{\\gamma}^S_j(t_i) = \\gamma_j^S(t_i)\\cdot\\mathds{1}[V_i(t_{ij})< Q_j+\\tau_i]$$\n for any set $S\\in [m]$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:hat gamma}\n\tFor every buyer $i$, type $t_i\\in T_i$, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities. Furthermore, for every set $S\\subseteq[m]$ and every subset $S'$ of $S$, $$\\hat{v}_i(t_i,S')\\geq \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\\end{lemma}\n\\begin{prevproof}{Lemma}{lem:hat gamma}\nBy Lemma~\\ref{lem:valuation v_i'} and Definition~\\ref{def:v hat}, $\\hat{v}_i(t_i,\\cdot)$ satisfies monotonicity, subadditivity and no externalities.\n\n\n\t$$\\hat{v}_i(t_i,S')= v_i\\left(t_i,S'\\cap Y_i(t_i)\\right)\\geq v_i\\left(t_i, \\left(S'\\cap Y_i(t_i)\\right)\\cap \\mathcal{C}_i(t_i)\\right) =v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right).$$\n\tSince $S'\\cap Y_i(t_i)\\subseteq S$,\n\t$$v'_i\\left(t_i, S'\\cap Y_i(t_i)\\right) \\geq \\sum_{j\\in S'\\cap Y_i(t_i)}\\gamma_j^S(t_i)= \\sum_{j\\in S'}\\hat{\\gamma}^S_j(t_i).$$\n\n\n\t\\end{prevproof}\n\n\\subsection{Anonymous Sequential Posted Price Mechanism with Entry Fee}\\label{sec:ASPE}\n\t\nConsider the sequential post-price mechanism with anonymous posted price $Q_j$ for item $j$. We visit the buyers in the alphabetical order\\footnote{We can visit the buyers in an arbitrary order. We use the the alphabetical order here just to ease the notations in the proof.} and charge every bidder an entry fee. We define the entry fee here.\n\n\\begin{definition}[Entry Fee]\\label{def:entry fee}\nFor any bidder $i$, any type $t_i\\in T_i$ and any set $S$, let $$\\mu_i(t_i,S) = \\max_{S'\\subseteq S} \\left(\\hat{v}_i(t_i, S') - \\sum_{j\\in S'} Q_j\\right).$$ For any type profile $t\\in T$ and any bidder $i$, let the entry fee for bidder $i$ be $$\\delta_i(S_i(t_{ \\beta_{ij}$.\n\t\n\tWe first bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]$.\n\t\\begin{equation}\\label{eq:second}\n\t\\begin{aligned}\n\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot \\beta_{ij}\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\\\\n\t\\leq &\\sum_i \\sum_{j\\in A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq Q_j+\\tau_i]+\\sum_i \\sum_{j\\notin A_i} \\beta_{ij}\\cdot \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\beta_{ij}]\/b\\\\\n\t\\leq & (1\/b) \\cdot \\sum_{i}\\sum_{j} \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\cdot \\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\max \\{\\beta_{ij},Q_j+\\tau_i\\}\\right]\\\\\n\t\\leq & \\frac{2}{b\\cdot(1-b)}\\cdot\\textsc{PostRev}\\end{aligned}\n\\end{equation}\nThe set $A_i$ in the first inequality is defined in Definition~\\ref{def:tau}. The second inequality is due to property (ii) in Lemma~\\ref{lem:requirement for beta}. The third inequality is due to Definition~\\ref{def:tau} and the last inequality is due to Lemma~\\ref{lem:beta_ij}.\n\nNext, we bound $\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]$.\n\n\\begin{equation}\\label{eq:third}\n\t\t\\begin{aligned}\n\t\t\t&\\sum_i \\sum_j \\sum_{t_i\\in T_i}f_i(t_i)\\cdot \\pi_{ij}^{(\\beta)}(t_i)\\cdot c_i\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j\\sum_{t_i}f_i(t_i)\\cdot \\mathds{1}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\\sum_j \\Pr_{t_{ij}}[V_i(t_{ij})\\geq \\max\\{Q_j+\\tau_i,\\beta_{ij}\\}]\\\\\n\t\t\t\\leq & \\sum_i c_i\/2\\\\\n\t\t\t\\leq& \\frac{2}{(1-b)}\\cdot\\textsc{PostRev}\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\nThe last inequality is due to Lemma~\\ref{lem:c_i}. Combining Inequality~(\\ref{eq:first}),~(\\ref{eq:second}) and~(\\ref{eq:third}), we have proved our claim.\n\t\\end{prevproof}\n\n\nLet $M_i^{(\\beta)}(t)$ be the set of items allocated to buyer $i$ by mechanism $M^{(\\beta)}$when the reported type profile is $t$. We argue that in expectation over all type profiles, we can provide a lower bound of the sum of $\\mu_i(t_i,S_i(t_{\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all {$U, V\\subseteq [m]$}, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\n\\end{prevproof}\n\nNow, we are ready to prove Lemma~\\ref{lem:concentration entry fee}.\n\n\\begin{prevproof}{Lemma}{lem:concentration entry fee}\nIt directly follows from Lemma~\\ref{lem:property of mu} and Corollary~\\ref{corollary:concentrate}. For any $i$ and $t_{\\mu_i(t_i',Y)$.\n\n\\begin{align*}\n&\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|= \\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',Y^{*})-\\sum_{j\\in Y^{*}}Q_j\\right)\\\\\n\\leq &\\left(\\hat{v}_i(t_i,X^{*})-\\sum_{j\\in X^{*}}Q_j\\right)-\\left(\\hat{v}_i(t_i',X^{*}\\cap H)-\\sum_{j\\in X^{*}\\cap H}Q_j\\right)\\quad\\text{(Optimality of $Y^{*}$ and $X^{*}\\cap H\\subseteq Y$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*})-\\hat{v}_i(t_i,X^{*}\\cap H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(No externalities of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\hat{v}_i(t_i,X^{*}\\backslash H)-\\sum_{j\\in X^{*}\\backslash H}Q_j\\qquad\\qquad\\text{(Subadditivity of $\\hat{v}_i(t_i,\\cdot)$)}\\\\\n\\leq &\\tau_i\\cdot |X^{*}\\backslash H|\\qquad\\qquad\\left(V_i(t_{ij})\\in [Q_j,Q_j+\\tau_i]\\text{ for all } j\\in X^{*}\\right)\\\\\n\\leq &\\tau_i\\cdot |X\\backslash H|\n\\end{align*}\n\nSimilarly, if $\\mu_i(t_i,X)\\leq \\mu_i(t_i',Y)$, $\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot |Y\\backslash H|$. Thus, $\\mu_i(\\cdot,\\cdot)$ is $\\tau_i$-Lipschitz as $$\\left|\\mu_i(t_i,X)-\\mu_i(t_i',Y)\\right|\\leq \\tau_i\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\leq \\tau_i\\cdot(|X\\Delta Y|+|X\\cap Y|-|H|).$$\n\nMonotonicity follows directly from the definition of $\\mu_i(t_i,\\cdot)$. Next, we argue subadditivity. For all {$U, V\\subseteq [m]$}, let $S^{*}\\in \\argmax_{S\\subseteq U\\cup V} \\left(\\hat{v}_i(t_i,S)-\\sum_{j\\in S} Q_j\\right)$, $X=S^{*}\\cap U\\subseteq U$, $Y=S^{*}\\backslash X\\subseteq V$. Since $\\hat{v}_i(t_i,\\cdot)$ is a subadditive valuation,\n\\begin{equation*}\n\\mu_i(t_i,U\\cup V)=\\hat{v}_i(t_i, S^{*}) -\\sum_{j\\in S^{*}} Q_j\\leq \\left(\\hat{v}_i(t_i, X) -\\sum_{j\\in X} Q_j\\right)+\\left(\\hat{v}_i(t_i, Y) -\\sum_{j\\in Y} Q_j\\right)\\leq \\mu_i(t_i,U)+\\mu_i(t_i,V)\n\\end{equation*}\n\nFinally, we argue that $\\mu_i(t_i,\\cdot)$ has no externalities. Consider a set $S$, and types $t_i, t_i'\\in T_i$ such that $t_{ij}'=t_{ij}$ for all $j\\in S$. For any $S'\\subseteq S$, since $\\hat{v}_i(t_i,\\cdot)$ has no externalities, $\\hat{v}_i(t_i,S')-\\sum_{j\\in S'}Q_j=\\hat{v}_i(t_i',S')-\\sum_{j\\in S'}Q_j$. Thus, $\\mu_i(t_i,S)=\\mu_i(t_i',S)$.\n\\end{prevproof}\n\n\\section{Efficient Approximation for Symmetric Bidders}\\label{sec:symmetric computation}\nIn this section, we sketch how to compute the RSPM and ASPE to approximate the optimal revenue in polynomial time for symmetric bidders\\footnote{Bidders are symmetric if for any two bidders $i$ and $i'$, we have $v_i(\\cdot,\\cdot) = v_{i'}(\\cdot,\\cdot)$ and $D_{ij}=D_{i'j}$ for all $j$.}. For any given BIC mechanism $M$, one can follow our proof to construct in polynomial time an RSPM and an ASPE such that the better of the two achieves a constant fraction of $M$'s revenue. We will describe the construction of the RSPM and the ASPE separately in this section. The difficulty of applying the method described above to construct the desired simple mechanisms is that we need to know an (approximately) revenue-maximizing mechanism $M^*$. We will show how to circumvent this difficulty when the bidders are symmetric.\n\n Indeed, we can directly construct an RSPM that approximates the $\\textsc{PostRev}$. As we have restricted the buyers to purchase at most one item in an RSPM, the $\\textsc{PostRev}$ is upper bounded by the optimal revenue of the unit-demand setting where buyer $i$ has value $V_i(t_{ij})$ for item $j$ when her type is $t_i$. By~\\cite{CaiDW16}, we know that the optimal revenue in this unit-demand setting is upper bounded by $4\\textsc{OPT}^{\\textsc{Copies-UD}}$, so one can simply use the RSPM constructed in~\\cite{ChawlaHMS10} to extract revenue at least $\\frac{\\textsc{PostRev} }{24}$. Note that the construction is independent of $M$.\n\n Unlike the RSPM, our construction for the ASPE heavily relies on $\\beta$ which depends on $M$ (Lemma~\\ref{lem:requirement for beta}). Given $\\beta$, we first compute $c_i$s according to Definition~\\ref{def:c_i}. Next, we compute the $Q_j$s (Definition~\\ref{def:posted prices}). Finally, we compute the $\\tau_i$s (Defintion~\\ref{def:tau}) and use them to compute the entry fee (Definition~\\ref{def:entry fee}). A few steps of the algorithm above requires sampling from the type distributions, but it is not hard to argue that a polynomial number of samples suffices. The main reason that the information about $M$ is necessary is because our construction crucially relies on the choice of $\\beta$. Next, we argue that for symmetric bidders, we can essentially choose a $\\beta$ that satisfies all requirements in Lemma~\\ref{lem:requirement for beta} for all mechanisms.\n\n When bidders are symmetric, the important observation is that the optimal mechanism must also be symmetric, and for any symmetric mechanism we can directly construct a $\\beta$ that satisfies all the requirements in Lemma~\\ref{lem:requirement for beta}. For every $i\\in [n], j\\in [m]$, choose $\\beta_{ij}$ such that $\\Pr_{t_{ij}}\\left[V_i(t_{ij})\\geq \\beta_{ij}\\right]=\\frac{b}{n}$. Clearly, this choice satisfies property (i) in Lemma~\\ref{lem:requirement for beta}. Furthermore, the ex-ante probability for any bidder $i$ to win item $j$ is the same in any symmetric mechanism, and therefore is no more than $1\/n$. Hence, property (ii) in Lemma~\\ref{lem:requirement for beta} is also satisfied. Given this $\\beta$, we can essentially follow the algorithm mentioned above to construct the ASPE. The only difference is that we no longer know the $\\sigma$, which is required when computing the $Q_j$s. This can be resolved by considering the welfare maximizing mechanism $M'$ with respect to $v'$. We compute the prices $Q_j$ using the allocation rule of $M'$ and construct our ASPE. As $M'$ is also symmetric, our $\\beta$ satisfies all requirements in Lemma~\\ref{lem:requirement for beta} with respect to $M'$. Therefore, Lemma~\\ref{lem:upper bounding Q} implies that either this ASPE or the RSPM constructed above has at least a constant fraction of $\\textsc{Core}(M',\\beta)$ as revenue. Since $M'$ is welfare maximizing, $\\textsc{Core}(M',\\beta)\\geq \\textsc{Core}(M^*,\\beta)$, where $M^*$ is the revenue optimal mechanism. Therefore, we construct in polynomial time a simple mechanism whose revenue is a constant fraction of the optimal BIC revenue.\n\n\n\\section{Analysis for the Single-Bidder Case}\\label{sec:single_appx}\n\n\n\\begin{comment}\n\\begin{prevproof}[lemma]\\ref{lem:single-single}\nRecall that $\\textsc{Single}(M)=\\sum_{t\\in T}f(t)\\cdot$ \\\\\n\\noindent$\\sum_{j\\in[m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\pi^{(\\beta)}_{j}(t)\\cdot {\\tilde{\\varphi}}_{j}(V(t_{j}))$.\n\nWe construct a new mechanism $M'$ in the copies setting based on $M^{(\\beta)}$. Whenever $M^{(\\beta)}$ allocates item $j$ to the buyer and $t\\in R_j^{(\\beta)}$, $M'$ serves the agent $j$. $M'$ is feasible in the copies setting as there is at most one agent being served, and $\\textsc{Single}(M)$ is the expected Myerson's ironed virtual welfare of $M'$. Since every agent's value is drawn independently, the optimal revenue in the copies setting is the same as the maximum Myerson's ironed virtual welfare in the same setting. Therefore, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is no less than $\\textsc{Single}(M)$.\n\nAs shown in~\\cite{ChawlaHMS10}, when there is a single buyer, a simple posted-price mechanism with the constraint that the buyer can only purchase one item achieves revenue at least $\\textsc{OPT}^{\\textsc{Copies-UD}}\/2$ in the original setting. Therefore, by the definition of $\\textsc{SRev}$ we have $2\\textsc{SRev}\\geq\\textsc{OPT}^{\\textsc{Copies-UD}}$.\n\\end{prevproof}\n\\end{comment}\n\n\n\\begin{prevproof}{lemma}{lem:single decomposition}\n{Recall that for all $t\\in T$ and $S\\subseteq [m]$, $v(t,S)\\leq v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$.} We replace every $v(t,S)$ in $\\textsc{Non-Favorite}(M)$ with $v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$. Also since $R^{\\beta}_0=\\emptyset$, the corresponding term is simply $0$. First, the contribution from $v\\left(t,S\\cap \\mathcal{C}(t)\\right)$ is upper bounded by the \\textsc{Core}(M).\n\n\\begin{align*}\n& \\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\left(\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot v\\left(t,(S\\backslash\\{j\\})\\cap \\mathcal{C}(t)\\right)+\\sum_{S:j\\not\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot v\\left(t,S\\cap \\mathcal{C}(t)\\right)\\right)\\\\\n\\leq& \\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{S}^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))\\quad(\\textsc{Core}(M))\n\\end{align*}\n\nThe inequality comes from the monotonicity of $v(t,\\cdot)$.\n\nNext, we upper bound the contribution from $\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$ by the $\\textsc{Tail}(M)$.\n\\begin{align*}\n&\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in [m]} \\mathds{1}\\left[t\\in R_j^{(\\beta)}\\right]\\cdot\\left(\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot \\sum_{k\\in (S\\backslash\\{j\\})\\cap \\mathcal{T}(t)}V(t_k)+\\sum_{S:j\\not\\in S}\\sigma_{S}^{(\\beta)}(t)\\cdot \\sum_{k\\in S\\cap \\mathcal{T}(t)}V(t_k)\\right) \\\\\n=&\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_j)\\cdot \\mathds{1}\\left[t\\not\\in R_j^{(\\beta)}\\right]\\cdot \\pi_j^{(\\beta)}(t)~~~~~~~\\text{{(Recall $\\pi_{j}^{(\\beta)}(t)=\\sum_{S:j\\in S}\\sigma_{S}^{(\\beta)}(t)$)}}\\\\\n\\leq &\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_{j})\\cdot\n{\\mathds{1}\\left[t \\not\\in R_j^{(\\beta)}\\right]}~~~\\text{($\\pi_j^{(\\beta)}(t)\\leq 1$)}\\\\\n\\leq &{\\sum_{t\\in T}f(t)\\cdot \\sum_{j\\in \\mathcal{T}(t)} V(t_{j})\\cdot \\mathds{1}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]~~~\\text{(Definition of $R_j^{(\\beta)}$)}}\\\\\n=&{\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_{j})\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]\\quad(\\textsc{Tail}(M))}\n\\end{align*}\n\\end{prevproof}\n\n\\begin{comment}\n\\subsubsection{Analyzing $\\textsc{Tail}(M)$~in the Single-Bidder Case}\n\\begin{prevproof}[lemma]{lem:single-tail}\nSince $\\textsc{Tail}(M)=\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$, for each type $t_j\\in T_j$ consider the mechanism that posts the same price $V(t_j)$ for each item but only allows the buyer to purchase at most one. Notice if there exists $k\\not= j$ such that $V(t_k)\\geq V(t_j)$, the mechanism is guaranteed to sell one item obtaining revenue $V(t_j)$. Thus, the revenue obtained by this mechanism\nis at least $V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$. By definition, this is no more than $\\textsc{SRev}$.\n\n\\begin{equation}\\label{equ:single-tail}\n\\textsc{Tail}(M)\\leq \\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot \\textsc{SRev}\\notshow{\\leq}{=} 2\\cdot \\textsc{SRev}\n\\end{equation}\n\n{\nThe last equality is because by the definition of $c$, \\\\\n\\noindent$\\sum_j \\Pr_{t_j}[V(t_j)\\geq c]=2$.\\footnote{This clearly holds if $V(t_j)$ is drawn from a continuous distribution. When $V(t_j)$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.}\n}\n\\end{prevproof}\n\\end{comment}\n\n\\begin{prevproof}{Lemma}{lem:single subadditive}\nWe argue the three properties one by one.\n\n\\begin{itemize}[leftmargin=0.7cm]\n\\item \\emph{Monotonicity:} For all $t\\in T$ and $U\\subseteq V\\subseteq [m]$, $U\\cap \\mathcal{C}(t)\\subseteq V\\cap \\mathcal{C}(t)$. Since $v(t,\\cdot)$ is monotone,\n$$v'(t,U)=v\\left(t,U\\cap \\mathcal{C}(t)\\right)\\leq v\\left(t,V\\cap \\mathcal{C}(t)\\right)=v'(t,V).$$ Thus, $v'(t,\\cdot)$ is monotone.\n\\item \\emph{Subadditivity:} For all $t\\in T$ and $U,V\\subseteq [m]$, notice $(U\\cup V)\\cap \\mathcal{C}(t)=\\left(U\\cap\\mathcal{C}(t)\\right)\\cup \\left(V\\cap\\mathcal{C}(t)\\right)$, we have\n$$v'(t,U\\cup V)=v\\left(\\left(t,(U\\cap\\mathcal{C}(t)\\right)\\cup \\left(V\\cap\\mathcal{C}(t)\\right)\\right)\\leq v\\left(t,U\\cap\\mathcal{C}(t)\\right)+v\\left(t,V\\cap\\mathcal{C}(t)\\right)=v'(t,U)+v'(t,V).$$\n\\item \\emph{No externalities:} For any $t\\in T$, $S\\subseteq [m]$, and any $t'\\in T$ such that $t_{j}=t_{j}'$ for all $j\\in S$, to prove $v'(t,S)=v'(t',S)$, it is enough to show $S\\cap \\mathcal{C}(t)=S\\cap \\mathcal{C}(t')$. Since $V(t_j)=V(t_j')$ for any $j\\in S$, $j\\in S\\cap \\mathcal{C}(t)$ if and only if $j\\in S\\cap \\mathcal{C}(t')$.\n\\end{itemize}\n\\end{prevproof}\n\n\\begin{prevproof}{Lemma}{lem:single Lipschitz}\nFor any $t,t'\\in T$, and set $X,Y\\subseteq [m]$, define set $H=\\left\\{j\\in X\\cap Y:t_j=t_j'\\right\\}$. Since $v'(\\cdot,\\cdot)$ has no externalities, $v'(t',H)=v'(t,H)$. Therefore,\n\\begin{align*}\n|v'(t,X)-v'(t',Y)|&=\\max\\left\\{v'(t,X)-v'(t',Y),v'(t',Y)-v'(t,X)\\right\\}\\\\\n&\\leq \\max\\left\\{v'(t,X)-v'(t',H),v'(t',Y)-v'(t,H)\\right\\}\\quad\\text{(Monotonicity)}\\\\\n&\\leq \\max\\left\\{v'(t,X\\backslash H),v'(t',Y\\backslash H)\\right\\}\\quad\\text{(Subadditivity)}\\\\\n& = \\max\\left\\{v\\left(t,(X\\backslash H)\\cap \\mathcal{C}(t)\\right),v\\left(t',(Y\\backslash H)\\cap\\mathcal{C}(t)\\right)\\right\\}\\quad\\text{(Definition of $v'(\\cdot,\\cdot)$)}\\\\\n&\\leq c\\cdot \\max\\left\\{|X\\backslash H|,|Y\\backslash H|\\right\\}\\\\\n&\\leq c\\cdot (|X\\Delta Y|+|X\\cap Y|-|H|)\n\\end{align*}\nThe second last inequality is because both $v(t,\\cdot)$ and $v(t',\\cdot)$ are subadditive and for any item $j\\in \\mathcal{C}(t)$ ($\\mathcal{C}(t')$) the single-item valuation $V(t_j)$ ($V(t'_j)$) is less than $c$.\n\\end{prevproof}\n\n\\section{Proof of Lemma~\\ref{lem:relaxed valuation}}\\label{sec:proof_relaxed_valuation}\nWe first prove some properties of $v^{(\\beta)}$, which will be useful for proving Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:relaxed larger}\n\tFor any $\\beta_i$, $t_i\\in T_i$ and $S\\in[m]$, $v_i^{(\\beta_i)}(t_i,S)\\geq v_i(t_i,S)$.\n\\end{lemma}\n\\begin{proof}\n\tThis follows from the fact that $v_i(t_i,\\cdot)$ is a subadditive function over bundles of items for all $t_i$.\n\\end{proof}\n\n\\begin{lemma}\n\tFor any $\\beta_i$ and $t_i\\in T_i$, $v_i^{(\\beta_i)}(t_i,\\cdot)$ is a monotone, subadditive function over the items.\n\\end{lemma}\n\\begin{proof}\nMonotonicity follows directly from the monotonicity of $v_i(t_i,\\cdot)$. We only argue subadditivity here. If $t_i$ belongs to $R_0^{(\\beta_i)}$, $v_i^{(\\beta_i)}(t_i,\\cdot)=v_i(t_i,\\cdot)$. So it is clearly a subadditive function. If $t_i$ belongs to $R_j^{(\\beta_i)}$ for some $j>0$ and $j$ is not in either $U$ or $V$, then clearly $v_i^{(\\beta_i)}(t_i,U\\cup V)\\leq v_i^{(\\beta_i)}(t_i,U)+v_i^{(\\beta_i)}(t_i,V)$. If $j$ is in one of the two sets, without loss of generality let's assume it is in $U$. Then $v_i^{(\\beta_i)}(t_i,U)+v_i^{(\\beta_i)}(t_i,V)=v_i(t_i,U\\backslash\\{j\\})+V_i(t_{ij})+v_i(t_i,V)\\geq v_i(t_i,V\\cup (U\\backslash\\{j\\}))+V_i(t_{ij})= v_i^{(\\beta_i)}(t_i,U\\cup V)$.\n\\end{proof}\n\n\nHere we prove a stronger version of Lemma~\\ref{lem:relaxed valuation}.\n\n\\begin{lemma}\\label{lem:relaxed valuation stronger}\nFor any $\\beta$, any absolute constant $\\eta\\in(0,1)$ and any BIC mechanism $M$ for subadditive valuations $\\{v_i(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, there exists a BIC mechanism $M^{(\\beta)}$ for valuations $\\{v_i^{(\\beta_i)}(t_i,\\cdot)\\}_{i\\in[n]}$ with $t_i\\sim D_i$ for all $i$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\t\t\\item $\\textsc{Rev}(M, v, D)\\leq$\\\\\n$~~~~\\frac{1}{1-\\eta}\\cdot{\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)}+\\frac{1}{\\eta}\\cdot\\sum_i\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$.\n\t\\end{enumerate}\n\t$\\textsc{Rev}(M, v, D)$ (or $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$) is the revenue of the mechanism $M$ (or $M^{(\\beta)}$) while the buyers' types are drawn from $D$ and buyer $i$'s valuation is $v_i(t_i,\\cdot)$ (or $v_i^{(\\beta_i)}(t_i,\\cdot)$). $\\sigma_{iS}(t_i)$ (or $\\sigma^{(\\beta)}_{iS}(t_i)$) is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i$ in mechanism $M$ (or $M^{(\\beta)}$).\n\\end{lemma}\n\n\n\\begin{prevproof}{lemma}{lem:relaxed valuation stronger}\nReaders who are familiar with the $\\epsilon$-BIC to BIC reduction~\\cite{HartlineKM11, BeiH11,DaskalakisW12} might have already realized that the problem here is quite similar. Our proof will follow essentially the same approach.\n\nFirst, we construct mechanism $M^{(\\beta)}$, which has two phases:\n\\vspace{.1in}\n\n\\noindent{\\bf Phase 1: Surrogate Sale}\n\\begin{enumerate}\n\t\\item For each buyer $i$, create $\\ell-1$ \\emph{replicas} and $\\ell$ \\emph{surrogates} sampled i.i.d. from $D_i$. The value of $\\ell$ will be specified later.\n\t\\item Ask each buyer to report her type $t_i$.\n\t\\item For each buyer $i$, create a weighted bipartite graph with the replicas and the buyer $i$ on the left and the surrogates on the right. The edge weight between a replica (or buyer $i$) with type $r_i$ and a surrogate with type $s_i$ is the expected value for a bidder with valuation $v_i^{(\\beta_i)}(r_i,\\cdot)$ to receive buyer $i$'s interim allocation in $M$ when she reported $s_i$ as her type subtract the interim payment of buyer $i$ multiplied by $(1-\\eta)$. Formally, the weight is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i^{(\\beta_i)}(r_i,S) - (1-\\eta)p_i(s_i)$, where $p_i(s_i)$ is the interim payment for buyer $i$ if she reported $s_i$.\n\t\\item Compute the VCG matching and prices on the bipartite graph created for each buyer $i$. If a replica (or bidder $i$) is unmatched in the VCG matching, match her to a random unmatched surrogate. The surrogate selected for buyer $i$ is whoever she is matched to.\n\\end{enumerate}\n\n\\vspace{.1in}\n\\noindent{\\bf Phase 2: Surrogate Competition}\n\\begin{enumerate}\n\t\\item Apply mechanism $M$ on the type profiles of the selected surrogates $\\vec{s}$. Let $M_i(\\vec{s})$ and $P_i(\\vec{s})$ be the corresponding allocated bundle and payment of buyer $i$.\n\t\\item If buyer $i$ is matched to her surrogate in the VCG matching, give her bundle $M_i(\\vec{s})$ and charge her $(1-\\eta)\\cdot P_i(\\vec{s})$ plus the VCG price. If buyer $i$ is not matched in the VCG matching, award them nothing and charge them nothing.\n\t\\end{enumerate}\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:same distribution}\n\tIf all buyers play $M^{(\\beta)}$ truthfully, then the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$.\n\\end{lemma}\n\\begin{proof}\nIn the mechanism, first the buyer $i$'s type is sampled from the distribution, then we sampled $\\ell-1$ replicas and $\\ell$ surrogates i.i.d. from the same distribution. Now, imagine a different order of sampling. We first sample the $\\ell$ replicas and $\\ell$ surrogates, then we pick one replica to be buyer $i$ uniformly at random. The two different orders above provide exactly the same joint distribution over the replicas, surrogates and buyer $i$. So we only need to argue that in the second order of sampling, the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$. Note that the perfect matching (VCG matching plus the uniform random matching with the leftover replicas\/surrogates) only depends on the types but not the identity of the node (replica or buyer $i$). So we can decide who is buyer $i$ after we have decided the perfect matching. Since buyer $i$ is chosen uniformly at random among the replicas, the chosen surrogate is also uniformly at random. Clearly, the distribution of the types of a surrogate chosen uniformly at random is also $D_i$. The assumption that buyer $i$ is reporting truthfully is crucial, because otherwise the distribution of buyer $i$'s reported type will be different from the type of a replica, and in that case, we cannot use the second sampling order.\n\\end{proof}\n\n\\begin{lemma}\n\t$M^{(\\beta)}$ is a BIC mechanism with respect to valuation $v^{(\\beta)}$.\n\\end{lemma}\n\\begin{proof}\n\tWe need to argue that for every buyer $i$ reporting truthfully is a best response, if every other buyer is truthful. In the VCG mechanism, buyer $i$ faces a competition with the replicas to win a surrogate. If buyer $i$ has type $t_i$, then her value for winning a surrogate with type $s_i$ in the VCG mechanism is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i^{(\\beta_i)}(t_i,S) - (1-\\eta)p_i(s_i)$ due to Lemma~\\ref{lem:same distribution}. Clearly, if buyer $i$ reports truthfully, the weights on the edges between her and all the surrogates will be exactly her value for winning those surrogates. Since buyer $i$ is in a VCG mechanism, reporting the true edge weights is a dominant strategy for her, therefore reporting truthfully is also a best response for her assuming the other buyers are truthful. It is critical that the other buyers are reporting truthfully, otherwise we cannot invoke Lemma~\\ref{lem:same distribution} and buyer $i$'s value for winning a surrogate with type $s_i$ may be different from the weight on the corresponding edge.\n\t\\end{proof}\n\t\n\\begin{lemma}\n\tFor any $i$ and $j$, $\\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S} \\sigma^{(\\beta)}_{iS}(t_i)\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe LHS is the ex-ante probability for buyer $i$ to win item $j$ in $M^{(\\beta)}$, and the RHS is the corresponding probability in $M$. By Lemma~\\ref{lem:same distribution}, we know the surrogate selected by buyer $i$ is participating in $M$ against all other surrogates whose types are drawn from $D_{-i}$. Therefore, the ex-ante probability for the surrogate chosen by buyer $i$ to win item $j$ is the same as RHS. Clearly, the chosen surrogate's ex-ante probability for winning any item should be at least as large as the ex-ante probability for buyer $i$ to win the item in $M^{(\\beta)}$.\n\t\\end{proof}\n\t\nNext, we want to compare $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)}, D)$ with $\\textsc{Rev}(M,v, D)$. The following simple Lemma relates both quantities to the expected prices charged to the surrogates by mechanism $M$. As in the proof of Lemma~\\ref{lem:same distribution}, we change the order of the sampling. We first sample $\\ell$ replicas and $\\ell$ surrogates then select a replica uniformly at random to be buyer $i$.\nLet $s_i^{k}$ and $r_i^{k}$ be the type of the $k$-th surrogate and replica, $\\bold{s_i}= (s_i^{1},\\ldots, s_i^{\\ell})$, $\\bold{r_i}=(r_i^{1},\\ldots, r_i^{\\ell})$ and $V(\\bold{s_i},\\bold{r_i})$ be the VCG matching between surrogates and replicas with types $\\bold{s_i}$ and $\\bold{r_i}$. We will slightly abuse notation by using $s_i^k$ (or $r_i^j$) $\\in V(\\bold{s_i},\\bold{r_i})$ to denote that $s_i^k$ (or $r_i^j$) is matched in the VCG matching $V(\\bold{s_i},\\bold{r_i})$.\n\\begin{lemma}\\label{lem:revenue by surrogates}\nFor every buyer $i$, her expected payments in $M^{(\\beta)}$ is at least $$(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right],$$ and her expected payments in $M$ is $${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right].$$\n\\end{lemma}\n\\begin{proof}\n\tThe revenue of $M^{(\\beta)}$ contains two parts -- the prices paid by the chosen surrogates and the revenue of the VCG mechanism. Let's compute the first part. For buyer $i$ and each realization of $\\bold{r_i}$ and $\\bold{s_i}$ only when the buyer $i$'s chosen surrogate is in $ V(\\bold{s_i},\\bold{r_i})$, she pays the surrogate price. Since each surrogate is selected with probability $1\/\\ell$, the expected surrogate price paid by buyer $i$ is exactly $(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]$. Since the VCG payments are nonnegative, we have proved our first statement.\n\t\n\tThe expected payment from buyer $i$ in $M$ is ${\\mathbb{E}}_{t_i\\sim D_i}\\left[p_i(t_i)\\right]$. Since all $s_i^k$ is drawn from $D_i$, this is exactly the same as ${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]$.\n\\end{proof}\n\nIf the VCG matching is always perfect, then Lemma~\\ref{lem:revenue by surrogates} already shows that the revenue of $M^{(\\beta)}$ is at least $(1-\\eta)$ fraction of the revenue of $M$. But since the VCG matching may not be perfect, we need to show that the total expected price from surrogates who are not in the VCG matching is small. We prove this in two steps. First, we consider another matching $X(\\bold{s_i},\\bold{r_i})$ -- a maximal matching that only matches replicas and surrogates that have the same type, and show that the expected cardinality of $X(\\bold{s_i},\\bold{r_i})$ is close to $\\ell$. Then we argue that for any realization $\\bold{r_i}$ and $\\bold{s_i}$ the total payments from surrogates that are in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$ is small.\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:equal type matching}\nFor every buyer $i$, the expected cardinality of a maximal matching that only matches replicas and surrogates with the same type is at least $\\ell-\\sqrt{|T_i|\\cdot \\ell}$.\n\\end{lemma}\n\n\nThe proof can be found in Hartline et al.~\\cite{HartlineKM11}.\n\\begin{corollary}\\label{cor:bound revenue by X}\nLet $\\mathcal{R} = \\max_{i,t_i\\in T_i}\\max_{S\\in[m]} v_i(t_i,S)$, then\n$${\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\geq {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]- \\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}.$$\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a IR mechanism when the buyers' valuations are $v$, $\\mathcal{R}\\geq p_i(t_i)$ for any buyer $i$ and any type $t_i$ of $i$. Our claim follows from Lemma~\\ref{lem:equal type matching}.\n\\end{proof}\n\nNow we implement the second step of our argument. The plan is to show the total prices from surrogates that are unmatched by going from $X(\\bold{s_i},\\bold{r_i})$ to $V(\\bold{s_i},\\bold{r_i})$. For any $\\bold{s_i},\\bold{r_i}$, $V(\\bold{s_i},\\bold{r_i})\\cup X(\\bold{s_i},\\bold{r_i})$ can be decompose into a disjoint collection augmenting paths and cycles. If a surrogate is matched in $X(\\bold{s_i},\\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$, then it must be the starting point of an augmenting path. The following Lemma upper bounds the price of this surrogate.\n\\begin{lemma}[Adapted from~\\cite{DaskalakisW12}]\\label{lem:bounding the price for each augmenting path}\n\tFor any buyer $i$ and any realization of $\\bold{s_i}$ and $\\bold{r_i}$, let $P$ be an augmenting path that starts with a surrogate that is matched in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$. It has the form of either (a) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}}\\right)$ when the path ends with a surrogate, or\\\\ (b) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}},r_i^{\\theta(k)}\\right)$ when the path ends with a replica, where $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$ and matched to $s_i^{\\rho(j+1)}$ in $V(\\bold{s_i},\\bold{r_i})$ (whenever $s_i^{\\rho(j+1)}$ exists) for any $j$.\n\t\\begin{align*}&\\sum_{s_i^{\\rho(j)}\\in P\\cap X(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)-\\sum_{s_i^{\\rho(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)\\leq\\\\\n\t &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S \\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}(r_i^{\\theta(j)},S)-v_i(r_i^{\\theta(j)},S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tSince $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$, $r_i^{\\theta(j)}$ must be equal to $s_i^{\\rho(j)}$. $M$ is a BIC mechanism when buyers valuations are $v$, therefore the expected utility for reporting the true type is better than lying. Hence, the following holds for all $j$:\n\t\\begin{equation}\\label{eq:BIC for M}\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\geq \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\t\n\t\\end{equation}\n\nThe VCG matching finds the maximum weight matching, so the total edge weights in path $P \\cap V(\\bold{s_i},\\bold{r_i})$ is at least as large as the total edge weights in path $P\\cap X(\\bold{s_i},\\bold{r_i})$. Mathematically, it is the following inequalities.\n\\begin{itemize}\n\\item If $P$ has format (a): \\begin{align}\\label{eq:VCG great a}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\\item If $P$ has format (b): \\begin{align}\\label{eq:VCG great b}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\n\\end{itemize}\n\nNext, we further relax the RHS of inequality~(\\ref{eq:VCG great a}) using inequality~(\\ref{eq:BIC for M}).\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great a})}\\\\\n\t\\geq& \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M})}\\\\\n\\end{align*}\nWe can obtain the following inequality by combining the relaxation above with the LHS of inequality~(\\ref{eq:VCG great a}) and rearrange the terms.\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq p_i\\left(s_i^{\\rho(1)}\\right)-p_i\\left(s_i^{\\rho(k)}\\right).$$\nThe inequality above is exactly the inequality in the statement of this Lemma when $P$ has format (a).\n\nSimilarly, we have the following relaxation when $P$ has format (b):\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great b})}\\\\\n\t\\geq& \\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M} and $M$ is IR)}\\\\\n\\end{align*}\nAgain, by combining the relaxation with the LHS of inequality~(\\ref{eq:VCG great b}), we can prove our claim when $P$ has format (b).\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i^{(\\beta_i)}\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq p_i\\left(s_i^{\\rho(1)}\\right).$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem: gap between X and V}\nFor any $\\beta$,\n\t\\begin{align*}\n&{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\leq\\\\\n&~~~~~~~~~~~~~~~{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]+\\frac{1}{\\eta}\\cdot\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tDue to Lemma~\\ref{lem:bounding the price for each augmenting path}, for any buyer $i$ and any realization of $\\bold{r_i}$ and $\\bold{s_i}$, we have\n\t$$\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}-\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\leq \\frac{1}{\\eta\\cdot\\ell}\\cdot\\sum_{s_i^k \\in V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{k}\\right)\\cdot \\left(v_i^{(\\beta_i)}(r_i^{\\omega(k)},S)-v_i(r_i^{\\omega(k)},S)\\right),$$ where $r_i^{\\omega(k)}$ is the replica that is matched to $s_i^k$ in $ V(\\bold{s_i},\\bold{r_i})$. If we take expectation over $\\bold{r_i}$ and $\\bold{s_i}$ on the RHS, the expectation means whenever mechanism $M^{(\\beta)}$ awards buyer $i$ (with type $t_i$) bundle $S$, $\\frac{1}{\\eta}\\cdot\\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$ is contributed to the expectation. Therefore, the expectation of the RHS is the same as $$\\frac{1}{\\eta}\\cdot\\left(\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\right).$$ This completes the proof of the Lemma.\n\\end{proof}\n\nNow, we are ready to prove Lemma~\\ref{lem:relaxed valuation stronger}.\n\\begin{align*}\n\t&\\textsc{Rev}(M, v, D)\\\\\n\t=& \\sum_i {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\\\\\n\t\\leq & \\sum_i\\left({\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right] +\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}\\right)~~\\text{(Corollary~\\ref{cor:bound revenue by X})}\\\\\n\t\\leq &\\sum_i {\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem: gap between X and V})}\\\\\n\t\\leq & \\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\n\\end{align*}\n\nSince $|T_i|$ and $\\mathcal{R}$ are finite numbers, we can take $\\ell$ to be sufficiently large, so that $\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} < \\epsilon$ for any $\\epsilon$. Let $P^{(\\beta)}$ be the set of all BIC mechanisms that satisfy the first condition in Lemma~\\ref{lem:relaxed valuation stronger}. Clearly, $P^{(\\beta)}$ is a compact set and contains all $M^{(\\beta)}$ we constructed (by choosing different values for $\\ell$). Notice that both $\\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)$ and $\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)$ are linear functions over the allocation\/price rules of mechanism $M^{(\\beta)}$. Therefore, \\begin{align*}\n \t&\\textsc{Rev}(M, v, D)\\\\\n \t\\leq &\\max_{M^{(\\beta)}\\in P^{(\\beta)}} \\left(\\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M^{(\\beta)},v^{(\\beta)},D)+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma^{(\\beta)}_{iS}(t_i)\\cdot \\left(v_i^{(\\beta_i)}(t_i, S)-v_i(t_i, S)\\right)\\right).\n \\end{align*}\nThis completes the proof of Lemma~\\ref{lem:relaxed valuation stronger}.\n\\end{prevproof}\n\n\\section{Proof of Lemma~\\ref{lem:relaxed valuation}}\n\\begin{lemma}\\label{lem:relaxed valuation stronger}\n\tIn a $n$-player $m$-item combinatorial auction, for any absolute constant $\\eta\\in(0,1)$ and $\\epsilon>0$, any two type profile distributions $D, D'$ on type profile set $T$ and $T'$ accordingly($T$ and $T'$ might be different), any two valuation functions $\\{v_i(\\cdot,\\cdot)\\}_{i\\in[n]}$, $\\{v_i'(\\cdot,\\cdot)\\}_{i\\in[n]}$, assume for every $i$ there exists a coupling $\\hat{D_i}$ for $D_i$ and $D_i'$ such that $\\forall t_i\\in T_i,t_i'\\in T_i', \\hat{D_i}(t_i,t_i')>0$, $v_i'(t_i',S)\\geq v_i(t_i,S)$ holds for subset $S$. Here $\\hat{D_i}(t_i,t_i')$ is the coupling probability. Then for any BIC mechanism $M$ for valuation functions $\\{v_i(\\cdot,\\cdot)\\}_{i\\in[n]}$ with respect to $D$, there exists a BIC mechanism $M'$ for valuation functions $\\{v_i'(\\cdot,\\cdot)\\}_{i\\in[n]}$ with respect to distribution $D'$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{t_i'\\in T_i'}f_i'(t_i')\\cdot\\sum_{S: j\\in S}\\sigma'_{iS}(t_i')\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$, for all $i$ and $j$,\n\t\t\\item $\\textsc{Rev}(M, v, D)\\leq$\\\\\n$~~~~\\frac{1}{1-\\eta}\\cdot{\\textsc{Rev}(M',v', D')}+\\frac{1}{\\eta}\\cdot \\sum_i\\sum_{t_i'\\in T_i'}\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]}\\hat{D_i}(t_i,t_i')\\cdot\\sigma'_{iS}(t_i')\\cdot \\left(v_i'(t_i', S)-v_i(t_i, S)\\right)+\\epsilon$.\n\t\\end{enumerate}\n\t$\\textsc{Rev}(M, v, D)$ is the revenue of the mechanism $M$ while the buyers' types are drawn from $D$ and their valuations are $v$ (similarly defined for $\\textsc{Rev}(M', v', D')$). $\\sigma'_{iS}(t_i')$ is the probability of buyer $i$ receiving exactly bundle $S$ when her reported type is $t_i'$ in mechanism $M'$ with respect to $D'$ and $\\sigma_{iS}(t_i)$ is the probability for the same event in mechanism $M$ with respect to $D$\n\\end{lemma}\n\n\\begin{prevproof}{lemma}{lem:relaxed valuation stronger}\nReaders who are familiar with the $\\epsilon$-BIC to BIC reduction~\\cite{HartlineKM11, BeiH11,DaskalakisW12} might have already realized that the problem here is quite similar. Our proof will follow essentially the same approach.\n\nFirst, we construct mechanism $M'$, which has two phases:\n\\vspace{.1in}\n\n\\noindent{\\bf Phase 1: Surrogate Sale}\n\\begin{enumerate}\n\t\\item For each buyer $i$, create $\\ell-1$ \\emph{replicas} sampled i.i.d. from $D_i'$ and $\\ell$ \\emph{surrogates} sampled i.i.d. from $D_i$. The value of $\\ell$ will be specified later.\n\t\\item Ask each buyer to report her type $t_i'$.\n\t\\item For each buyer $i$, create a weighted bipartite graph with the replicas and the bidder $i$ on the left and the surrogates on the right. The edge weight between a replica (bidder $i$) with type $r_i$ and a surrogate with type $s_i$ is the expected value for a bidder with valuation $v_i'(r_i,\\cdot)$ to receive bidder $i$'s interim allocation in $M$ when she reported $s_i$ as her type subtract the expected payment of bidder $i$ multiplied by $(1-\\eta)$. Formally, the weight is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i'(r_i,S) - (1-\\eta)p_i(s_i).$\n\t\\item Compute the VCG matching and prices on the bipartite graph created for each buyer $i$. If a replica (or bidder $i$) is unmatched in the VCG matching, match her to a random unmatched surrogate. The surrogate selected for buyer $i$ is whoever she is matched to.\n\\end{enumerate}\n\n\\vspace{.1in}\n\\noindent{\\bf Phase 2: Surrogate Competition}\n\\begin{enumerate}\n\t\\item Apply mechanism $M$ on the type profiles of the selected surrogates $\\vec{s}$. Let $M_i(\\vec{s})$ and $P_i(\\vec{s})$ be the corresponding allocated bundle and payment of buyer $i$.\n\t\\item If buyer $i$ is matched to her surrogate in the VCG matching, give her bundle $M_i(\\vec{s})$ and charge her $(1-\\eta)\\cdot P_i(\\vec{s})$ plus the VCG price. If buyer $i$ is not matched in the VCG matching, award them nothing and charge them nothing.\n\t\\end{enumerate}\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:same distribution}\n\tIf all buyers play $M'$ truthfully, then the distribution of types of the surrogate chosen for buyer $i$ is exactly $D_i$.\n\\end{lemma}\n\\begin{proof}\nIn the mechanism, first the buyer $i$'s type and $\\ell-1$ replicas are sampled i.i.d. from the distribution $D_i'$, while $\\ell$ surrogates are sampled i.i.d. from the distribution $D_i$. Now, imagine a different order of sampling. We first sample the $\\ell$ replicas and $\\ell$ surrogates, then we pick one replica to be buyer $i$ uniformly at random. The two different orders above provide exactly the same joint distribution over the replicas, surrogates and buyer $i$. So we only need to argue that in the second order of sampling, the distribution of types of the surrogate chosen by buyer $i$ is exactly $D_i$. Note that the perfect matching (VCG matching plus the uniform random matching with the leftover replicas\/surrogates) only depends on the types but not the identity of the node (replica or buyer $i$). So we can decide who is buyer $i$ after we have decided the perfect matching. Since buyer $i$ is chosen uniformly at random among the replicas, the chosen surrogate is also uniformly at random. Clearly, the distribution of the types of a surrogate chosen uniformly at random is also $D_i$. The assumption that buyer $i$ is reporting truthfully is crucial, because otherwise the distribution of buyer $i$'s reported type will be different from the type of a replica, and in that case, we cannot use the second sampling order.\n\\end{proof}\n\n\\begin{lemma}\n\t$M'$ is a BIC mechanism with respect to valuation $v'$.\n\\end{lemma}\n\\begin{proof}\n\tWe need to argue that for every buyer $i$ reporting truthfully is a best response, if every other buyer is truthful. In the VCG mechanism, buyer $i$ faces a competition with the replicas to win a surrogate. If buyer $i$ has type $t_i'$, then her value for winning a surrogate with type $s_i$ in the VCG mechanism is $\\sum_{S} \\sigma_{iS}(s_i)\\cdot v_i'(t_i',S) - (1-\\eta)p_i(s_i)$ due to Lemma~\\ref{lem:same distribution}. Clearly, if buyer $i$ reports truthfully, the weights on the edges between her and all the surrogates will be exactly her value for winning those surrogates. Since buyer $i$ is in a VCG mechanism, reporting the true edge weights is a dominant strategy for her, therefore reporting truthfully is also a best response for her assuming the other buyers are truthful.\n\t\\end{proof}\n\t\n\\begin{lemma}\n\tFor any $i$ and $j$, $\\sum_{t_i'\\in T_i'}f_i'(t_i')\\cdot\\sum_{S: j\\in S} \\sigma'_{iS}(t_i')\\leq \\sum_{t_i\\in T_i}f_i(t_i)\\cdot\\sum_{S: j\\in S}\\sigma_{iS}(t_i)$.\n\\end{lemma}\n\n\\begin{proof}\n\tThe LHS is the ex-ante probability for buyer $i$ to win item $j$ in $M'$, and the RHS is the corresponding probability in $M$. By Lemma~\\ref{lem:same distribution}, we know the surrogate selected by buyer $i$ is participating in $M$ against all other surrogates whose types are drawn from $D_{-i}$. Therefore, the ex-ante probability for the surrogate chosen by buyer $i$ to win item $j$ is the same as RHS. Clearly, this surrogate's ex-ante probability for winning any item should be at least as large as the ex-ante probability for $i$ to win the item in $M'$.\n\t\\end{proof}\n\t\nNext, we want to compare $\\textsc{Rev}(M',v', D')$ with $\\textsc{Rev}(M,v, D)$. The following simple Lemma relates both quantities to the expected prices charged to the surrogates by mechanism $M$. As in the proof of Lemma~\\ref{lem:same distribution}, we change the order of the sampling. We first sample $\\ell$ replicas and $\\ell$ surrogates then select a replica uniformly at random to be buyer $i$.\nLet $s_i^{k}\\in T_i$ and $r_i^{k}\\in T_i'$ be the type of the $k$-th surrogate and replica, $\\bold{s_i}= (s_i^{1},\\ldots, s_i^{\\ell})$, $\\bold{r_i}=(r_i^{1},\\ldots, r_i^{\\ell})$ and $V(\\bold{s_i},\\bold{r_i})$ be the VCG matching between surrogates and replicas with types $\\bold{s_i}$ and $\\bold{r_i}$. \t\n\\begin{lemma}\\label{lem:revenue by surrogates}\nFor every buyer $i$, her expected payments in $M'$ is at least $$(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right],$$ and her expected payments in $M$ is $${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right].$$\n\\end{lemma}\n\\begin{proof}\n\tThe revenue of $M'$ contains two parts -- the prices paid by the chosen surrogates and the revenue of the VCG mechanism. Let's compute the first part. For buyer $i$ and each realization of $\\bold{r_i}$ and $\\bold{s_i}$ only when the buyer $i$'s chosen surrogate is in $ V(\\bold{s_i},\\bold{r_i})$, $i$ pays the surrogate price. Since each surrogate is selected with probability $1\/\\ell$, the expected surrogate price paid by buyer $i$ is exactly $(1-\\eta)\\cdot{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]$. Since the VCG payments are nonnegative, we have proved our first statement.\n\t\n\tThe expected payment from buyer $i$ in $M$ is ${\\mathbb{E}}_{t_i\\sim D_i}\\left[p_i(t_i)\\right]$. Since all $s_i^k$ is drawn from $D_i$, this is exactly the same as ${\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]$.\n\\end{proof}\n\nIf the VCG matching is always perfect, then Lemma~\\ref{lem:revenue by surrogates} already shows that the revenue of $M'$ is at least $(1-\\eta)$ fraction of the revenue of $M$. But since the VCG matching may not be perfect, we need to show that the total expected price from surrogates who are not in the VCG matching is small. We prove this in two steps. First, we consider a different type of matching $X(\\bold{s_i},\\bold{r_i})$ -- a maximal matching that only matches replicas and surrogates that have the same type, and show that the expected cardinality of $X(\\bold{s_i},\\bold{r_i})$ is close to $\\ell$. Then we argue that for any realization $\\bold{r_i}$ and $\\bold{s_i}$ the total payments from surrogates that are in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$ is small.\n\n\\begin{lemma}[\\cite{HartlineKM11}]\\label{lem:equal type matching}\nFor every buyer $i$, the expected cardinality of a maximal matching that only matches replicas and surrogates with the same type is at least $\\ell-\\sqrt{|T_i|\\cdot \\ell}$.\n\\end{lemma}\n\n\nThe proof can be found in Hartline et. al.~\\cite{HartlineKM11}.\n\\begin{corollary}\\label{cor:bound revenue by X}\nLet $\\mathcal{R} = \\max_{i,t_i\\in T_i}\\max_{S\\in[m]} v_i(t_i,S)$, then\n$${\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\geq {\\mathbb{E}}_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]- \\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}.$$\n\\end{corollary}\n\\begin{proof}\n\tSince $M$ is a IR mechanism when the buyers' valuations are $v$, $\\mathcal{R}\\geq p_i(t_i)$ for any buyer $i$ and any type $t_i$ of $i$. Our claim follows from Lemma~\\ref{lem:equal type matching}.\n\\end{proof}\n\nNow we implement the second step of our argument. The plan is to show the total prices from surrogates that are unmatched by going from $X(\\bold{s_i},\\bold{r_i})$ to $V(\\bold{s_i},\\bold{r_i})$. For any $\\bold{s_i},\\bold{r_i}$, $V(\\bold{s_i},\\bold{r_i})\\cup X(\\bold{s_i},\\bold{r_i})$ can be decompose into a disjoint collection augmenting paths and cycles. If a surrogate is matched in $X(\\bold{s_i},\\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$, then it must be the starting point of an augmenting path. The following Lemma upper bounds the price of this surrogate.\n\\begin{lemma}\\label{lem:bounding the price for each augmenting path}\n\tFor any buyer $i$ and any realization of $\\bold{s_i}$ and $\\bold{r_i}$, let $P$ be an augmenting path that starts with a surrogate that is in $X(\\bold{s_i}, \\bold{r_i})$ but not in $V(\\bold{s_i},\\bold{r_i})$. It has the form of either (a) $\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}}\\right)$ when the path ends with a surrogate, or (b)$\\left(s_i^{\\rho(1)},r_i^{\\theta(1)},s_i^{\\rho(2)},r_i^{\\theta(2)},\\ldots, s_i^{\\rho{(k)}},r_i^{\\theta(k)}\\right)$ when the path ends with a replica, where $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$ and matched to $s_i^{\\rho(j+1)}$ (whenever $s_i^{\\rho(j+1)}$ exists) for any $j$.\n\t\\begin{align*}&\\sum_{s_i^{\\rho(j)}\\in P\\cap X(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)-\\sum_{s_i^{\\rho(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} p_i \\left(s_i^{\\rho(j)}\\right)\\leq\\\\\n\t &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{\\eta}\\cdot\\sum_{r_i^{\\theta(j)}\\in P\\cap V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'(r_i^{\\theta(j)},S)-v_i(r_i^{\\theta(j)},S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tSince $r_i^{\\theta(j)}$ is matched to $s_i^{\\rho(j)}$ in $X(\\bold{s_i}, \\bold{r_i})$, $r_i^{\\theta(j)}$ must be equal to $s_i^{\\rho(j)}$. $M$ is a BIC mechanism when buyers valuations are $v$, therefore the expected utility for reporting the true type is better than lying. Hence, the following holds for all $j$:\n\t\\begin{equation}\\label{eq:BIC for M}\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\geq \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\t\n\t\\end{equation}\n\nThe VCG matching finds the maximum weight matching, so the total edge weights in path $P$ and $V(\\bold{s_i},\\bold{r_i})$ is at least as large as the total edge weights in path $P$ and $X(\\bold{s_i},\\bold{r_i})$. Mathematically, it is the following inequalities.\n\\begin{itemize}\n\\item If $P$ has format (a): \\begin{align}\\label{eq:VCG great a}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\\item If $P$ has format (b): \\begin{align}\\label{eq:VCG great b}\n&\\sum_{j=1}^{k-1} \\left(\\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j+1)}\\right)\\right) \\geq\t\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i'\\left(r_i^{\\theta(j)},S\\right)-(1-\\eta)\\cdot p_i\\left(s_i^{\\rho(j)}\\right)\\right) \\nonumber\n\\end{align}\n\n\\end{itemize}\n\nNext, we further relax the RHS of inequality~(\\ref{eq:VCG great a}) using inequality~(\\ref{eq:BIC for M}).\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great a})}\\\\\n\t\\geq& \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k-1}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M})}\\\\\n\\end{align*}\nWe can obtain the following inequality by combining the relaxation above with the LHS of inequality~(\\ref{eq:VCG great a}) and rearrange the terms.\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq p_i\\left(s_i^{\\rho(1)}\\right)-p_i\\left(s_i^{\\rho(k)}\\right).$$\nThe inequality above is exactly the inequality in the statement of this Lemma when $P$ has format (a).\n\nSimilarly, we have the following relaxation when $P$ has format (b):\n\\begin{align*}\n\t&\\text{RHS of inequality~(\\ref{eq:VCG great b})}\\\\\n\t\\geq& \\sum_{j=1}^{k}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Lemma~\\ref{lem:relaxed larger})}\\\\\n\t\\geq & \\sum_{j=1}^{k-1}\\left( \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot v_i\\left(r_i^{\\theta(j)},S\\right)-p_i\\left(s_i^{\\rho(j+1)}\\right)\\right)+ \\eta\\cdot\\sum_{j=1}^{k}p_i\\left(s_i^{\\rho(j)}\\right)~~\\text{(Inequality~\\ref{eq:BIC for M} and $M$ is IR)}\\\\\n\\end{align*}\nAgain, by combining the relaxation with the LHS of inequality~(\\ref{eq:VCG great b}), we can prove our claim when $P$ has format (b).\n$$\\frac{1}{\\eta}\\cdot\\sum_{j=1}^{k-1} \\sum_S\\sigma_{iS}\\left(s_i^{\\rho(j+1)}\\right)\\cdot \\left(v_i'\\left(r_i^{\\theta(j)},S\\right)-v_i\\left(r_i^{\\theta(j)},S\\right)\\right)\\geq p_i\\left(s_i^{\\rho(1)}\\right).$$\n\\end{proof}\n\n\\begin{lemma}\\label{lem: gap between X and V}\n\t\\begin{align*}\n&{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\leq\\\\\n&~~~~~~~~~~~~~~~{\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]+\\frac{1}{\\eta}\\cdot\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right).\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}\n\tDue to Lemma~\\ref{lem:bounding the price for each augmenting path}, for any buyer $i$ and any realization of $\\bold{r_i}$ and $\\bold{s_i}$, we have\n\t$$\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}-\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\leq \\frac{1}{\\eta\\cdot\\ell}\\cdot\\sum_{s_i^k \\in V(\\bold{s_i},\\bold{r_i})} \\sum_S \\sigma_{iS}\\left(s_i^{k}\\right)\\cdot \\left(v_i'(r_i^{\\omega(k)},S)-v_i(r_i^{\\omega(k)},S)\\right),$$ where $r_i^{\\omega(k)}$ is the replica that is matched to $s_i^k$ in $ V(\\bold{s_i},\\bold{r_i})$. If we take expectation over $\\bold{r_i}$ and $\\bold{s_i}$ on the RHS, the expectation means whenever mechanism $M'$ awards buyer $i$ (with type $t_i$) bundle $S$, $\\frac{1}{\\eta}\\cdot\\left(v_i'(t_i, S)-v_i(t_i, S)\\right)$ is contributed to the expectation. Therefore, the expectation of the RHS is the same as $$\\frac{1}{\\eta}\\cdot\\left(\n\\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)\\right).$$ This completes the proof of the Lemma.\n\\end{proof}\n\nNow, we are ready to prove Lemma~\\ref{lem:relaxed valuation stronger}.\n\\begin{align*}\n\t&\\textsc{Rev}(M, v, D)\\\\\n\t=& \\sum_i E_{\\bold{s_i}}\\left[\\sum_{k\\in[\\ell]} \\frac{ p_i(s_i^k)}{\\ell}\\right]~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\\\\\n\t\\leq & \\sum_i\\left({\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in X(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right] +\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R}\\right)~~\\text{(Corollary~\\ref{cor:bound revenue by X})}\\\\\n\t\\leq &\\sum_i {\\mathbb{E}}_{\\bold{s_i},\\bold{r_i}}\\left[\\sum_{s_i^k\\in V(\\bold{s_i},\\bold{r_i})} \\frac{ p_i(s_i^k)}{\\ell}\\right]\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem: gap between X and V})}\\\\\n\t\\leq & \\frac{1}{1-\\eta}\\cdot \\textsc{Rev}(M',v',D)\\\\\n\t&~~~~~~~~~~~~+\\frac{1}{\\eta}\\cdot\\sum_i \\sum_{t_i\\in T_i}\\sum_{S\\subseteq[m]} f_i(t_i)\\cdot\\sigma'_{iS}(t_i)\\cdot \\left(v_i'(t_i, S)-v_i(t_i, S)\\right)+\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} ~~~\\text{(Lemma~\\ref{lem:revenue by surrogates})}\n\\end{align*}\n\nSince $|T_i|$ and $\\cal{R}$ are finite numbers, we can take $\\ell$ to be sufficiently large, so that $\\sum_i\\sqrt{\\frac{|T_i|}{\\ell}}\\cdot\\mathcal{R} < \\epsilon\/(1-\\eta)$. Therefore, we finished the proof of Lemma~\\ref{lem:relaxed valuation stronger}.\n\\end{prevproof}\n\n\n\n\n\n\n\\section{Paper Organization}\\label{sec:roadmap}\nIn this section, we provide the roadmap to our paper. In Section~\\ref{sec:duality}, we review the Duality framework of~\\cite{CaiDW16}.\n\nIn Section~\\ref{sec:flow}, we derive an upper bound of the optimal revenue for subadditive bidders by combining the duality framework with our new techniques, i.e. valuation relaxation and adaptive dual variables. Our main result in this section, Theorem~\\ref{thm:revenue upperbound for subadditive}, shows that the revenue can be upper bounded by two terms -- $\\textsc{Non-Favorite}$ and $\\textsc{Single}$ defined in Lemma~\\ref{lem:upper bound the revenue of the relaxed mechanism}.\n\nIn Section~\\ref{sec:single}, we use the single bidder case to familiarize the readers with some basic ideas and techniques used to bound $\\textsc{Single}$ and $\\textsc{Non-Favorite}$. The main result of this section, Theorem~\\ref{thm:single}, shows that the optimal revenue for a single subadditive bidder is upper bounded by $24\\textsc{SRev}$ and $16\\textsc{BRev}$.\n\nSection~\\ref{sec:multi} contains the main result of this paper. We show how to upper bound the optimal revenue for XOS (or subadditive) bidders with a constant number of (or $O(\\log m)$) $\\textsc{PostRev}$ (the optimal revenue obtainable by an RSPM) and $\\textsc{APostEnRev}$ ((the optimal revenue obtainable by an ASPE). In particular, $\\textsc{Single}$ can be upper bounded by the optimal revenue $\\textsc{OPT}^{\\textsc{Copies-UD}}$ in the copies setting which is again upper bounded by $6\\textsc{PostRev}$. We further decompose $\\textsc{Non-Favorite}$ into two terms $\\textsc{Tail}$ and $\\textsc{Core}$, and show how to bound $\\textsc{Tail}$ in Section~\\ref{subsection:tail} and how to bound $\\textsc{Core}$ in Section~\\ref{subsection:core}.\n\n\n\n\\section{Warm Up: Single Bidder}\\label{sec:single}\nTo warm up, we first study the case where there is a single subadditive buyer and show how to improve the approximation ratio from $338$ to $40$. Since there is only one buyer, we will drop the subscript $i$ in the notations. As specified in Section~\\ref{sec:choice of beta}, we use a $\\beta$ that satisfies both properties in Lemma~\\ref{lem:requirement for beta}. For a single buyer, we can simply set $\\beta_{j}$ to be $0$ for all $j$. We use $\\textsc{Single}(M), \\textsc{Non-Favorite}(M)$ in the following proof to denote the corresponding terms in Theorem~\\ref{thm:revenue upperbound for subadditive} for $\\beta=\\textbf{0}$. Notice $R_0^{(\\textbf{0})}=\\emptyset$. Theorem~\\ref{thm:single} shows that the optimal revenue is within a constant factor of the better of selling separately and grand bundling.\n\n\\begin{theorem}\\label{thm:single}\nFor a single buyer whose valuation distribution is subadditive over independent items, \n\\[\\textsc{Rev}(M,v,D)\\leq 24\\cdot\\textsc{SRev}+16\\cdot\\textsc{BRev}\\]\nfor any BIC mechanism $M$.\n\\end{theorem}\n\nRecall that the revenue for mechanism $M$ is upper bounded by $4\\cdot \\textsc{Non-Favorite}(M)+2\\cdot\\textsc{Single}(M)$ (Theorem~\\ref{thm:revenue upperbound for subadditive}). We first upper bound $\\textsc{Single}(M)$ by $\\textsc{OPT}^{\\textsc{Copies-UD}}$. Since $\\sigma^{(\\beta)}_{S}(t)$ is a feasible allocation in the original setting, $ \\mathds{1}[t\\in R_j^{(\\beta)}]\\cdot\\pi^{(\\beta)}_{j}(t)$ with $\\pi^{(\\beta)}_j(t)=\\sum_{S:j\\in S}\\sigma^{(\\beta)}_{S}(t)$ is a feasible allocation in the copies setting, and therefore $\\textsc{Single}(M)$ is the Myerson Virtual Welfare of a certain allocation in the copies setting, which is upper bounded by $\\textsc{OPT}^{\\textsc{Copies-UD}}$. By~\\cite{ChawlaHMS10}, $\\textsc{OPT}^{\\textsc{Copies-UD}}$ is at most $2\\cdot\\textsc{SRev}$.\n\\begin{lemma}\\label{lem:single-single}\nFor any BIC mechanism $M$, $\\textsc{Single} (M)\\leq \\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 2\\cdot\\textsc{SRev}.$\n\\end{lemma}\n\nWe decompose $\\textsc{Non-Favorite}(M)$ into two terms $\\textsc{Core}(M)$ and $\\textsc{Tail}(M)$, and bound them separately. For every $t\\in T$, define $\\mathcal{C}(t)=\\{j:V(t_j)< c\\}$, $\\mathcal{T}(t)=[m]\\backslash \\mathcal{C}(t)$. Here the threshold $c$ is chosen as\n\\begin{equation}\\label{equ:single-def of c}\nc:=\\inf\\left\\{x\\geq 0:\\ \\sum_j \\Pr_{t_j}\\left[V(t_j)\\geq x\\right]\\leq 2\\right\\}.\n\\end{equation}\nSince $v(t,\\cdot)$ is subadditive for all $t\\in T$ , we have for every $S\\subseteq [m]$, $v(t,S)\\leq v\\left(t,S\\cap \\mathcal{C}(t)\\right)+\\sum_{j\\in S\\cap \\mathcal{T}(t)}V(t_j)$. {We decompose $\\textsc{Non-Favorite}(M)$ based on the inequality above.} Proof of Lemma~\\ref{lem:single decomposition} can be found in Appendix~\\ref{sec:single_appx}.\n\n\n\n\n\\begin{lemma}\\label{lem:single decomposition}\n\\begin{align*}\t\\textsc{Non-Favorite}(M)\n\\leq &\t\\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq[m]}\\sigma_{S}^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))~~~~~~~~~\\quad(\\textsc{Core}(M))\\\\\n+&\\sum_j\\sum_{t_{j}:V(t_{j})\\geq c}f_{j}(t_{j})\\cdot V(t_{j})\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]~~~~\\quad(\\textsc{Tail}(M))\n\\end{align*}\n\\end{lemma}\n\n\nUsing the definition of $c$ and $\\textsc{SRev}$, we can upper bound $\\textsc{Tail}(M)$ with a similar argument as in~\\cite{CaiDW16}. \n\\begin{lemma}\\label{lem:single-tail}\nFor any BIC mechanism $M$, $\\textsc{Tail}(M)\\leq 2\\cdot\\textsc{SRev}$.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\textsc{Tail}(M)=\\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$, for each type $t_j\\in T_j$ consider the mechanism that posts the same price $V(t_j)$ for each item but only allows the buyer to purchase at most one. Notice if there exists $k\\not= j$ such that $V(t_k)\\geq V(t_j)$, the mechanism is guaranteed to sell one item obtaining revenue $V(t_j)$. Thus, the revenue obtained by this mechanism\nis at least $V(t_j)\\cdot \\Pr_{t_{-j}}\\left[\\exists k\\not=j, V(t_k)\\geq V(t_j)\\right]$. By definition, this is no more than $\\textsc{SRev}$.\n\n\\begin{equation}\\label{equ:single-tail}\n\\textsc{Tail}(M)\\leq \\sum_j\\sum_{t_j:V(t_j)\\geq c}f_j(t_j)\\cdot \\textsc{SRev}\\notshow{\\leq}{=} 2\\cdot \\textsc{SRev}\n\\end{equation}\n\n{\nThe last equality is because by the definition of $c$,\n\\noindent$\\sum_j \\Pr_{t_j}[V(t_j)\\geq c]=2$.\\footnote{This clearly holds if $V(t_j)$ is drawn from a continuous distribution. When $V(t_j)$ is drawn from a discrete distribution, see the proof of Lemma~\\ref{lem:requirement for beta} for a simple fix.}\n}\n\\end{proof}\n\n\nThe $\\textsc{Core}(M)$ is upper bounded by ${\\mathbb{E}}_{t}[v'(t,[m])]$ where $v'(t,S)$\n$= v(t,S\\cap \\mathcal{C}(t))$. We argue that $v'(t,\\cdot)$ is drawn from a distribution that is subadditive over independent items and $v'(\\cdot,\\cdot)$ is $c$-Lipschitz (see Definition~\\ref{def:Lipschitz}). Using a concentration bound by Schechtman~\\cite{Schechtman2003concentration}, we show ${\\mathbb{E}}_{t}[v'(t,[m])]$ is upper bounded by the median of random variable $v'(t,[m])$ and $c$, which are upper bounded by $\\textsc{BRev}$ and $\\textsc{SRev}$ respectively.\n\\begin{lemma}\\label{lem:single-core}\nFor any BIC mechanism $M$, $\\textsc{Core}(M) \\leq 3\\cdot\\textsc{SRev}+4\\cdot\\textsc{BRev}$.\n\\end{lemma}\n\nRecall that\n\\begin{equation}\n\\textsc{Core}(M)=\\sum_{t\\in T}f(t)\\cdot \\sum_{S\\subseteq [m]}\\sigma_S^{(\\beta)}(t)\\cdot v(t,S\\cap \\mathcal{C}(t))\n\\end{equation}\n\nWe will bound $\\textsc{Core}(M)$ with a concentration inequality from~\\cite{Schechtman2003concentration}. It requires the following definition:\n\n\\begin{definition}\\label{def:Lipschitz}\nA function $v(\\cdot,\\cdot)$ is \\textbf{$a$-Lipschitz} if for any type $t,t'\\in T$, and set $X,Y\\subseteq [m]$,\n$$\\left|v(t,X)-v(t',Y)\\right|\\leq a\\cdot \\left(\\left|X\\Delta Y\\right|+\\left|\\{j\\in X\\cap Y:t_j\\not=t_j'\\}\\right|\\right),$$ where $X\\Delta Y=\\left(X\\backslash Y\\right)\\cup \\left(Y\\backslash X\\right)$ is the symmetric difference between $X$ and $Y$.\n\\end{definition}\n\nDefine a new valuation function for the bidder as $v'(t,S)=v(t,S\\cap \\mathcal{C}(t))$, for all $t\\in T$ and $S\\subseteq [m]$. Then $v'(\\cdot,\\cdot)$ is $c-$ Lipschitz, and when $t$ is drawn from the product distribution $D=\\prod_j D_j$, $v'(t,\\cdot)$ remains to be a valuation drawn from a distribution that is subadditive over independent items. See Appendix~\\ref{sec:single_appx} for the proof of Lemma~\\ref{lem:single subadditive} and Lemma~\\ref{lem:single Lipschitz}.\n\n\\begin{lemma}\\label{lem:single subadditive}\nFor all $t\\in T$, $v'(t,\\cdot)$ satisfies monotonicity, subadditivity and no externalities defined in Definition~\\ref{def:subadditive independent}.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:single Lipschitz}\n$v'(\\cdot,\\cdot)$ is $c-$Lipschitz.\n\\end{lemma}\n\nNext, we apply the following concentration inequality to derive Corollary~\\ref{corollary:concentrate}, which is useful to analyze the $\\textsc{Core}(M)$. \n\n\\begin{lemma}~\\cite{Schechtman2003concentration}\\label{lem:schechtman}\nLet $g(t,\\cdot)$ with $t\\sim D=\\prod_j D_j$ be a function drawn from a distribution that is subadditive over independent items of ground set $I$. If $g(\\cdot,\\cdot)$ is $c$-Lipschitz, then for all $a>0, k\\in \\{1,2,...,|I|\\}, q\\in \\mathbb{N}$,\n$$\\Pr_t[g(t,I)\\geq (q+1)a+k\\cdot c]\\leq \\Pr_t[g(t,I)\\leq a]^{-q}q^{-k}.$$\n\\end{lemma}\n\n\\begin{corollary}\\label{corollary:concentrate}\nLet $g(t,\\cdot)$ with $t\\sim D=\\prod_j D_j$ be a function drawn from a distribution that is subadditive over independent items of ground set $I$. If $g(\\cdot,\\cdot)$ is $c$-Lipschitz, then if we let $a$ be the median of the value of the grand bundle $g(t,I)$, i.e. $a=\\inf\\left\\{x\\geq 0: \\Pr_t[g(t,I)\\leq x]\\geq \\frac{1}{2}\\right\\}$,\n$$\\mathds{E}_t[g(t,I)]\\leq 2a+\\frac{5c}{2}.$$\n\\end{corollary}\n\n\\begin{proof}\nLet $Y$ be $g(t,I)$. If we apply Lemma~\\ref{lem:schechtman} to the case where $a$ is the median and $q=2$, we have\n\n\\begin{align*}\n\\Pr_t[Y\\geq 3a]\\cdot{\\mathbb{E}}_{t}[Y|Y\\geq 3a]&= 3a\\cdot \\Pr_t[Y\\geq 3a]+\\int_{y=0}^{\\infty}\\Pr_t[Y\\geq 3a+y]dy\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+c\\cdot\\sum_{k=0}^{|I|} \\Pr_t[Y\\geq 3a+k\\cdot c] \\quad(Y\\leq |I|\\cdot c)\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+c\\cdot \\sum_{k=0}^2 \\Pr_t[Y > a]+ c\\cdot\\sum_{k=3}^{|I|} 4\\cdot 2^{-k}\\quad(\\text{Lemma~\\ref{lem:schechtman}})\\\\\n&\\leq 3a\\cdot \\Pr_t[Y\\geq 3a]+\\frac{5}{2}c\\\\\n\\end{align*}\n\nWith the inequality above, we can upper bound the expected value of $Y$.\n\\begin{align*}\n{\\mathbb{E}}_{t}[Y]&\\leq a\\cdot \\Pr_t[Y\\leq a]+3a\\cdot \\Pr_{t}[Y\\in (a,3a)]+\\Pr_t[Y\\geq 3a]\\cdot{\\mathbb{E}}_{t}[Y|Y\\geq 3a]\\\\\n&\\leq a\\cdot \\Pr_t[Y\\leq a]+3a\\cdot \\Pr_{t}[Y\\in (a,3a)]+3a\\cdot \\Pr_t[Y\\geq 3a]+\\frac{5}{2}c\\\\\n&= a+2a\\cdot \\Pr_{t}[Y>a]+\\frac{5}{2}c\\\\\n&\\leq 2a+\\frac{5}{2}c\n\\end{align*}\n\\end{proof}\n\n\nNow, we are ready to prove Lemma~\\ref{lem:single-core}.\n\n\\begin{prevproof}{Lemma}{lem:single-core}\nLet $\\delta$ be the median of $v'(t,[m])$ when $t$ is sampled from distribution $D$. Now consider the mechanism that sells the grand bundle with price $\\delta$. Notice that the bidder's valuation for the grand bundle is $v(t,[m])\\geq v'(t,[m])$. Thus with probability at least $\\frac{1}{2}$,\n the bidder purchases the bundle. Thus, $\\textsc{BRev}\\geq \\frac{1}{2}\\delta$.\n\nAccording to Corollary~\\ref{corollary:concentrate},\n\n\\begin{comment}\n\\begin{equation}\\label{equ:single-core}\n\\begin{aligned}\n\\textsc{Core}(M)&\\leq \\mathds{E}_{t\\sim D}[v'(t,[m])]\\leq 2\\delta+\\frac{5c}{2}\\\\\n&< 4\\cdot\\textsc{BRev}+3\\cdot\\textsc{SRev} \\text{(Lemma~\\ref{lem:single-bound for c}, Inequality~\\ref{equ:bound for delta})}\n\\end{aligned}\n\\end{equation}\n\\end{comment}\n\n\\begin{equation}\\label{equ:single-core-prev}\n\\textsc{Core}(M)\\leq \\mathds{E}_{t\\sim D}[v'(t,[m])]\\leq 2\\delta+\\frac{5c}{2}\n\\end{equation}\n\nIt remains to argue that the Lipchitz constant $c$ can be upper bounded using $\\textsc{SRev}$. Notice that by AM-GM Inequality,\n\\begin{align*}\n&\\Pr_t\\left[\\exists j\\in [m], V(t_j)\\geq c\\right]= 1-\\prod_{j}\\Pr_{t_j}[V(t_j)< c]\\\\\n\\geq& 1-(\\frac{\\sum_j \\Pr_{t_j}[V(t_j)< c]}{m})^m\n= 1-(1-\\frac{2}{m})^m\n\\geq 1-e^{-2}\n\\end{align*}\n\n\n\nConsider the mechanism that posts price $c$ for each item but only allow the buyer to purchase one item. Then with probability at least $1-e^{-2}$, the mechanism sells one item obtaining expected revenue $(1-e^{-2})\\cdot c$. Thus $c\\leq \\frac{1}{1-e^{-2}}\\cdot\\textsc{SRev}$. Inequality~\\eqref{equ:single-core-prev} becomes\n\n\\begin{equation}\\label{equ:single-core}\n\\textsc{Core}(M)\\leq 2\\delta+\\frac{5c}{2}<4\\cdot\\textsc{BRev}+3\\cdot\\textsc{SRev}\n\\end{equation}\n\n\\end{prevproof}\n\n\\begin{prevproof}{Theorem}{thm:single}\nSince $\\textsc{OPT}^{\\textsc{Copies-UD}}\\leq 2 \\textsc{SRev}$ (Lemma~\\ref{lem:single-single}) and $\\textsc{Non-Favorite}(M)\\leq 5\\textsc{SRev}+4\\textsc{BRev}$ (Lemma~\\ref{lem:single-tail} and~\\ref{lem:single-core}), $\\textsc{Rev}(M,v,D)\\leq 24\\cdot\\textsc{SRev}+16\\cdot\\textsc{BRev}$ according to Theorem~\\ref{thm:revenue upperbound for subadditive}.\n\\end{prevproof}\n\n\n\n\n\n\n\n\n\n\\section{Sequentially Posted-Price Mechanism with Entry Fee}\\label{sec:spm}\nHere is the formal specification of the Sequential Posted Price with Entry Fee Mechanism.\\\\\n\n\\begin{algorithm}[ht]\n\\begin{algorithmic}[1]\n\\REQUIRE $\\xi_{ij}$ is the price for bidder $i$ to purchase item $j$ and $\\delta_i(\\cdot)$ is bidder $i$'s entry fee function.\n\\STATE $S\\gets [m]$\n\\FOR{$i \\in [n]$}\n\t\\STATE Show bidder $i$ {the} set of available items $S$, and define entry fee as ${\\delta_i}(S)$.\n \\IF{Bidder $i$ pays the entry fee ${\\delta_i}(S)$}\n \\STATE $i$ receives her favorite bundle $S_i^{*}$, paying $\\sum_{j\\in S_i^{*}}\\xi_{ij}$.\n \\STATE $S\\gets S\\backslash S_i^{*}$.\n \\ELSE\n \\STATE $i$ gets nothing and pays $0$.\n \\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\caption{{\\sf Sequential Posted Price with Entry Fee Mechanism}}\n\\label{alg:seq-mech}\n\\end{algorithm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}