{"text":"\\section{Introduction}\n\t\nThe purpose of inverse time-harmonic elastic scattering is to recover the position, shape and physical properties of an elastic body by using information of the scattered wave generated by time-harmonic plane and point source waves. We refer to \\cite{Ku1979, MC2005, Habib2015} for a comprehensive introduction of mathematical theory and inverse problems in linear elasticity.\n\n \nOver the last twenty years,\n sampling-type methods have attracted much attention, because forward solvers and good initial approximations of the target are no longer required,  in contrast with the iterative approaches.  The multi-wave sampling methods do not require a priori information on physical and geometrical properties of the scatterer, but usually need  far-field data for a large number of incident waves. Here we give an incomplete list of the applications to the Navier equation, including linear sampling and factorization methods \\cite{Arens, kress2002, Cha2003, Cha2007, HuKirsch2012}, singular source method \\cite{Das2008}, orthogonal\/direct sampling method \\cite{Ji2018}, enclosure method\\cite{GS2012} and the Reverse time-migration method in the frequency domain \\cite{HC}. On the other hand, there also exists the so-called one-wave sampling methods, which are usually designed to test the analytic extensibility of the scattered field; see the monograph \\cite[Chapter 15]{NP2013} for detailed discussions on scalar equations, for instance, range test  and no-response test \\cite{K2003, Lin2021, L2003} and enclosure method \\cite{I1999, II2009}. The one-wave method requires only a single far-field pattern or one-pair Cauchy data, but one must pre-assume the absence of an analytical continuation across the scattering interface.\n\n\nIf a single far-field pattern is available only, the inverse scattering problems become severely ill-posed and thus more challenging. \n This paper is concerned with the one-wave factorization method for recovering a convex rigid elastic body of polygonal type from a single elastic far-field pattern.  Such a method \n was earlier discussed in \\cite{el-hu19} for inverse elastic scattering from rigid polygonal bodies but without too much details. It  \n is closet to the extended linear sampling method \\cite{sun2019, S2018} and the one-wave range test method \\cite{K2003}.\nIn the authors' previous work \\cite{M-H}, the one-wave factorization method for the Helmholtz equation was rigorously established with the help of corner scattering theory. The connections to the range test and extended linear sampling were also discussed there. \nThe one-wave factorization method is a both data-driven and model driven method, and could lead to an explicit characterization of  an arbitrary convex scatterer of polygonal type if the testing scatterers are chosen as disks. \nIn this sense, it inherits merits of the classical factorization method for precisely characterizing targets \\cite{K2008} but restricted to convex polygonal scatterers\/sources. The purpose of this paper is to generalize the mathematical theory of \\cite{M-H} to the Navier equation.   The following items  can be considered as  complementary contributions to the previous work \\cite{el-hu19}: i) One-wave factorization method using only compressional or shear waves in linear elasticity; ii) Explicit expression of the spectral data for elastic far-field operators corresponding to rigid disks and a straightforward verification of the one-wave factorization method by using testing disks. \n \n \n \n\n\n \n \n\n\n\n\n\n This paper is organized as follows. In Section \\ref{sec:2}, we introduce basic concepts of the direct and inverse elastic scattering problems. In Section \\ref{sec:3}, the multi-wave factorization method for recovering a rigid scatterers will be briefly reviewed. In Section \\ref{sec:4}, we present a rigorous justification of the one-wave method by combining the classical factorization method and elastic corner scattering theory. Explicit examples by using testing  disks will be presented in Section \\ref{sec:5}, including \n derivation of an eigensystem of the far-field operator for a rigid disk.\n Finally, we describe our imaging schemes in Section \\ref{sec6}. \n\t\n\t\\section{Preliminaries}\\label{sec:2}\n\tIn this paper, we will consider the scattering of elastic waves in two-dimensional space $\\mathbb{R}^2$. Let $D \\subset \\mathbb{R}^2$ be a bounded rigid elastic body with connected exterior $D^c \\coloneqq \\mathbb{R}^2 \\backslash \\bar{D}$. Let $D^c$ be filled with a homogeneous and isotropic elastic medium. Suppose that a time-harmonic elastic plane wave of the form \n\t\\begin{equation}\\label{2.1}\n\t\tu^{i}\\left(x; d, c_p, c_s \\right) = c_p d \\text{e}^{ik_px\\cdot d} + c_s d^{\\bot} \\text{e}^{ik_sx\\cdot d}, \\quad c_p,c_s \\in \\mathbb{C}, \\quad |c_p|+|c_s|\\neq 0,\n\t\\end{equation} \n\tis incident onto the scatterer $D$. Here $d = (\\cos \\theta_d,\\sin \\theta_d )^T, \\theta_d \\in \\left[0,2\\pi\\right)$ is the incident direction; $d^{\\bot} \\coloneqq (-\\cos \\theta_d, \\sin \\theta_d)^T$ is a vector orthogonal to $d$; $\\omega >0$ is the frequency;  $ k_p \\coloneqq \\omega \/ \\sqrt{\\lambda+2\\mu}$ and $k_s \\coloneqq \\omega \/ \\sqrt{\\mu}$ are the compressional and shear wave numbers, respectively. Note that for simplicity the density of the background medium has been normalized to be one and the Lame constants \n$\\lambda$ and $\\mu$ satisfy $\\mu > 0$ and $\\lambda +2 \\mu > 0$ in two dimensions. The propagation of time-harmonic elastic waves in $D^c$ is governed by the Navier equation (or system)\n\t\\begin{equation}\\label{lame-eqn}\n\t\t\\Delta^{\\ast}u + \\omega^2 u \\coloneqq \\mu \\vartriangle u + \\left( \\lambda +\\mu \\right) \\triangledown \\left( \\triangledown \\cdot u\\right) + \\omega ^2 u = 0 \\quad \\text{in} \\quad D^c, \\quad u = \\left( u_1, u_2\\right)^T,\n\t\\end{equation}\n\twhere $u = u^{i} + u^{s}$ donotes the total displacement field. By Hodge decomposition, any solution $u$ to \\eqref{lame-eqn} can be decomposed into the from\n\t\\begin{equation}\\label{Hodgedecom}\n\t\tu = u_p + u_s, \\quad u_p \\coloneqq -\\frac{1}{k^2_p} \\ \\text{grad} \\ \\text{div} \\ u, \\quad u_s \\coloneqq \\frac{1}{k^2_s} \\ \\text{curl} \\ \\overrightarrow{\\text{curl}} \\ u, \n\t\\end{equation}\n\twhere $u_p$ and $u_s$ are called compressional and shear waves respectively. Note that in \\eqref{Hodgedecom} the two curl operators are defined as \n\t\\begin{equation}\n\t\t\\overrightarrow{\\text{curl}} \\ u \\coloneqq \\partial_2 u_1 - \\partial_1 u_2, \\quad \\text{curl} \\ f = \\left(-\\partial_2 f,\\partial_1 f\\right)^T,\n\t\\end{equation}\n\tsatisfying the relation\n\t\\begin{equation}\n\t\t\\text{curl} \\  \\overrightarrow{\\text{curl}} \\ u = -\\Delta u + \\text{grad} \\ \\text{div} \\  u.\n\t\\end{equation}\n\tMoreover, $u_{\\alpha}\\left(\\alpha=p,s\\right)$ satisfies the vector Helmholtz equations $\\left(\\Delta + k^2_{\\alpha}\\right) u_{\\alpha}=0$ and $\\overrightarrow{\\text{curl}} \\ u_p = \\text{div} \\ u_s = 0$ in $D^c$.\n\tIn this paper, we require $u^{s}$ to fulfill the Kupradze radiation condition \n\t\\begin{equation}\\label{Kupr-cd}\n\t\t\\partial_r u^{s}_{\\alpha} - ik_{\\alpha}u^{s}_{\\alpha} = o\\left(r^{-\\frac{1}{2}}\\right) \\quad \\text{as} \\quad r=|x| \\to \\infty, \\quad \\alpha = p, s,\n\t\\end{equation}\n\tuniformly in all directions $\\hat{x} = x\/|x|$ on the unit circle $\\mathbb{S} \\coloneqq \\left\\{x \\in \\mathbb{R}^2 : |x| = 1\\right\\}.$ \nIt is well known that the direct scattering problem admits one solution $u\\in C^2({\\mathbb R}^2\\backslash\\overline{D})\\cap C^1({\\mathbb R}^2\\backslash D)$ if $\\partial D$ is of $C^2$-smooth (see \\cite{Ku1979}) and $u\\in (H^1_{loc}({\\mathbb R}^2\\backslash\\overline{D}))^2$ if $\\partial D$ is Lipschitz (see e.g., \\cite{Bao2018, Li2016}).\n\t\nThis paper is concerned with the inverse scattering problem of recovering $\\partial D$ from the information of the far-field pattern of a single incoming plane wave.\nThe compressional and shear parts $u^{s}_{\\alpha} \\left( \\alpha =p,s \\right)$ of the radiating solution $u^{s}$ admit an asymptotic behavior of the form \\cite{el-hu19,PGC1993}\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tu^{s}_{p}\\left( x\\right) &= \\frac{\\text{e}^{ik_pr}}{\\sqrt{r}} \\left\\{ u^{\\infty}_p \\left( \\hat{x} \\right) \\hat{x} + \\mathcal{O}\\left( \\frac{1}{r}\\right)\\right\\}, \\\\\n\t\t\tu^{s}_{s}\\left( x\\right) &= \\frac{\\text{e}^{ik_sr}}{\\sqrt{r}} \\left\\{ u^{\\infty}_s \\left( \\hat{x} \\right) \\hat{x}^{\\perp} + \\mathcal{O}\\left( \\frac{1}{r}\\right)\\right\\}\n\t\t\\end{split}\n\t\\end{equation}\n\tas $r = |x| \\to \\infty$, where $u^{\\infty}_p$ and $u^{\\infty}_s$ are both scalar functions defined on $\\mathbb{S}$. Hence, a Kupradze radiating solution has the asymptotic behavior \n\t\\begin{equation}\n\t\tu^{s}\\left( x \\right) = \\frac{\\text{e}^{ik_pr}}{\\sqrt{r}} u^{\\infty}_p \\left( \\hat{x} \\right) \\hat{x} + \\frac{\\text{e}^{ik_sr}}{\\sqrt{r}} u^{\\infty}_s \\left( \\hat{x} \\right) \\hat{x}^{\\perp } + \\mathcal{O}\\left( \\frac{1}{r^{\\frac{3}{2}}}\\right) \\quad \\text{as} \\quad r \\to \\infty.\n\t\\end{equation}\n\tThe far-field pattern $u^{\\infty}$ of $u^{s}$ is defined as \n\t\\begin{equation}\n\t\tu^{\\infty} \\left( \\hat{x}\\right) \\coloneqq u^{\\infty}_p \\left( \\hat{x}\\right) \\hat{x} +\n\t\tu^{\\infty}_s \\left( \\hat{x}\\right) \\hat{x}^{\\perp }.\n\t\\end{equation}\n\tThen the compressional and shear parts of the far field are uniquely determined by $u^{\\infty}$ as \n\t\\begin{equation}\n\t\tu^{\\infty}_p \\left( \\hat{x}\\right) = u^{\\infty} \\left( \\hat{x}\\right) \\cdot \\hat{x}, \\quad u^{\\infty}_s \\left( \\hat{x}\\right) = u^{\\infty} \\left( \\hat{x}\\right) \\cdot \\hat{x}^{\\perp}.\n\t\\end{equation}\nIntroduce the compressional and shear parts of $u^{i}$ by\n\t\\begin{equation}\n\t\tu^{i}_p\\left(x; d\\right)  \\coloneqq u^{i}\\left(x; d, 1,0 \\right), \\  u^{i}_s\\left(x; d\\right)  \\coloneqq u^{i}\\left(x; d, 0, 1 \\right).\n\t\\end{equation}\n\tObviously, \n\t\\begin{eqnarray}\n\t\tu^{i}(x; d, c_p, c_s) = c_p u^{i}_p(x; d) + c_s u^{i}_s(x; d).\n\t\\end{eqnarray}\n\tCorrespondingly, for the far-filed pattern, we have\n\t\\begin{equation}\\label{u-inftypp}\n\t\t\\begin{split}\n\t\t\tu^{\\infty}_{pp} \\left( \\hat{x};d\\right) & \\coloneqq u^{\\infty}_p \\left( \\hat{x}; d, 1,0\\right), \\\\\n\t\t\tu^{\\infty}_{sp} \\left( \\hat{x};d\\right) & \\coloneqq u^{\\infty}_s \\left( \\hat{x}; d, 1,0\\right),\\\\\n\t\t\tu^{\\infty}_{ps} \\left( \\hat{x};d\\right) & \\coloneqq u^{\\infty}_p \\left( \\hat{x}; d, 0,1\\right),\\\\\n\t\t\tu^{\\infty}_{ss} \\left( \\hat{x};d\\right) & \\coloneqq u^{\\infty}_s \\left( \\hat{x};d,  0,1\\right).\n\t\t\\end{split}\n\t\\end{equation}\n\tThus, we obtain\n\t\\begin{equation}\\label{u-infty}\n\t\tu^{\\infty} (\\hat{x}; d, c_p, c_s) = \\left(c_p u^{\\infty}_{pp}(\\hat{x}) + c_s u^{\\infty}_{ps}(\\hat{x})\\right) \\hat{x} + \\left(c_p u^{\\infty}_{sp}(\\hat{x}) + c_s u^{\\infty}_{ss}(\\hat{x})\\right)\n\t\t\\hat{x}^{\\perp},\n\t\\end{equation}\nwhere the dependence of $u^\\infty_{\\alpha\\beta}$ $(\\alpha, \\beta=p,s)$ on $d$ has been omitted for simplicity.\n\tIn this paper, the following inverse elastic scattering problems will be considered:\n\n\t\\textbf{IP-P:} Reconstruct the shape and position of the scattere $D$ from  knowledge of the compressional part $u^{\\infty}_{pp}(\\hat{x})$ of the far-field pattern due to one incident compressional wave $u^{i}_p$.\n\t\n\t\\textbf{IP-S:} Reconstruct the shape and position of  $D$ from knowledge of the shear part $u^{\\infty}_{ss}(\\hat{x})$ of the far-field pattern due to one incident shear wave $u^{i}_s$.\n\n\t\\textbf{IP-F:} Reconstruct the shape and position of  $D$ from using the entire far-field pattern $u^{\\infty}(\\hat{x})$ due to one incident wave $u^{i}$.\n\n\n\t\\section{Factorization method with infinitely many plane waves}\\label{sec:3}\n\t\n\t\\subsection{Review of the classical Factorization method for inverse elastic scattering}\n\n\tGiven a vector field $g(d)=g_p(d) d + g_s(d) d^{\\bot} \\in \\left(L^2(\\mathbb{S})\\right)^2$, the superposition of plane waves\n\t\\begin{equation}\n\t\tv_g(x):= \\text{ e}^{-i\\frac{\\pi}{4}} \\int_{{\\mathbb S}} \\left\\{\\sqrt{\\frac{k_p}{\\omega}}d\\text{e}^{ik_px \\cdot d}g_p(d) + \\sqrt{\\frac{k_s}{\\omega}}d^{\\bot}\\text{e}^{ik_sx \\cdot d}g_s(d) \\right\\} ds(d)\n\t\\end{equation}\n\tis denoted as the elastic Herglotz wave function with density $g$. \n\tThe Green's tensor of the Navier equation in free space, also called Kupradze's tensor (see e.g., \\cite{Arens}), is denoted by\n\t\\begin{equation}\n\t\t\\Gamma (x,y) \\coloneqq \\frac{i}{4\\mu} H^{(1)}_0(k_s |x-y|) {\\bf I} + \\frac{i}{4\\omega^2}\\triangledown^{\\bot }_x \\triangledown_x\\left(H^{(1)}_0(k_s|x-y|) - H^{(1)}_0(k_p|x-y|)\\right), \\ x,y\\in \\mathbb{R}^2, \\ x \\neq y,\n\t\\end{equation}\n\twhere $H^{(1)}_0$ is the Hankel function of the first kind and of order $n$. For any $y \\in {\\mathbb R}^2$ and any direction $a \\in {\\mathbb S}$, an elastic point source in $y$ with the polarization $a$ is given by \n\t\\begin{equation}\n\t\tu(x) = \\Gamma(x,y)a, \\quad x \\in {\\mathbb R}^2\\backslash\\{y\\}.\n\t\\end{equation}\n\tThe far-field pattern $\\Gamma^{\\infty}(\\cdot,y;a)$ of this point source is given by \n\t\\begin{equation}\n\t\t\\Gamma^{\\infty}(\\hat{x},y;a) = \\frac{k^2_p}{\\omega^2} \\frac{\\text{e}^{i\\frac{\\pi}{4}}}{\\sqrt{8\\pi k_p}} \\text{e}^{-ik_p\\hat{x}\\cdot y} (\\hat{x}\\cdot a) \\hat{x} + \\frac{k^2_s}{\\omega^2} \\frac{\\text{e}^{i\\frac{\\pi}{4}}}{\\sqrt{8\\pi k_s}} \\text{e}^{-ik_s\\hat{x}\\cdot y} (\\hat{x}^{\\bot}\\cdot a) \\hat{x}^{\\bot},\n\t\\end{equation}\nwith the compressional and shear parts:\n\t\\begin{equation}\n\t    \\Gamma^{\\infty}_p(\\hat{x},y;a)= \\Gamma^{\\infty}(\\hat{x},y;a) \\cdot \\hat{x}, \\quad \\Gamma^{\\infty}_s(\\hat{x},y;a) = \\Gamma^{\\infty}(\\hat{x},y;a) \\cdot \\hat{x}^{\\bot}.\n\t\\end{equation}\nIn this paper we define the elastic far-field operator as follows.\n\t\\begin{definition}\n\t\tThe far-field operator $F_D : \\left(L^2(\\mathbb{S})\\right)^2 \\to \\left(L^2(\\mathbb{S})\\right)^2$ is defined by \n\t\t\\begin{equation}\n\t\t\t\\begin{split}\n\t\t\t\t(F_D g)(\\hat{x})  & \\coloneqq \\text{ e}^{-i\\frac{\\pi}{4}} \\int_{\\mathtt{S}} u^{\\infty}_D \\left(\\hat{x}; d, \\sqrt{\\frac{k_p}{\\omega}}g_p(d), \\sqrt{\\frac{k_s}{\\omega}}g_s(d)\\right) ds(d) \\\\ \n\t\t\t\t&= \\text{ e}^{-i\\frac{\\pi}{4}} \\int_{\\mathtt{S}} \\left\\{\\sqrt{\\frac{k_p}{\\omega}}u^{\\infty}_D(\\hat{x}; d, 1, 0)g_p(d) + \\sqrt{\\frac{k_s}{\\omega}}u^{\\infty}_D(\\hat{x}; d, 0, 1)g_s(d)\\right\\} ds(d).\n\t\t\t\\end{split}\n\t\t\\end{equation}\n\t\\end{definition}\nFor rigid elastic bodies, it is well known that $F_D$ is a normal operator. It was proved in \\cite[Theorem 4.3]{Arens} that the operator $F_D$ can be decomposed into the form\n\t\\begin{equation}\\label{fd-svd}\n\t\tF_D = -\\sqrt{8\\pi \\omega} G_D S^{\\ast}_D G^{\\ast}_D.\n\t\\end{equation}\n\tHere the data-to-patten operator $G_D : \\left(H^{1\/2}(\\partial D)\\right)^2 \\to \\left(L^2(\\mathbb{S})\\right)^2$ is defined by $G_D h \\coloneqq u^{\\infty}$, where $u^{\\infty}$ is the far-field pattern of the solution to the Dirichlet boundary value problem of the Navier equation with the boundary value $h$. \n\tThe operator $S^{\\ast}_D:\\left(H^{-1\/2}(\\partial D)\\right)^2 \\to \\left(H^{1\/2}(\\partial D)\\right)^2$ is the adjoint of the elastic single layer potential operator $S_D$, given by \n\t\\begin{equation}\n\t\tS_D \\phi (x) \\coloneqq \\int_{\\partial D} \\Gamma (x,y) \\phi(y) ds(y), \\quad x \\in \\partial D. \n\t\\end{equation}\n\tBy the Factorization method, the far-field pattern $\\Gamma^{\\infty}(\\cdot,z;a)$ belongs to the range of $G_D$ if and only if $z \\in D$ (see \\cite[Theorem 4.7]{Arens}). Moreover, the $(F^{\\ast}F)^{1\/4}$-method (see \\cite[Theorem 4.8]{Arens}) verifies the relation $\\text{Range} (G_D)= \\text{Range} ((F^{\\ast}_D F_D)^{1\/4})$ if $\\omega^2$ is not an eigenvalue of $-\\Delta^{\\ast}$ over $D$. Hence, by the Picard theorem, the scatterer $D$ can be characterized by the spectra of $F_D$ as follows.\n\t\\begin{theorem}\\label{classfm}\n\t\t(\\cite[Theorem 4.8]{Arens}) Assume that $\\omega^2$ is not a Dirichlet eigenvalue of $-\\Delta^{\\ast}$ over $D$. Denote by $(\\lambda^{(n)}_{D}, \\varphi^{(n)}_{D})$ a spectrum system of the \n\t\tfar-field operator $F_D : \\left(L^{2}(\\mathbb{S})\\right)^2 \\rightarrow \\left(L^{2}(\\mathbb{S})\\right)^2$.\n\t\tThen,\n\t\t  \\begin{equation}\\label{FD}\n\t\t\tz \\in D  \n\t\t\t\\Longleftrightarrow\n\t\t  \tI(z) \\coloneqq \\sum\\limits_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle \\Gamma^{\\infty}(\\cdot,z;a), \\varphi_{D}^{(n)}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda_{D}^{(n)}\\right|} < +\\infty.\n\t\t  \\end{equation}\n\t\\end{theorem}\nBy Theorem \\ref{classfm}, the sign of the indicator function $I(z)$ can be regarded as the characteristic function of $D$. We note that in \\eqref{FD}, $z\\in {\\mathbb R}^2$ are the sampling variables\/points and the spectral data $(\\lambda^{(n)}_{D}, \\varphi^{(n)}_{D})$ are determined by the far-field patterns $u_D^\\infty(\\hat{x},d)$ over all observation and incident directions $\\hat{x}, d\\in {\\mathbb S}$.\n\nBelow we state the Factorization method which involves only the compressional or shear plane waves.\nIntroduce the projection space $L^2_p({\\mathbb S}) \\coloneqq \\left\\{g_p : {\\mathbb S} \\to {\\mathbb C}, g_p(d) = g(d) \\cdot d, |g_p| \\in L^2({\\mathbb S})\\right\\}$ and $L^2_s({\\mathbb S}) \\coloneqq \\left\\{g_s : {\\mathbb S} \\to {\\mathbb C}, g_s(d) = g(d) \\cdot d^{\\bot}, |g_s| \\in L^2({\\mathbb S})\\right\\}$.  Define the projection operators $P_p : \\left(L^2({\\mathbb S})\\right)^2 \\to L^2_p({\\mathbb S})$ and $P_s : \\left(L^2({\\mathbb S})\\right)^2 \\to L^2_s({\\mathbb S})$ (see e.g., \\cite{HuKirsch2012}) by\n\t\\begin{equation}\n\t\tP_pg(d) := g_p(d), \\quad P_sg(d):=g_s(d).\n\t\\end{equation}\n\tThus, we can define the P-part and S-part of the far-field operator $F_D$.\n\t\\begin{definition}\\label{def-ps}\n\t\tThe far-field operator $F^{(p)}_D : L^2_p({\\mathbb S}) \\to L^2_p({\\mathbb S})$ is defined by \n\t\t\\begin{equation}\n\t\t\tF^{(p)}_D g_p(d) \\coloneqq P_p F_D P^{\\ast}_p g_p(d).\t\n\t\t\\end{equation}\n\tAnalogously,\tthe far-field operator $F^{(s)}_D : L^2_s({\\mathbb S}) \\to L^2_s({\\mathbb S})$ is defined by\n\t\t\\begin{equation}\n\t\t\tF^{(s)}_D g_s(d) \\coloneqq P_s F_D P^{\\ast}_s g_s(d).\t\n\t\t\\end{equation}\nHere\t$P^{\\ast}_p$ and $P^{\\ast}_s$ are the adjoint operators of $P_p$ and $P_s$, respectively.\n\t\\end{definition}\n By \\eqref{fd-svd}, we have the factorization \n\t\\begin{equation}\\label{3.13}\n\t\tF^{(\\alpha)}_D = -\\sqrt{8\\pi \\omega}(P_{\\alpha}G_D)S^{\\ast}_D(P_{\\alpha}G_D)^{\\ast}, \n\t\\end{equation}\n\twhere $\\alpha = p, s$.\nBased on (\\ref{3.13}), the $F_{\\#}$-method (see \\cite[Lemma 3.5]{HuKirsch2012}) verifies the relation $\\text{Range} (P_{\\alpha}G_D)= \\text{Range} ((F^{(\\alpha)}_{D,\\#} )^{1\/2})$, provided $\\omega^2$ is not a Dirichlet eigenvalue of $-\\Delta^{\\ast}$ over $D$. Here the operator $F_{\\#}$ is defined by \n\t\\begin{equation}\n\t\tF_{\\#} \\coloneqq |\\text{Re}F| + |\\text{Im}F|, \\quad \\text{Re}F \\coloneqq \\frac{1}{2}[F+F^{\\ast}], \\quad \\text{Im}F \\coloneqq \\frac{1}{2i}[F-F^{\\ast}].\n\t\\end{equation}\n\tHence, the scatterer $D$ can be characterized by the spectra of $F^{(\\alpha)}_D$ as follows.\n\t\\begin{theorem\n\t\t(\\cite[Theorem 3.7, 3.8]{HuKirsch2012}) Assume that $\\omega^2$ is not a Dirichlet eigenvalue of $-\\Delta^{\\ast}$ over $D$. Denote by $(\\lambda^{(n)}_{D,\\alpha}, \\varphi^{(n)}_{D,\\alpha})$ a spectrum system of the positive operator $F^{(\\alpha)}_{D,\\#}$.\n\t\tThen,\n\t\t  \\begin{equation}\n\t\t\tz \\in D  \n\t\t\t\\Longleftrightarrow\n\t\t  \tI^{(\\alpha)}(z) \\coloneqq \\sum\\limits_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle \\Gamma^{\\infty}_{\\alpha}(\\cdot,z;a), \\varphi_{D,\\alpha}^{(n)}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda_{D,\\alpha}^{(n)}\\right|} < +\\infty, \\quad \\alpha=p, s.\n\t\t  \\end{equation}\n\t\\end{theorem}\n\n\t\\subsection{Further discussions on Factorization method}\nBefore stating the one-wave version of the factorization method for inverse elastic scattering, we first present a corollary of Theorems \\ref{classfm}. Denote by $\\Omega\\subset {\\mathbb R}^2$ a convex and bounded Lipschitz domain which represents a rigid elastic scatterer.  Here we use a new notation $\\Omega$ in order to distinguish from our target scatterer $D$. The far-field operator $F_{\\Omega} : \\left(L^{2}(\\mathbb{S})\\right)^2 \\rightarrow \\left(L^{2}(\\mathbb{S})\\right)^2$ corresponding to $\\Omega$ is therefore defined by\n\t\\begin{equation}\\label{op-domain}\n\t\t(F_{\\Omega} g)(\\hat{x}) \\coloneqq \\text{ e}^{-i\\frac{\\pi}{4}} \\int_{\\mathbb{S}} \\left\\{\\sqrt{\\frac{k_p}{\\omega}}u^{\\infty}_{\\Omega}(\\hat{x}; d, 1, 0)g_p(d) + \\sqrt{\\frac{k_s}{\\omega}}u^{\\infty}_{\\Omega}(\\hat{x}; d, 0, 1)g_s(d)\\right\\} ds(d),\n\t\\end{equation}\n\twhere $u^{\\infty}_{\\Omega} (\\hat{x}; d, 1, 0)$ and $u^{\\infty}_{\\Omega} (\\hat{x}; d, 0, 1)$ are the far-field patterns corresponding to the elastic plane waves $u^{i}_p(x; d)$ and $u^{i}_s(x; d)$ incident onto $\\Omega$, respectively. The eigenvalues and eigenfunctions of $F_{\\Omega}$ will be denoted by $(\\lambda^{(n)}_{\\Omega}, \\varphi^{(n)}_{\\Omega})$.\nThe following Corollaries can be derived straightforwardly from the classical Factorization method in the previous subsection. \n\n\n\t\\begin{corollary}\\label{fm_convex_domain} Let $v^\\infty\\in \\left(L^2({\\mathbb S})\\right)^2$ and assume that\n\t\t$\\omega^2$ is not a Dirichlet eigenvalue of $-\\Delta^{\\ast}$ over $\\Omega$. Then \n\t   \\begin{equation}\n\t\t   I(\\Omega) = \\sum_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle v^{\\infty}, \\varphi^{(n)}_{\\Omega}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{\\Omega}\\right|} < + \\infty\n\t   \\end{equation}\n\t\t   if and only if $v^{\\infty}$ is the far-filed pattern of some Kupradze radiating solution $v^{s}$, where $v^{s}$ satisfies the Navier equation\n\t   \\begin{equation}\\label{Navier}\n\t\t   \\Delta^{\\ast} v^{s} + \\omega^{2} v^{s} = 0 \\qquad \\text{in} \\quad  \\mathbb{R}^2 \\backslash \\overline{\\Omega},\n\t   \\end{equation}\n\t\twith the boundary data $v^{s} |_{\\partial \\Omega}\\in \\left(H^{1\/2}(\\partial \\Omega)\\right)^2.$\n\t   \\end{corollary}\n\n\t   \\begin{proof}\n\t\tBy \\eqref{fd-svd}, we have\n\t\t$F_{\\Omega} = -\\sqrt{8\\pi \\omega} G_{\\Omega} S^*_{\\Omega} G^*_{\\Omega}$,\n\t\twhere $G_{\\Omega}: \\left(H^{1\/2}(\\partial \\Omega)\\right)^2 \\to \\left(L^2(\\mathbb{S})\\right)^2$ is the data-to-pattern operator corresponding to $\\Omega$.\n\t\tObviously, $I(\\Omega) <+\\infty$ if and only if $v^{\\infty} \\in \\text{Range}((F^*_{\\Omega} F_{\\Omega})^{1\/4})$. Since $\\text{Range}((F^*_{\\Omega} F_{\\Omega})^{1\/4}) = \\text{Range}(G_{\\Omega})$, we get $v^{\\infty} \\in \\text{Range}(G_{\\Omega})$ if and only if $I(\\Omega) <+\\infty$.\n\t\tRecalling the definition of $G_{\\Omega}$, it follows that $v^{s}$ satisfies the Navier equation \\eqref{Navier} and the Kupradze radiation condition \\eqref{Kupr-cd} with the boundary data $v^{s} |_{\\partial \\Omega}\\in \\left(H^{1\/2}(\\partial \\Omega)\\right)^2.$\n\t\t\\end{proof}\n\n\t\n\t\t\\begin{corollary}\\label{fm_convex_domainpp} \n\t\t\tLet $w^\\infty_{\\alpha\\alpha} \\in L^2_{\\alpha}({\\mathbb S})$($\\alpha = p, s$) and assume that\n\t\t\t$\\omega^2$ is not a Dirichlet eigenvalue of $-\\Delta^{\\ast}$ over $\\Omega$. Denote by $(\\lambda^{(n)}_{\\Omega,\\alpha}, \\varphi^{(n)}_{\\Omega,\\alpha})$ a spectrum system of the positive operator $F^{(\\alpha)}_{\\Omega,\\#}$. Then \n\t\t   \\begin{equation}\\label{OP}\n\t\t\t   I^{(\\alpha)}(\\Omega) = \\sum_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle w^\\infty_{\\alpha\\alpha}, \\varphi^{(n)}_{\\Omega, \\alpha}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{\\Omega, \\alpha}\\right|} < + \\infty\n\t\t   \\end{equation}\n\t\t\t   if and only if $w^\\infty_{\\alpha\\alpha}= P_{\\alpha} v^{\\infty}$, where $v^{\\infty}$ is the far-field pattern of some Kupradze radiating solution $v^{s}$, which is defined in $\\mathbb{R}^2 \\backslash \\overline{\\Omega}$ and $v^{s} |_{\\partial \\Omega}\\in \\left(H^{1\/2}(\\partial \\Omega)\\right)^2$. \t\nThat is,\t$w^\\infty_{\\alpha\\alpha}$ is the far-field pattern of some Sommerfeld radiating solution $w^s_{\\alpha\\alpha}$, fulfilling the relations\n \\begin{equation}\\label{wsp}\n w_{pp}^{s} = -\\frac{1}{k_p^2}{\\rm div\\,} v^s\\quad \\mbox{if}\\quad\\alpha = p;\\quad w_{ss}^{s} = \\frac{1}{k_s^2}{\\rm curl\\,} v^s\\quad\\mbox{if} \\quad \\alpha = s, \n \\end{equation}\n where $v^s$ satisfies the boundary value problem of the Navier equation \n\t\t   \\begin{equation}\\label{navierbp}\n\t\t\t   \\Delta^{\\ast} v^{s} + \\omega^{2} v^{s} = 0 \\ \\text{in} \\  \\mathbb{R}^2 \\backslash \\overline{\\Omega}, \\quad v^{s} |_{\\partial \\Omega}\\in \\left(H^{1\/2}(\\partial \\Omega)\\right)^2.\n\t\t   \\end{equation}\n\t\t\\end{corollary}\nSimilar to Corollary \\ref{fm_convex_domain},  Corollary \\ref{fm_convex_domainpp} can be proved by using the projection operator $P_{\\alpha}$ and the $F_{\\#}$-method.\n\t\n\t\t\\begin{proof}\n\t\tWithout loss of generality, we assume that the relation (\\ref{OP}) holds with $\\alpha=p$. By the Picard theorem,   $w^\\infty_{pp} \\in  \\text{Range}(F^{(p)}_{\\Omega, \\#}) = \\text{Range}(P_pG_{\\Omega})$ if and only if the indicator function $I^{(p)}(\\Omega)  < + \\infty$. This means that $w^\\infty_{pp}=P_p v^{\\infty}$, where $v^{\\infty}$ is the far-field pattern of a Kupradze's radiating solution $v^s$ to the boundary value problem \\eqref{navierbp}. Using the Hodge decomposition, we see $w^\\infty_{pp}$ is the far-field pattern of $w^s_{pp}$, where $w^s_{pp}=-\\frac{1}{k_p^2}\\text{div} v^s$. \n\n\t\tOn the other hand, let $v^s$ be a solution to the boundary value problem \\eqref{navierbp} and define  $w_{pp}^{s} := -\\frac{1}{k_p^2}{\\rm div\\,} v^s$. Suppose that \n $w^\\infty_{pp}$ is the far-field pattern of $w^s_{pp}$. Then, by the Hodge decomposition  it follows that $w^\\infty_{pp} = P_p v^{\\infty}=P_p G_\\Omega(v^s|_{\\partial \\Omega})$, implying that $w^\\infty_{pp} \\in \\text{Range}(P_pG_{\\Omega})=\\text{Range}(F^{(p)}_{\\Omega, \\#}) $. Applying the Picard theorem yields $I^{(p)}(\\Omega) < + \\infty$.\t\t\n\t\t\\end{proof}\n\\begin{remark}\nIt follows from (\\ref{wsp}) that the restrictions of $w^s_{pp}$, $w^s_{ss}$ to $\\partial \\Omega$ lie in the space $H^{-1\/2}(\\partial\\Omega)$.\n\\end{remark}\n\n\t\\section{Factorization method with one plane waves}\\label{sec:4}\n\n\n\n\t\tIn our applications of Corollary \\ref{fm_convex_domain} (resp. Corollary \\ref{fm_convex_domainpp}), we will take $v^\\infty$ to be the measurement data $u_D^\\infty(\\hat x; d_0, c_p, c_s)$ (resp. $u_{D,\\alpha\\alpha}^\\infty(\\hat x; d_0)$, $\\alpha=p,s$)\tcorresponding to our target elastic scatterer $D$ and the incident elastic plane wave $u^{i}(x; d_0, c_p, c_s)$ (resp. $u^{i}_\\alpha(\\hat{x};d_0)$) for some fixed $d_0\\in{\\mathbb S}$. We shall omit the dependance on $d_0$ if it is always clear from the context.  Our purpose is to extract the geometrical information on $D$ from the domain-defined indicator functions $I(\\Omega)$ and $I^{(\\alpha)}(\\Omega)$ ($\\alpha=p,s$). By Corollary \\ref{fm_convex_domain}, $I(\\Omega)<\\infty$ if the scattered field $u^{s}_D(x)=u^{s}(x; d_0, D)$ can be extended to the domain ${\\mathbb R}^2\\backslash\\overline{\\Omega}$ as a solution to the Navier equation.\nBelow we shall discuss the absence of the analytic extension of $u^{s}_D$, $u^{s}_{D, p}$ and $u^{s}_{D,s}$ around a planer corner point of $D$.\n\n\n\t\n\\begin{lemma}\\label{ana-ext}\n\t\t\tAssume that $D$ is a rigid elastic scatterer occupying a convex polygon.\nThen the scattered field $u^{s}_D(x; d_0)$, $u^{s}_{D, p}(x; d_0)$ and $u^{s}_{D,s}(x; d_0)$\n cannot be analytically extended from $\\mathbb{R}^{2} \\backslash \\overline{D}$ into $D$ across any corner of $D$.\n\t\t\\end{lemma}\n\n\n\n\t\t\\begin{proof} We shall carry out the proof by contradiction.\n\t\t \n\t(i)\tAssume on the contrary that $u^{s}_D(x;d_0)$ can be analytically continued across a corner of $\\partial D$. By coordinate translation and rotation, we may suppose that this corner coincides with the origin, so that $u^{s}_D(x;d_0)$ and also the total field $u_D=u^{s}_D(x;d_0)+u^{i}(x; d_0, c_p, c_s)$ satisfy the Navier equation \\eqref{lame-eqn} in $B_\\epsilon(O)$ for some $\\epsilon>0$. Since $u_D$ is real analytic in $({\\mathbb R}^2\\backslash\\overline{D})\\cup B_\\epsilon(O)$ and $D$ is a convex polygon, $u_D$ satisfies the Navier equation on the closure of an infinite sector $\\Sigma\\subset {\\mathbb R}^2\\backslash\\overline{D}$ which extends the finite sector $B_\\epsilon(O)\\cap D$ to ${\\mathbb R}^2\\backslash\\overline{D}$. In particular, the total field $u_D$ fulfills the Dirichlet boundary condition on the two half lines $\\partial \\Sigma$ starting from the corner point $O$. \nSince $u^{s}_D(x;d_0)$ fulfills the Kupradze's  radiation condition, it holds that\n\\[\n\\lim_{|x|\\rightarrow\\infty} u^{i}(x; d_0, c_p, c_s)=-\n\\lim_{|x|\\rightarrow\\infty} u^{s}_D(x; d_0, c_p, c_s)=0,\\quad x\\in \\partial \\Sigma.\n\\]\nHowever, this is impossible for an elastic plane wave incidence of the form \\eqref{2.1}.\t\n\t\n\n\n(ii) Suppose that $u^{s}_{D,p}$ can be analytically extended from ${\\mathbb R} \\backslash \\bar{D}$ into $D$ across a corner $O$ of $D$. \nThat is,  $u^{s}_{D,p}$ extends to a function which satisfies the vector Helmholtz equation $\\Delta u^{s}_{D,p} + k_p^2 u^{s}_{D,p} = 0$ in $B_\\epsilon(O)$ for some $\\epsilon>0$.\nBy the Hodge decomposition of the total field $u_D = u_{D,p} + u_{D,s} = \\triangledown \\varphi + {\\rm curl\\,} \\psi$,  the function $\\varphi$ can be also extended to $B_\\epsilon(O)$ as a solution to the scalar Helmholtz equation. Here we have used the fact that the incident wave $u^{i}$ is an entire solution to the Navier equation. \nIn particular, this implies that the normal and tangential derivatives $(\\partial_n \\varphi, \\partial_\\tau \\varphi)$ of $\\varphi$ are both piecewise analytic on $\\partial D\\cap B_\\epsilon(O)$. Recalling the Dirichlet boundary condition of $u_D$, we have\n\\begin{equation}\n\t\t\t\\left\\{\n\t\t\t\t\\begin{split}\n\t\t\t\t\tu_D\\cdot n &= \\frac{\\partial \\varphi}{\\partial n} + \\frac{\\partial \\psi}{\\partial \\tau} = 0 \\\\\n\t\t\t\t\tu_D\\cdot \\tau &= \\frac{\\partial \\varphi}{\\partial \\tau} - \\frac{\\partial \\psi}{\\partial n} = 0\n\t\t\t\t\\end{split}\n\t\t\t\\right. \\quad \\text{on} \\ \\partial D\\cap B_\\epsilon(O).\n\t\t\\end{equation}\nHence, $(\\partial_\\tau\\psi, \\partial_n \\psi)=(-\\partial_n\\varphi, \\partial_\\tau\\varphi)$ on $\\partial D\\cap B_\\epsilon(O)$ and thus the Cauchy data of $\\psi$ are also piecewise analytic. By the Cauchy-Kovalevskaya theorem,  the function $\\psi$ admits an extension from $B_\\epsilon(O)\\cap ({\\mathbb R}^2\\backslash\\overline{D})$ to $B_\\epsilon(O)\\cap D$, as a solution to the scalar Helmholtz equation with the wave number $k_s$.\nHence, the total field can be continued to $B_\\epsilon(O)\\cap D$, which however is impossible by the first part of the proof. \n\n(iii) The case of $u^{s}_{D,s}$ can be proved similarly to the second step for \n$u^{s}_{D,p}$.\n\t\t\\end{proof}\n\nAs in the acoustic case \\cite{M-H},  our approach applies to inverse source problems as well. For this purpose, we need to justify the absence of analytical extension for elastic source scattering problems in a corner domain, which is closet to studies of non-radiating elastic sources given in \\cite{Bla2018}.\n\t\t\\begin{lemma\nLet $\\chi_D$ be the characteristic function for the convex polygon $D$. If $u\\in \\left(H^2_{loc}(\\mathbb{R}^2)\\right)^2$ is a radiating solution to\n\t\t\t\\begin{equation}\n\t\t\t\\Delta^{\\ast} u(x)+ \\omega^2u(x)=\\chi_D(x)f(x)\\quad\\mbox{in}\\quad \\mathbb{R}^2,\n\t\t\t\\end{equation}\n\t\t\twhere $f\\in L^\\infty({\\mathbb R}^2)$ is H\\\"older continuous near the corner point $O$ of $D$ satisfying $f(O)\\neq 0$. Then $u$ cannot be analytically extended from $\\mathbb{R}^2\\backslash\\overline{D}$ to $D$ across the  corner $O$. \n\t\t\\end{lemma}\n\n\t\t\\begin{figure}[ht]\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=0.3]{pic2.eps}  \n\t\t\t\\caption{Illustration of a convex polygonal source term where $O$ is corner point of $D$.}\\label{pic2}\n\t\t\\end{figure}\n\n\t\t\\begin{proof}Without loss of generality, the corner point $O$ is supposed to coincide with the origin. Set $v^\\pm=u|_{D^\\pm}$ where $D^+:=B_\\delta(O)\\cap ({\\mathbb R}^2\\backslash\\overline{D})$ and $D^-:=B_\\delta(O)\\cap D$. \n\t\tAssume that $v^+$ can be analytically extended from ${\\mathbb R}^2\\backslash\\overline{D}$ to $B_\\delta(O)$ for some $\\delta > 0$ (see Figure \\ref{pic2}), as a solution of the Navier equation. This implies that\n$\\Delta^{\\ast} v^+ + \\omega^2v^+ = 0$ in $B_\\delta(O)$,\nwith the Cauchy data \n\\[\nv^-=v^+, \\quad \\mathcal{T}_{\\nu} v^-=\\mathcal{T}_{\\nu} v^+\\quad \\mbox{on}\\quad\\Gamma:=\\partial D\\cap B_\\delta(O).\n\\] Here, the boundary traction operator $\\mathcal{T}_{\\nu}$ is defined as \n\\[\n\\mathcal{T}_{\\nu}u = 2\\mu \\frac{\\partial u}{\\partial \\nu} + \\lambda \\nu \\triangledown \\cdot u + \\mu \\nu^{\\bot}(\\partial_2 u_1 - \\partial_1 u_2)\n\\] in the two-dimensional case. \nSince $\\Delta^{\\ast} v^- + \\omega^2v^- = f$ in $B_\\delta(O)\\cap D$, the difference $w:=u^--u^+$ is a solution to \n\t\t\t\\begin{equation}\n\t\t\t\t\t\\Delta^{\\ast} w + \\omega^2w = f\\quad\\mbox{in}\\quad B_\\delta(O)\\cap D,\\quad\n\t\t\t\t\tw=\\mathcal{T}_{\\nu}w=0\\quad\\mbox{on}\\quad \\Gamma.\n\t\t\t\\end{equation}\nBy \\cite[Proposition 3.2]{Bla2018}, it follows that $f(O)=0$, which is in contradiction with our assumption  that $f(O)\\neq 0$.\n\t\t\\end{proof}\n\n\nTo state the one-wave factorization method, we shall restrict our discussions to a convex polygonal rigid elastic scatterer $D$. \nLet $\\Omega$ be another convex rigid scatterer for detecting $D$ such that $\\omega^2$ is not the Dirichlet eigenvalue of $-\\Delta^{\\ast}$ in $\\Omega$. Denote by $(\\lambda^{(n)}_{\\Omega}, \\varphi^{(n)}_{\\Omega})$ the eigenvalues and eigenfunctions of the far-field operator $F_{\\Omega}$. Below we characterize the inclusion relationship  between our target scatterer $D$ and the test domain $\\Omega$ by the measurement data $u_D^\\infty$ and the spectra of $F_{\\Omega}$.\n\t\t\\begin{theorem}\\label{One-FM}\n\t\t\tDefine\t\n\t\t\t\\begin{equation}\\label{svd-omega}\n\t\t\t\tW(\\Omega) := \\sum_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle u_D^{\\infty}, \\varphi^{(n)}_{\\Omega}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{\\Omega}\\right|}.\n\t\t\t\\end{equation}\n\t\t\tThen $W(\\Omega)<\\infty$\tif and only if $D \\subseteq \\Omega$.\n\t\t\\end{theorem}\n\n\t\t\\begin{proof}\n\t\tBy Corollary \\ref{fm_convex_domain}, $W(\\Omega) < +\\infty$ implies that $u^{s}_D(x;d_0)$ is analytic in $\\mathbb{R}^2 \\backslash \\overline{\\Omega}$.\n\t\tIf  $D \\nsubseteq \\Omega$, three cases might happen: (i) $\\Omega\\subset D$; (ii) $\\Omega\\cap D=\\emptyset$; (iii) $\\Omega\\cap D\\neq \\emptyset$ and $\\Omega\\cap ({\\mathbb R}^2\\backslash\\overline{D})\\neq \\emptyset$.\n\t\tIn either of these cases, we observe that there is always a corner $O \\in D$ and $O \\notin \\Omega$ by the convexity of both $\\Omega$ and $D$. Then, $u^{s}_D(x;d_0)$ can be analytically continued from ${\\mathbb R}^2\\backslash\\overline{D}$ to $D$ across the corner $O$ of $\\partial D$, which however is impossible by  Lemma \\ref{ana-ext}. This proves the relationship $D \\subseteq \\Omega$.\n\n\tNow assume that $D \\subseteq \\Omega$. Then the scattered field $u^{s}_D(x;d_0)$ satisfies the Navier equation $ \\Delta^{\\ast} u^{s}_D(x;d_0) + \\omega^{2} u^{s}_D(x;d_0) = 0$ in  $\\mathbb{R}^2 \\backslash \\overline{\\Omega}$ with the boundary data  $f:=u^{s}_D(x;d_0)|_{\\partial \\Omega} \\in \\left(H^{1\/2}(\\partial \\Omega)\\right)^2$. \n\tThis implies that $u_D^\\infty=G_\\Omega(f)$. Hence,\nwe get $W(\\Omega) < +\\infty$ by applying Corollary \\ref{fm_convex_domain}.\n\t\t\\end{proof}\n\n\t\tFor the operator $F^{(\\alpha)}_{\\Omega}$($\\alpha=p, s$), denote by $(\\lambda^{(n)}_{\\Omega,\\alpha}, \\varphi^{(n)}_{\\Omega,\\alpha})$ the eigenvalues and eigenfunctions of the positive operator $F^{(\\alpha)}_{\\Omega,\\#}$. \nUsing Corollary \\ref{fm_convex_domainpp} and arguing analogously to the proof of Theorem \\ref{One-FM}, we obtain\n\t\\begin{theorem}\\label{One-FMpp}\n\t\t\tDefine\t\n\t\t\t\\begin{equation\n\t\t\t\tW^{(\\alpha)}(\\Omega) := \\sum_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle u_{D,\\alpha \\alpha}^{\\infty}, \\varphi^{(n)}_{\\Omega,\\alpha}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{\\Omega,\\alpha}\\right|},\n\t\t\t\\end{equation}\n\t\t\twhere $u_{D,\\alpha \\alpha}^{\\infty}$ is defined the same as \\eqref{u-inftypp} for the scatterer $D$, $\\alpha = p, s$. Then $W^{(\\alpha)}(\\Omega)<\\infty$\tif and only if $D \\subseteq \\Omega$.\n\t\t\\end{theorem}\n\t\n\n\n\t\\section{Explicit examples when $\\Omega$ is a disk}\\label{sec:5}\n\t\n\tTheorems \\ref{One-FM} and \\ref{One-FMpp} rely essentially on the factorization form (see e.g., \\eqref{fd-svd}) of the far-field operator for the elastic scatterer $\\Omega$.  Below we show that the results of Theorems \\ref{One-FM} and \\ref{One-FMpp} can be justified independently of the factorization form, as long as the test domain $\\Omega$ is chosen to be a rigid elastic disk. This is mainly due to the explicit form of  the far-field pattern for a rigid disk in terms of special functions; see Subsection \\ref{sec5.1} below. Then we can get an explicit spectral system of the far-field operator $F_{B_R}$ in Subsection \\ref{sec5.2}. The proofs \n Corollaries \\ref{fm_convex_domain} and  \\ref{fm_convex_domainpp}  will be shown in Subsection \\ref{sec5.3}.\t\nNote that this section is of independent interests, since as shown in the subsequent sections, the derivation of eigenvalues and eigenfunctions of $F_{B_R}$ turns out to be non-trivial, which is in contrast to the acoustic case of the Helmholtz equation. \n\t\n\t\\subsection{Far-field pattern of a rigid disk $B_R$}\\label{sec5.1}\nAssume that $B_R \\coloneqq \\left\\{x : |x|<R \\right\\}$ is a rigid disk centered at the origin with radius $R>0$. Let $\\hat{x}=\\left(\\cos \\theta_x, \\sin \\theta_x \\right)^T\\in {\\mathbb S}$ be the observation direction (or variable) of the elastic far-field pattern.\nAccording to the Hodge decomposition \\eqref{Hodgedecom}, we can introduce scalar functions $\\varphi\\left(r,\\theta_x\\right)$ and $\\psi\\left(r,\\theta_x\\right)$ such that in polar coordinates $x=(r, \\theta_x)$,\n\t\\begin{equation}\\label{5.1}\n\t\tu^{s}(r,\\theta_x) = \\text{grad}\\  \\varphi (r, \\theta_x) + \\text{curl}\\  \\psi (r, \\theta_x). \n\t\\end{equation}\nRecall the relationship between the Cartesian and polar coordinates for gradient: \n\t\\begin{equation}\n\t\t\\begin{pmatrix}\n\t\t\t\\partial_1 \\\\ \\partial_2\n\t\t\\end{pmatrix}\n\t\t= \n\t\t\\begin{pmatrix}\n\t\t\t\\cos \\theta & -\\frac{1}{r}\\sin \\theta \\\\\n\t\t\t\\sin \\theta & \\frac{1}{r} \\cos \\theta\n\t\t\\end{pmatrix}\n\t\t\\begin{pmatrix}\n\t\t\t\\partial_r \\\\ \\partial_{\\theta}\n\t\t\\end{pmatrix}.\n\t\\end{equation}\nNoting that $\\varphi$ and $\\psi$ are both Sommerfeld radiating solutions, we make the following ansatz: \n \\[\n \\varphi = \\sum_{n \\in \\mathbb{Z}} \\frac{1}{\\sqrt{k_p}} A_n H^{(1)}_n(k_p r)\\text{e}^{in\\theta_x},\\quad\n \\psi = \\sum_{n \\in \\mathbb{Z}} \\frac{1}{\\sqrt{k_s}} B_n H^{(1)}_n(k_s r)\\text{e}^{in\\theta_x}.\n \\] We can get  from (\\ref{5.1}) that\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tu^{s}(r,\\theta_x) &= \n\t\t\t\\begin{pmatrix}\n\t\t\t\t\\cos \\theta_x \\partial_r \\varphi - \\frac{1}{r} \\sin \\theta_x \\partial_{\\theta_x} \\varphi \\\\\n\t\t\t\t\\sin \\theta_x \\partial_r \\varphi + \\frac{1}{r} \\cos \\theta_x \\partial_{\\theta_x} \\varphi\n\t\t\t\\end{pmatrix}\n\t\t\t+\n\t\t\t\\begin{pmatrix}\n\t\t\t\t-\\sin \\theta_x \\partial_r \\psi - \\frac{1}{r} \\cos \\theta_x \\partial_{\\theta_x} \\psi \\\\\n\t\t\t\t\\cos \\theta_x \\partial_r \\psi - \\frac{1}{r} \\sin \\theta_x \\partial_{\\theta_x} \\psi\n\t\t\t\\end{pmatrix} \\\\\n\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{\\text{e}^{in\\theta_x}}{r} \\left[\n\t\t\t\t\\left(\n\t\t\t\t\t\\sqrt{k_p} r H^{(1)\\prime}_n(k_p r) A_n - \\frac{in}{\\sqrt{k_s}} H^{(1)}_n(k_s r)B_n\n\t\t\t\t\\right) \n\t\t\t\\begin{pmatrix}\n\t\t\t\t\\cos \\theta_x \\\\ \\sin \\theta_x\n\t\t\t\\end{pmatrix}\n\t\t\t\\right. \\\\\n\t\t\t & \\quad \\left.   + \\left(\n\t\t\t\t\\frac{in}{\\sqrt{k_p}} H^{(1)}_n(k_p r)A_n + \\sqrt{k_s} r H^{(1)\\prime}_n(k_s r)B_n\n\t\t\t\t\\right)\n\t\t\t\\begin{pmatrix}\n\t\t\t\t-\\sin \\theta_x \\\\ \\cos \\theta_x\n\t\t\t\\end{pmatrix}\n\t\t\t\\right].\n\t\t\\end{split}\n\t\\end{equation}\n\tUsing the asymptotic property of Hankel functions \n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tH_n^{(1)}(z) &= \\sqrt{\\frac{2}{\\pi z}} \\text{e}^{i(z-\\frac{n\\pi}{2}-\\frac{\\pi}{4})} \\left( 1+\\mathcal{O}(\\frac{1}{z})\\right), \\ z \\to \\infty, \\\\\n\t\t\tH_n^{(1)\\prime}(z) &= \\sqrt{\\frac{2}{\\pi z}} \\text{e}^{i(z-\\frac{n\\pi}{2}+\\frac{\\pi}{4})} \\left( 1+\\mathcal{O}(\\frac{1}{z})\\right), \\ z \\to \\infty,\n\t\t\\end{split}\n\t\\end{equation}\n\twe have \n\t\\begin{equation}\n\t\tu^{s}(x) = \\frac{\\text{e}^{ik_pr}}{\\sqrt{r}} \\sqrt{\\frac{2}{\\pi}} \\sum_{n \\in \\mathbb{Z}} \\text{e}^{-i(\\frac{n\\pi}{2}-\\frac{\\pi}{4})} A_n \\text{e}^{in\\theta_x}\\hat{x} + \\frac{\\text{e}^{ik_sr}}{\\sqrt{r}} \\sqrt{\\frac{2}{\\pi}} \\sum_{n \\in \\mathbb{Z}} \\text{e}^{-i(\\frac{n\\pi}{2}-\\frac{\\pi}{4})} B_n \\text{e}^{in\\theta_x}\\hat{x}^{\\bot} + \\mathcal{O}(\\frac{1}{r^{3\/2}}).\n\t\\end{equation}\n\tThus, \n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tu^{\\infty}_p(\\hat{x}) = \\sqrt{\\frac{2}{\\pi}} \\text{e}^{i\\frac{\\pi}{4}} \\sum_{n \\in \\mathbb{Z}} i^{-n} A_n \\text{e}^{in\\theta_x}, \\quad\n\t\t\tu^{\\infty}_s(\\hat{x}) = \\sqrt{\\frac{2}{\\pi}} \\text{e}^{i\\frac{\\pi}{4}} \\sum_{n \\in \\mathbb{Z}} i^{-n} B_n \\text{e}^{in\\theta_x}.\n\t\t\\end{split}\n\t\\end{equation}\n\tNow, we set \n\t\\begin{equation}\n\t\tt_{\\alpha} = k_{\\alpha} R, \n\t\t\\ \\alpha = p, s,\n\t\\end{equation}\n\tto get \n\t\\begin{equation}\\label{uscR}\n\t\t\\begin{split}\n\t\t\tu^{s}(r,\\theta_x)|_{r=R} &= \\sum_{n \\in \\mathbb{Z}} \n\t\t\t\\left[ \n\t\t\t\t\\left(\n\t\t\t\t\t\\frac{t_p}{\\sqrt{k_p}} H^{(1)\\prime}_n(t_p) A_n - \\frac{in}{\\sqrt{k_s}} H^{(1)}_n(t_s)B_n\n\t\t\t\t\\right)\n\t\t\t\t\\begin{pmatrix}\n\t\t\t\t\t\\cos \\theta_x \\\\ \\sin \\theta_x\n\t\t\t\t\\end{pmatrix} \n\t\t\t\\right. \\\\\n\t\t\t&\\quad +\\left. \\left(\n\t\t\t\t\\frac{in}{\\sqrt{k_p}} H^{(1)}_n(t_p) A_n + \\frac{t_s}{\\sqrt{k_s}} H^{(1)\\prime}_n(t_s)B_n\n\t\t\t\t\\right)\n\t\t\t\t\\begin{pmatrix}\n\t\t\t\t\t-\\sin \\theta_x \\\\ \\cos \\theta_x\n\t\t\t\t\\end{pmatrix}\n\t\t\t\\right]\n\t\t\t\\frac{\\text{e}^{in \\theta_x}}{R} \\\\\n\t\t\t&=\\sum_{n \\in \\mathbb{Z}} \n\t\t\t\\begin{pmatrix}\n\t\t\t\t\\nu & \\tau\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\tt_p H^{(1)\\prime}_n(t_p) & -in H^{(1)}_n(t_s)\\\\\n\t\t\t\tin H^{(1)}_n(t_p) & t_s H^{(1)\\prime}_n(t_s)\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\t\\frac{1}{\\sqrt{k_p}} & 0 \\\\\n\t\t\t\t0 & \\frac{1}{\\sqrt{k_s}}\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\tA_n \\\\ B_n\n\t\t\t\\end{pmatrix}\n\t\t\t\\frac{\\text{e}^{in \\theta_x}}{R},\n\t\t\\end{split}\n\t\\end{equation}\n\twhere $\\nu = (\\cos \\theta_x, \\sin \\theta_x)^T$ and $\\tau = (-\\sin \\theta_x, \\cos \\theta_x)^T$ are tangential and normal directions of $\\partial B_R$ respectively, and that $(\\nu\\quad  \\tau)$ is thus a 2-by-2 matrix.\nSet\n\t\\begin{equation}\\label{Hn}\n\t\t\\mathcal{H}_n \\coloneqq \t\\begin{pmatrix}\n\t\t\tt_p H^{(1)\\prime}_n(t_p) & -in H^{(1)}_n(t_s)\\\\\n\t\t\tin H^{(1)}_n(t_p) & t_s H^{(1)\\prime}_n(t_s)\n\t\t\\end{pmatrix} \\text{ and }\n\t\tQ \\coloneqq \\begin{pmatrix}\n\t\t\t\\frac{1}{\\sqrt{k_p}} & 0 \\\\\n\t\t\t0 & \\frac{1}{\\sqrt{k_s}}\t\n\t\t\\end{pmatrix}.\n\t\\end{equation}\n\tFrom \\eqref{uscR} and \\eqref{Hn}, we have \n\t\\begin{equation}\\label{scfield}\n\t\tu^{s}(r,\\theta_x)|_{r=R} = \\sum_{n \\in \\mathbb{Z}} \n\t\t\\begin{pmatrix}\n\t\t\t\\nu & \\tau\n\t\t\\end{pmatrix}\n\t\t\\mathcal{H}_n Q\n\t\t\\begin{pmatrix}\n\t\t\tA_n \\\\ B_n\n\t\t\\end{pmatrix}\n\t\t\\frac{\\text{e}^{in \\theta_x}}{R}.\n\t\\end{equation}\n\n\t\\begin{remark}\\label{hn-determin} By \n\t\\cite[Lemma 2.11]{Bao2018} and \\cite[Remark 3.2]{Li2016}, \n\tit follows that $|\\mathcal{H}_n| = t_p t_s H_n^{(1)\\prime}(t_p)H_n^{(1)\\prime}(t_s) - n^2 H_n^{(1)}(t_p)H_n^{(1)}(t_s) \\neq 0$ for any $n \\in \\mathbb{Z}$. Hence\nthe matrix $\\mathcal{H}_n$ is always invertible.\n\\end{remark}\n\nHaving expanded the scattered field into the series (\\ref{scfield}), now we need to represent an elastic plane wave in terms of the special functions on $|x|=R$.\nLet $d =\\left(\\cos \\theta_d, \\sin \\theta_d\\right)^T\\in {\\mathbb S}$ be the incident direction. By the Jacobi-Anger expansion (see e.g., \\cite[Formula (3.89)]{kress.1998}), we see\n\t\\begin{equation}\\label{JAe}\n\t\t\\text{e}^{ikx \\cdot d} = \\sum_{n \\in \\mathbb{Z}} i^n J_n\\left(k|x|\\right) \\text{e}^{in \\theta}, \\quad x \\in \\mathbb{R}^2,\n\t\\end{equation}\n\twhere $\\theta = \\theta_x - \\theta_d$ denotes the angle between $\\hat{x}$ and $d$.\n\tLet $d^\\bot = (-\\sin \\theta_d, \\cos \\theta_d)^T$ be a vector perpendicular to $d$. Recalling the compressional part $u^{i}_p$ of the incident wave $u^{i}$, \n\t\\begin{equation}\\label{uinp}\n\t\tu^{i}_p (x) = d \\  \\text{e}^{i k_p x \\cdot d},\n\t\\end{equation}\nand inserting \\eqref{JAe} to \\eqref{uinp}, we get the form of $u^{i}_p$ over $\\partial B_R$ as\n\t\\begin{equation}\\label{5.14}\n\t\tu^{i}_p|_{r=R} = \\sum_{n \\in \\mathbb{Z}} d J_n(t_p) i^n \\text{e}^{in \\theta}.\n\t\\end{equation}\nUsing the formulas \n\t\\begin{equation}\n\t\t\\cos \\theta = \\frac{1}{2}\\left( \\text{e}^{i \\theta} + \\text{e}^{-i \\theta}\\right), \\ \\sin \\theta = \\frac{1}{2i} \\left( \\text{e}^{i \\theta} - \\text{e}^{-i \\theta}\\right)\n\t\\end{equation}\n\tand the following properties of Bessel functions\n\t\\begin{equation}\n\t\t2 J^{\\prime}_n(x) = J_{n-1}(x) - J_{n+1}(x), \\ J_{n-1}(x) + J_{n+1}(x) = \\frac{2n}{x} J_n(x),\n\t\\end{equation}\n\twe can get from (\\ref{5.14}) that \n\t\\begin{equation}\\label{5.16}\n\t\t\\begin{split}\n\t\t\tu^{i}_p \\cdot \\nu |_{r=R} &= \\frac{1}{2} \\sum_{n \\in \\mathbb{Z}} \\left(\\text{e}^{i \\theta} + \\text{e}^{-i \\theta}\\right) J_n(t_p ) i^n \\text{e}^{in \\theta} \\\\\n\t\t\t&= \\frac{1}{2} \\sum_{n \\in \\mathbb{Z}} \\left( J_n(t_p) i^n \\text{e}^{i(n+1) \\theta} + J_n(t_p) i^n \\text{e}^{i(n-1) \\theta} \\right) \\\\\n\t\t\t&= \\frac{1}{2} \\sum_{n \\in \\mathbb{Z}} \\left(J_n(t_p) i^{n-1} \\text{e}^{in \\theta} + J_n(t_p) i^{n+1} \\text{e}^{in \\theta} \\right) \\\\\n\t\t\t&= \\frac{1}{2} \\sum_{n \\in \\mathbb{Z}} \\left(J_{n-1}(t_p) - J_{n+1}(t_p) \\right) i^{n-1} \\text{e}^{in \\theta} \\\\\n\t\t\t&= \\sum_{n \\in \\mathbb{Z}} J^{\\prime}_n(t_p) i^{n-1} \\text{e}^{in \\theta}\n\t\t\\end{split}\n\t\\end{equation}\n\tand \n\t\\begin{equation}\\label{5.17}\n\t\t\\begin{split}\n\t\t\tu^{i}_p \\cdot \\tau |_{r=R} &= \\frac{1}{2i} \\sum_{n \\in \\mathbb{Z}} \\left(\\text{e}^{-i \\theta} - \\text{e}^{i \\theta}\\right) J_n(t_p) i^n \\text{e}^{in \\theta} \\\\\n\t\t\t&= \\frac{1}{2} \\sum_{n \\in \\mathbb{Z}} \\left( J_n(t_p) i^{n-1} \\text{e}^{i(n-1) \\theta} - J_n(t_p) i^{n-1} \\text{e}^{i(n+1) \\theta} \\right) \\\\\n\t\t\t&= \\frac{1}{2} \\sum_{n \\in \\mathbb{Z}} \\left(J_{n+1}(t_p) i^{n} \\text{e}^{in \\theta} + J_{n-1}(t_p) i^{n} \\text{e}^{in \\theta} \\right) \\\\\n\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{n}{t_p} J_n(t_p) i^{n} \\text{e}^{in \\theta}.\n\t\t\\end{split}\n\t\\end{equation}\n\t\nWe use the notation $v^{s}$ to denote the scattered field produced by the compressional part $u^{i}_p$ of the incident wave $u^{i}$. Note that $v^{s}$  is usually different from the compressional part of the scattered field for $u^{i}$.\nBy \\eqref{scfield}, we can represent $v^{s}(r,\\theta_x)$ on $|x|=R$\tas the series\n \\begin{equation}\n\t\tv^{s}(r,\\theta_x)|_{r=R} =\\sum_{n \\in \\mathbb{Z}} \n\t\t\\begin{pmatrix}\n\t\t\t\\nu & \\tau\n\t\t\\end{pmatrix}\n\t\t\\mathcal{H}_n Q\n\t\t\\begin{pmatrix}\n\t\t\tA_{n,v} \\\\ B_{n,v}\n\t\t\\end{pmatrix}\n\t\t\\frac{\\text{e}^{in \\theta_x}}{R},\n\t\\end{equation}\n\twhere the coefficients $A_{n, v}, B_{n, v}\\in{\\mathbb C}$  are associated with $v^{s}$. Making use of the Dirichlet boundary condition $u^{i}_p = -v^{s}$ on $\\partial B_R$, we have \n\t\\begin{equation*}\n\t\t\\left.\n\t\t\\begin{pmatrix}\n\t\t\tu^{i}_p \\cdot \\nu \\\\ u^{i}_p \\cdot \\tau\n\t\t\\end{pmatrix} \\right\\rvert_{r=R} \n\t\t= - \\left.\n\t\t\\begin{pmatrix}\n\t\t\tv^{s} \\cdot \\nu \\\\ v^{s} \\cdot \\tau\n\t\t\\end{pmatrix} \\right\\rvert_{r=R},\n\t\\end{equation*}\n\twhich together with (\\ref{5.16}) and (\\ref{5.17}) leads to \n\t\\begin{equation}\n\t\t\\sum_{n \\in \\mathbb{Z}}\n\t\t\\begin{pmatrix}\n\t\t\tJ^{\\prime}_n(t_p) \\\\ \\frac{in}{t_p} J_n(t_p)\n\t\t\\end{pmatrix}\n\t\ti^{n-1} \\text{e}^{in \\theta_x} \\text{e}^{-in \\theta_d} = -\\sum_{n \\in \\mathbb{Z}}\n\t\t\\mathcal{H}_n Q\n\t\t\\begin{pmatrix}\n\t\t\tA_{n,v} \\\\ B_{n,v}\n\t\t\\end{pmatrix}\n\t\t\\frac{\\text{e}^{in \\theta_x}}{R}. \n\t\\end{equation}\n\tBy the arbitrariness of $\\theta_x$, we can get\n\t\\begin{equation*}\n\t\t\\begin{pmatrix}\n\t\t\ti J^{\\prime}_n(t_p) \\\\ -\\frac{n}{t_p} J_n(t_p)\n\t\t\\end{pmatrix}\n\t\ti^{n} \\text{e}^{-in \\theta_d} \n\t\t= \n\t\t\\frac{\\mathcal{H}_n Q}{R}\n\t\t\\begin{pmatrix}\n\t\t\tA_{n,v} \\\\ B_{n,v}\n\t\t\\end{pmatrix},\n\t\\end{equation*}\n\timplying the relation \n\t\\begin{equation}\n\t\t\\begin{pmatrix}\n\t\t\tA_{n,v} \\\\ B_{n,v}\n\t\t\\end{pmatrix}\n\t\t= Q^{-1} \\mathcal{H}_n^{-1}\n\t\t\\begin{pmatrix}\n\t\t\ti J^{\\prime}_n(t_p) \\\\ -\\frac{n}{t_p} J_n(t_p)\n\t\t\\end{pmatrix}\n\t\tR i^{n} \\text{e}^{-in \\theta_d}  \\,.\n\t\\end{equation}\n\tThus, we obtain the P-part and S-part of the far-field patterns of $v^{s}$ as follows:\n\t\\begin{equation}\\label{ppsp-infty}\n\t\t\\begin{pmatrix}\n\t\t\tu^{\\infty}_{pp}(\\hat{x}) \\\\ \n\t\t\tu^{\\infty}_{sp}(\\hat{x})\n\t\t\\end{pmatrix}\n\t\t= \\sqrt{\\frac{2}{\\pi}} \\text{e}^{i\\frac{\\pi}{4}}iR \\sum_{n \\in \\mathbb{Z}} Q^{-1} \\mathcal{H}_n^{-1} \n\t\t\\begin{pmatrix}\n\t\t\t J^{\\prime}_n(t_p) \\\\ \\frac{in}{t_p} J_n(t_p)\n\t\t\\end{pmatrix}\n\t\t\\text{e}^{in \\theta}. \n\t\\end{equation}\n\nThe far-field pattern excited by the S-part of an elastic plane wave can be treated similarly.\nThe shear part $u^{i}_s$ of the incident wave $u^{i}$ takes the form \n\t\\begin{equation}\n\t\tu^{i}_s (x) = d^{\\bot } \\text{e}^{i k_s x \\cdot d},\n\t\\end{equation}\nwhich together with  \\eqref{JAe} gives arise to\n\t\\begin{equation}\n\t\tu^{i}_s|_{r=R} = \\sum_{n \\in \\mathbb{Z}} d^{\\bot } J_n(t_s) i^n \\text{e}^{in \\theta}.\n\t\\end{equation}\n\tCorrespondingly, we have\n\t\\begin{equation*}\n\t\n\t\t\tu^{i}_s \\cdot \\nu |_{r=R} = -\\sum_{n \\in \\mathbb{Z}} \\frac{n}{t_s} J_n(t_s) i^{n} \\text{e}^{in \\theta}, \\quad\n\t\t\tu^{i}_s \\cdot \\tau |_{r=R}= \\sum_{n \\in \\mathbb{Z}} J^{\\prime}_n(t_s) i^{n-1} \\text{e}^{in \\theta}.\n\n\t\\end{equation*}\nAgain using \\eqref{scfield}, we can represent by $w^{s}$ the scattered field produced by $u^{i}_s$ in the form of\n\t\\begin{equation}\n\t\tw^{s}(r,\\theta_x)|_{r=R} =\\sum_{n \\in \\mathbb{Z}} \n\t\t\\begin{pmatrix}\n\t\t\t\\nu & \\tau\n\t\t\\end{pmatrix}\n\t\t\\mathcal{H}_n Q\n\t\t\\begin{pmatrix}\n\t\t\tA_{n,w} \\\\ B_{n,w}\n\t\t\\end{pmatrix}\n\t\t\\frac{\\text{e}^{in \\theta_x}}{R}.\n\t\\end{equation}\nCombining the proves two identities together with the boundary condition $u^{i}_s = -w^{s}$ on $\\partial B_R$, we arrive at  \n\t\\begin{equation}\n\t\t\\begin{pmatrix}\n\t\t\tA_{n,w} \\\\ B_{n,w}\n\t\t\\end{pmatrix}\n\t\t= Q^{-1} \\mathcal{H}_n^{-1}\n\t\t\\begin{pmatrix}\n\t\t\t\\frac{n}{t_s} J_n(t_s) \\\\ i J^{\\prime}_n(t_s)\n\t\t\\end{pmatrix}\n\t\tR i^{n} \\text{e}^{-in \\theta_d}.  \n\t\\end{equation}\n\tThus, we obtain $u^{\\infty}_{ps}$ and $u^{\\infty}_{ss}$ as follows:\n\t\\begin{equation}\\label{psss-infty}\n\t\t\\begin{pmatrix}\n\t\t\tu^{\\infty}_{ps}(\\hat{x})\\\\ \n\t\t\tu^{\\infty}_{ss}(\\hat{x})\n\t\t\\end{pmatrix}\n\t\t= \\sqrt{\\frac{2}{\\pi}} \\text{e}^{i\\frac{\\pi}{4}}iR \\sum_{n \\in \\mathbb{Z}} Q^{-1} \\mathcal{H}_n^{-1} \n\t\t\\begin{pmatrix}\n\t\t\t-\\frac{in}{t_s} J_n(t_s) \\\\  J^{\\prime}_n(t_s)\n\t\t\\end{pmatrix}\n\t\t\\text{e}^{in \\theta}. \n\t\\end{equation}\n This enables us to define the matrix\n\t\\begin{equation}\n\t\tU_{B_R}^{\\infty} \\coloneqq\n\t\t\\begin{pmatrix}\n\t\t\tu^{\\infty}_{pp}(\\hat{x}) & u^{\\infty}_{ps}(\\hat{x}) \\\\ \n\t\t\tu^{\\infty}_{sp}(\\hat{x}) & u^{\\infty}_{ss}(\\hat{x})\n\t\t\\end{pmatrix}\n\t\t= \\sqrt{\\frac{2}{\\pi}} \\text{e}^{i\\frac{\\pi}{4}} i \\sum_{n \\in \\mathbb{Z}} Q^{-1} \\mathcal{H}_n^{-1}\n\t\t\\begin{pmatrix}\n\t\t\tR J^{\\prime}_n(t_p) & -\\frac{in}{k_s} J_n(t_s) \\\\\n\t\t    \\frac{in}{k_p} J_n(t_p) & R J^{\\prime}_n(t_s)\n\t\t\\end{pmatrix}\n\t\t\\text{e}^{in \\theta}.\n\t\\end{equation}\nSetting\n\t\\begin{equation\n\t\t\\mathcal{J}_n \\coloneqq \\begin{pmatrix}\n\t\t\tt_p J^{\\prime}_n(t_p) & -in J_n(t_s)\\\\\n\t\t\tin J_n(t_p) & t_s J^{\\prime}_n(t_s)\n\t\t\\end{pmatrix},\n\t\\end{equation}\nwe can rewrite $U_{B_R}^{\\infty}$ as\n\t\\begin{equation}\\label{matrix-uinf}\n\t\tU_{B_R}^{\\infty}(\\hat{x}) = \\sqrt{\\frac{2}{\\pi}} \\text{e}^{i\\frac{\\pi}{4}} i \\sum_{n \\in \\mathbb{Z}} Q^{-1} \\mathcal{H}_n^{-1} \\mathcal{J}_n Q^2\n\t\t\\text{e}^{in \\theta}.\n\t\\end{equation}\n\n\t\nTo sum up,  for the elastic plane wave  $u^{i}$ of the general form (\\ref{2.1}), by linear superposition its far-field pattern $u^{\\infty}$ takes the form of \n\t\\eqref{u-infty}, where the P-part and S-part for $u^{i}_p$ and \n\t$u^{i}_s$\n\t are given in the matrix $U_{B_R}^\\infty$ (see also \\eqref{ppsp-infty} and \\eqref{psss-infty}).\t\n\t\n\t\n\t\\subsection{Spectral system of  the far-field operator $F_{B_R}$}\\label{sec5.2}\n\t\n\n\tNow, we need to derive eigenvalues and the associated eigenfunctions  of the far-field operator $F_{B_R}$ defined by (e.g. (\\ref{op-domain}))\n\t\\begin{equation}\\label{op-br}\n\t\t\t(F_{B_R}g)(\\hat{x}) = \\text{ e}^{-i\\frac{\\pi}{4}} \\int_{\\mathtt{S}} \\left\\{\\sqrt{\\frac{k_p}{\\omega}}u^{\\infty}_{B_R}(\\hat{x}; d, 1, 0)g_p(d) + \\sqrt{\\frac{k_s}{\\omega}}u^{\\infty}_{B_R}(\\hat{x}; d, 0, 1)g_s(d)\\right\\} ds(d).\n\t\t\\end{equation}\nObviously, the spectral system of $F_{B_R}$ should be connected to the spectral system of the matrix $U_{B_R}$. To disclose this relation, we retain the notations from the previous subsection to define $\\widetilde{\\Sigma}_n  := Q^{-1} \\mathcal{H}_n^{-1} \\mathcal{J}_n Q$.\n\n\t\\begin{lemma}\\label{gn}\n\t\tIf $(\\lambda_n, \\widetilde{X}_n)$ is the spectral pair of $\\widetilde{\\Sigma}_n$, that is, $\\widetilde{\\Sigma}_n\\widetilde{X}_n = \\lambda_n \\widetilde{X}_n,\\ \\widetilde{X}_n = (\\widetilde{X}_n^{(1)}, \\widetilde{X}_n^{(2)})^T$. \n\t\tThen \n\t\t\\begin{equation}\\label{L5.2}\n\t\t(F_{B_R}g)(\\hat{x}) = \\sqrt{\\frac{8\\pi}{\\omega}}i \\lambda_n\\; g(\\hat{x}),\\quad g(\\hat{x}):=(\\widetilde{X}_n^{(1)} \\hat{x} + \\widetilde{X}_n^{(2)} \\hat{x}^{\\bot })\\text{e}^{in \\theta_x}.\n\t\t\\end{equation}\n\t\\end{lemma}\n\t\\begin{proof}  Let $g\\in \\left(L^2({\\mathbb S})\\right)^2$ be given as in \\eqref{L5.2}.\nIt is easy to see the P- and S-component of $g$ as $g_p(d)=\\widetilde{X}_n^{(1)} \\text{e}^{in \\theta_d}$, $g_s(d)=\\widetilde{X}_n^{(2)} \\text{e}^{in \\theta_d}$. It then follows from the definition of $F_{B_R}$ in \\eqref{op-br} that\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\left(F_{B_R}g\\right)(\\hat{x}) &= \\frac{1}{\\sqrt{\\omega}} \\text{ e}^{-i\\frac{\\pi}{4}}\n\t\t\t\\int_{\\mathbb{S}} g_p(d) \\left[\\sqrt{k_p}u^{\\infty}_{pp}(\\hat{x}; d) \\hat{x} + \\sqrt{k_p}u^{\\infty}_{sp}(\\hat{x}; d) \\hat{x}^{\\bot}\\right] \\\\ \n\t\t\t& \\quad + g_s(d) \\left[\\sqrt{k_s}u^{\\infty}_{ps}(\\hat{x}; d) \\hat{x} + \\sqrt{k_s}u^{\\infty}_{ss}(\\hat{x}; d) \\hat{x}^{\\bot}\\right] ds(d) \\\\\n\t\t\t&= \\frac{1}{\\sqrt{\\omega}} \\text{ e}^{-i\\frac{\\pi}{4}} \\int_{\\mathbb{S}} \\left[\\widetilde{X}_n^{(1)} \\sqrt{k_p}u^{\\infty}_{pp}(\\hat{x}; d) \\hat{x} + \\widetilde{X}_n^{(1)} \\sqrt{k_p}u^{\\infty}_{sp}(\\hat{x}; d) \\hat{x}^{\\bot}\\right]\\text{e}^{in \\theta_d} \\\\ \n\t\t\t& \\quad + \\left[\\widetilde{X}_n^{(2)} \\sqrt{k_s}u^{\\infty}_{ps}(\\hat{x}; d) \\hat{x} + \\widetilde{X}_n^{(2)} \\sqrt{k_s}u^{\\infty}_{ss}(\\hat{x}; d) \\hat{x}^{\\bot}\\right]\\text{e}^{in \\theta_d} ds(d) \\\\\n\t\t\t&= \\frac{1}{\\sqrt{\\omega}} \\text{ e}^{-i\\frac{\\pi}{4}} \\int_{\\mathbb{S}} \\left[ \\left(\\widetilde{X}_n^{(1)} \\sqrt{k_p}u^{\\infty}_{pp}(\\hat{x}; d) + \\widetilde{X}_n^{(2)} \\sqrt{k_s}u^{\\infty}_{ps}(\\hat{x}; d) \\right)\\hat{x}\\right. \\\\\n\t\t\t& \\quad + \\left. \\left(\\widetilde{X}_n^{(1)} \\sqrt{k_p}u^{\\infty}_{sp}(\\hat{x}; d) + \\widetilde{X}_n^{(2)} \\sqrt{k_s}u^{\\infty}_{ss}(\\hat{x}; d)\\right)\n\t\t\t\\hat{x}^{\\bot}\\right]\\text{e}^{in \\theta_d} ds(d) \\\\\n\t\t\t&= \\frac{1}{\\sqrt{\\omega}} \\text{ e}^{-i\\frac{\\pi}{4}} \\int_{\\mathbb{S}} \\begin{pmatrix}\n\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\\end{pmatrix}\n\t\t\tU_{B_R}^{\\infty} Q^{-1} \\widetilde{X}_n \\text{e}^{in \\theta_d} ds(d)\\\\\n\t\t\t&= \\sqrt{\\frac{2}{\\omega \\pi}}i \\int_{\\mathbb{S}} \\sum_{m \\in \\mathbb{Z}} \\begin{pmatrix}\n\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\\end{pmatrix} \\widetilde{\\Sigma}_m \\widetilde{X}_n \\text{e}^{im \\theta_x} \\text{e}^{in \\theta_d} ds(d).\n\t\t\\end{split}\n\t\\end{equation*}\nUsing the orthogonality of $\\text{e}^{in \\theta_d}$ for $n\\in {\\mathbb Z}$ and the fact that  $\\widetilde{\\Sigma}_n$ and $\\widetilde{X}_n$ are independent of $d$, we arrive at\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t\\left(F_{B_R}g\\right)(\\hat{x}) &= \\sqrt{\\frac{2}{\\omega \\pi}}i \\int_{\\mathbb{S}} \\begin{pmatrix}\n\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\\end{pmatrix} \\widetilde{\\Sigma}_n \\widetilde{X}_n \\text{e}^{in \\theta_x} ds(d)\\\\\n\t\t\t&= \\sqrt{\\frac{8\\pi}{\\omega}}i  \\begin{pmatrix}\n\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\\end{pmatrix} \\widetilde{\\Sigma}_n \\widetilde{X}_n \\text{e}^{in \\theta_x} \\\\\n\t\t\t&= \\sqrt{\\frac{8\\pi}{\\omega}}i  \\begin{pmatrix}\n\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\\end{pmatrix} \\lambda_n \\widetilde{X}_n \\text{e}^{in \\theta_x}\\\\\n\t\t\t&= \\sqrt{\\frac{8\\pi}{\\omega}}i \\lambda_n g(\\hat{x}).\n\t\t\\end{split}\n\t\\end{equation*}\n    \\end{proof}\nAs a consequence of Lemma \\ref{gn}, we obtain the spectral pair of $F_{B_R}$ as follows. \n\t\\begin{lemma}\\label{eig-FBR}\n\t\tThe spectral pair of $F_{B_R}$ is given by\n\t\\[\n\t(\\lambda^{(n)}_{B_R}, X^{(n)}_{B_R}) = \\left(\\sqrt{\\frac{8\\pi}{\\omega}}i \\lambda_n,\\; \\begin{pmatrix}\n\t\t\t\\hat{x} & \\hat{x}^{\\bot }\n\t\t\\end{pmatrix}\n\t\t Q^{-1}X_n \\text{e}^{in \\theta_x} \\right),\n\t\t \\]\n\t\t  where $(\\lambda_n, X_n)$ is the spectral pair of $\\Sigma_n \\coloneqq \\mathcal{H}_n^{-1} \\mathcal{J}_n$.\n\t\\end{lemma}\n\t\\begin{proof} Suppose that $(\\lambda_n, \\widetilde{X}_n)$ is the spectral pair of $\\widetilde{\\Sigma}_n$.\tWriting $X_n := Q \\widetilde{X}_n$, we have\n\t\t\\begin{equation}\n\t\t\t\\lambda_n \\widetilde{X}_n =\\widetilde{\\Sigma}_n \\widetilde{X}_n = Q^{-1}\\Sigma_n Q \\widetilde{X}_n = Q^{-1}\\Sigma_n X_n.\n\t\t\\end{equation}\nThis implies that $\\lambda_n  X_n = \\Sigma_n  X_n$.\nUsing Lemma \\ref{gn}, we get\n\\[\n\\lambda^{(n)}_{B_R}= \\sqrt{\\frac{8\\pi}{\\omega}}i \\lambda_n,\n\\quad\nX^{(n)}_{B_R}(\\hat{x})=\\left(\\hat{x} \\;\\;  \\hat{x}^{\\bot }\\right) Q^{-1}X_n \\text{e}^{in \\theta_x}.\n\\]\n\t\\end{proof}\nSince the eigenvalues of $F_{B_R}$ have appeared in the denominator of the indicator \\eqref{svd-omega} with $\\Omega=B_R$, \nit is necessary to show  $\\lambda^{(n)}_{B_R}\\neq 0$ for all $n\\in {\\mathbb N}$ under an additional assumption of the frequency. \n\t\\begin{lemma}\n\t\tIf $\\omega^2$ is not a Dirichlet eigenvalue of $-\\Delta^{\\ast}$ in $B_R$, then the eigenvalue $\\lambda_n$ of the matrix $\\Sigma_n$ cannot vanish for any $n \\in \\mathbb{Z}$.\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tSuppose that there exists  $n\\in \\mathbb{Z}$ such that $\\lambda_n = 0$ is the eigenvalue of the matrix $\\Sigma_n$ and that $X_n \\neq 0$ is the corresponding eigenvector. Then, \n\t\t\\begin{equation*}\n\t\t\t\\mathcal{H}_n^{-1} \\mathcal{J}_n X_n=\\Sigma_n X_n = \\lambda_n X_n = 0,\n\t\t\\end{equation*}\n\t\n\t\n\t\n\t\n\t\tSince $\\mathcal{H}_n^{-1}$ is invertible\n\t\t(see Remark \\ref{hn-determin}), \n\t\twe have \n\t\t\t$\\mathcal{J}_n X_n = 0$, implying that\n$|\\mathcal{J}_n|=0$.  Since\n$X_n = (X_n^{(1)},X_n^{(2)})^T \\neq 0$, we may define the non-trivial  function $u = {\\rm grad\\,} \\varphi + {\\rm curl\\,} \\psi$ where\n\\[\n\\varphi = \\sum_{n \\in \\mathbb{Z}}  X_n^{(1)} J_n(k_p r)\\text{e}^{in\\theta_x}, \\quad \\psi = \\sum_{n \\in \\mathbb{Z}} X_n^{(2)} J_n(k_s r)\\text{e}^{in\\theta_x}.\n\\]  Then it is easy to check that\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t& \\quad u|_{\\partial B_R} \\\\\n\t\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{\\text{e}^{in\\theta_x}}{R} \\left[\n\t\t\t\t\t\\left(\n\t\t\t\t\t\tk_p R J^{\\prime}_n(k_p R) X_n^{(1)} - in J_n(k_s R)X_n^{(2)}\n\t\t\t\t\t\\right) \\hat{x} + \n\t\t\t\t\t\\left(\n\t\t\t\t\t\tin J_n(k_p R)X_n^{(1)} + k_s R J^{\\prime}_n(k_s R)X_n^{(2)}\n\t\t\t\t\t\\right) \\hat{x}^{\\bot}\\right]\\\\\n\t\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{\\text{e}^{in\\theta_x}}{R} \n\t\t\t\t\\begin{pmatrix}\n\t\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\t\\end{pmatrix}\n\t\t\t\t\\begin{pmatrix}\n\t\t\t\t\tt_p J^{\\prime}_n(t_p) & - in J_n(t_s ) \\\\\n\t\t\t\t\tin J_n(t_p) & t_s J^{\\prime}_n(t_s)\n\t\t\t\t\\end{pmatrix}\n\t\t\t\t\\begin{pmatrix}\n\t\t\t\t\tX_n^{(1)} \\\\ X_n^{(2)}\n\t\t\t\t\\end{pmatrix}\\\\\n\t\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{\\text{e}^{in\\theta_x}}{R} \n\t\t\t\t\\begin{pmatrix}\n\t\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\t\\end{pmatrix}\n\t\t\t\t\\mathcal{J}_n X_n\\\\\n\t\t\t\t&= 0.\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\tOn the other hand, it is obvious that  $u$ satisfies the Navier equation $-\\Delta^{\\ast} u = \\omega^2 u $ in  $B_R$. Hence, it is a Dirichlet eigenfunction of $-\\Delta^*$ over $B_R$, which is impossible.\n\t\n\t\n\t\\end{proof}\n\n\n\n\n\nTo calculate the spectra of $F_{B_R}$, by Lemma \\ref{eig-FBR} we need to consider the generalized eigenvalue problem \n\t\\begin{equation}\\label{gev}\n\t\t\\mathcal{J}_n X_n = \\lambda_n \\mathcal{H}_n X_n,\n\t\\end{equation}\n\twhere $\\lambda_n$ and $X_n$ represent eigenvalues and eigenvectors of $\\Sigma_n$.\nRecalling the Hankel functions and its derivatives,\n\t\\begin{equation*}\n\t\tH^{(1)}_n(z) = J_n(z) + i Y_n(z), \\  H^{(1)\\prime}_n(z) = J^{\\prime}_n(z) + i Y^{\\prime}_n(z),\n\t\\end{equation*}\nand  setting\n\t\\begin{equation*}\n\t\t\\mathcal{Y}_n \\coloneqq \\begin{pmatrix}\n\t\t\tt_p Y^{\\prime}_n(t_p) & -in Y_n(t_s)\\\\\n\t\t\tin Y_n(t_p) & t_s Y^{\\prime}_n(t_s)\n\t\t\\end{pmatrix},\n\t\\end{equation*}\nwe can rephrase the matrix $\\mathcal{H}_n$ as\n\t\\begin{equation}\\label{jyh}\n\t\t\\mathcal{H}_n = \\mathcal{J}_n + i \\mathcal{Y}_n.\n\t\\end{equation}\nBelow we describe an eigensystem of the generalized eigenvalue problem (\\ref{gev}) with the help of the decomposition \\eqref{jyh}.\n\n\t\\begin{lemma}\\label{eig-hn}\nA normalized eigensystem $(\\lambda_{n,j}, X_{n,j})$ with $n\\in {\\mathbb N}, j=1,2$ to the generalized eigenvalue problem (\\ref{gev}) is given by \n\\begin{equation}\\label{5.34}\n\\lambda_{n,j} = \\frac{t_p J_n^{\\prime}(t_p) + in J_n(t_s)\\sigma_j^{(n)}}{t_p H_n^{(1)\\prime}(t_p) + in H_n^{(1)}(t_s)\\sigma_j^{(n)}}, \\quad\nX_{n,j} = \\frac{(1,\\sigma_j^{(n)})^T}{\\sqrt{1+|\\sigma_j^{(n)}|^2}},\n\\end{equation}\t\nwith\n\\begin{equation*}\n\\begin{split}\n&\\sigma_1^{(n)} = \\frac{-\\beta_n + \\sqrt{\\beta_n^2 -4}}{2}, \\quad\n\t\t\\sigma_2^{(n)} = \\frac{-\\beta_n - \\sqrt{\\beta_n^2 -4}}{2},\\\\\n&\\beta_n = \\frac{\\pi}{2in}\\left[n^2(J_n(t_s)Y_n(t_p)-J_n(t_p)Y_n(t_s))+t_pt_s(J_n^{\\prime}(t_p)Y_n^{\\prime}(t_s)-J_n^{\\prime}(t_s)Y_n^{\\prime}(t_p))\\right].\n\\end{split}\n\\end{equation*}\n\t\t\n\t\\end{lemma}\n\n\t\\begin{proof}\n\t\tLet $X_n = (1,\\sigma^{(n)})^T$ be an eigenvector of the generalized  eigenvalue problem $\\mathcal{J}_n X_n = \\eta_n \\mathcal{Y}_n X_n$, where $\\eta_n$ is the eigenvalue. \nUsing the Wronskian \n\t\t\\begin{equation*}\n\t\t\tJ_n(t)Y_n^{\\prime}(t)-J_n^{\\prime}(t)Y_n(t) = \\frac{2}{\\pi t},\n\t\t\\end{equation*}\nsimple calculations show that $\\sigma^{(n)}$ should satisfy the algebraic equation\n\t\t\\begin{equation}\\label{x2}\n\t\t\t\\sigma^{(n)2}+ \\beta_n \\sigma^{(n)} +1 =0,\n\t\t\\end{equation}\n\t\twhere $\\beta_n$ is defined as in the lemma.\nThe two roots of \\eqref{x2} are given by\n\t\t\\begin{equation*}\n\t\t\t\\sigma_1^{(n)} = \\frac{-\\beta_n + \\sqrt{\\beta_n^2 -4}}{2}, \\quad\n\t\t\t\\sigma_2^{(n)} = \\frac{-\\beta_n - \\sqrt{\\beta_n^2 -4}}{2}.\n\t\t\\end{equation*}\nOn the other hand, one can also calculate the corresponding eigenvalues \n\t\t\\begin{equation*}\n\t\t\t\\eta_{n,j} = \\frac{t_p J_n^{\\prime}(t_p) + in J_n(t_s)\\sigma_j^{(n)}}{t_p Y_n^{\\prime}(t_p) + in Y_n(t_s)\\sigma_j^{(n)}},\\quad  j=1,2.\n\t\t\\end{equation*}\nUsing the decomposition (\\ref{jyh}), we get\n\t\t\\begin{equation*}\n\t\t\t\\mathcal{H}_n X_n = (\\mathcal{J}_n + i \\mathcal{Y}_n) X_n = \\frac{\\eta_n+i}{\\eta_n} \\mathcal{J}_n X_n.\n\t\t\\end{equation*}\nTherefore,  $(\\frac{\\eta_n}{\\eta_n+i},X_n)$ is the eigensystem of the generalized eigenvalue problem $\\mathcal{J}_n X_n = \\lambda_n \\mathcal{H}_n X_n$.\n\t\tFurther, we get the eigenvalues \n\t\t\\begin{equation\n\t\t\t\\lambda_{n,j} =\\frac{\\eta_{n,j}}{\\eta_{n,j}+i}= \\frac{t_p J_n^{\\prime}(t_p) + in J_n(t_s)\\sigma_j^{(n)}}{t_p H_n^{(1)\\prime}(t_p) + in H_n^{(1)}(t_s)\\sigma_j^{(n)}}, \\  j=1,2.\n\t\t\\end{equation}\n\t\n\t\\end{proof}\nCombining Lemma \\ref{eig-FBR} and \\ref{eig-hn}, we obtain an eigensystem of the far-field operator $F_{B_R}$ by\n\t\\begin{equation}\\label{eigsys-fbr}\n\t\t\\begin{split}\n\t\t\t\\lambda^{(n)}_{B_R,j} &= \\sqrt{\\frac{8\\pi}{\\omega}}\\frac{it_p J_n^{\\prime}(t_p) - n J_n(t_s)\\sigma_j^{(n)}}{t_p H_n^{(1)\\prime}(t_p) + in H_n^{(1)}(t_s)\\sigma_j^{(n)}}, \\\\\n\t\t\tX^{(n)}_{B_R,j} &= \\left(\\sqrt{k_p}X_{n,j}^{(1)} \\hat{x} + \\sqrt{k_s}X_{n,j}^{(2)} \\hat{x}^{\\bot} \\right) \\text{e}^{in\\theta_x}\\\\\n\t\t\t&= \\left(\\sqrt{k_p}\\hat{x}+\\sqrt{k_s}\\sigma_j^{(n)}\\hat{x}^{\\bot}\\right)\\frac{\\text{e}^{in \\theta_x}}{\\sqrt{1+|\\sigma_j^{(n)}|^2}}\n\t\t\\end{split}\n\t\\end{equation}\nfor $n\\in {\\mathbb N}, j=1,2$. \n\n\\begin{remark}\\label{asy-beta}\nThe asymptotics of $\\sigma_j^{(n)}$ can be derived as follows.\nRecall the asymptotic behavior of \t\nBessel functions (see \\cite{kress.1998}) \n\t\\begin{equation\n\t\t\\begin{split}\n\t\t\tJ_{n}(z) &= \\frac{z^{n}}{2^{n} n!} \\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\\quad n \\rightarrow +\\infty, \\\\\n\t\t\tJ_{n}^{\\prime}(z) &= \\frac{z^{n-1}}{2^{n} (n-1)!} \\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\\quad n \\rightarrow +\\infty,\n\t\t\\end{split}\n\t\\end{equation}\n\tand those of Neumann functions:\n\t\\begin{equation\n\t\t\\begin{split}\n\t\t\tY_{n}(z) &= -\\frac{2^{n}(n-1)!}{\\pi z^{n}}\n\t\t\t\\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\\quad n \\rightarrow +\\infty,\\\\\n\t\t\tY_{n}^{\\prime}(z) &= \\frac{2^{n} n!}{\\pi z^{n+1}} \n\t\t\t\\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\\quad n \\rightarrow +\\infty.\n\t\t\\end{split}\n\t\\end{equation}\nThen we get from the definition of $\\beta_n$ stated in Lemma \\ref{eig-hn} that\n\\begin{equation*}\\begin{split}\n\\beta_n = i\\left(\\frac{t_s^n}{t_p^n}-\\frac{t_p^n}{t_s^n}\\right) \\left(1+\\mathcal{O}\\left(\\frac{1}{n}\\right)\\right),\\quad\n\\sqrt{\\beta_n^2-4} = i\\left(\\frac{t_s^n}{t_p^n}+\\frac{t_p^n}{t_s^n}\\right) \\left(1+\\mathcal{O}\\left(\\frac{1}{n}\\right)\\right),\n\\end{split}\n\\end{equation*}\nwhence it follows as $n\\to\\infty$ that , \n\\[ \n\\sigma_1^{(n)}=i\\frac{t_p^n}{t_s^n}\\left(1+\\mathcal{O}\\left(\\frac{1}{n}\\right)\\right),\\quad \\sigma_2^{(n)}=-i\\frac{t_s^n}{t_p^n}\\left(1+\\mathcal{O}\\left(\\frac{1}{n}\\right)\\right).\n\\]\n\\end{remark}\n\nNow we can get the asymptotic behavior of the eigenvalues of $F_{B_R}$ as $n\\rightarrow\\infty$.\nUsing \\eqref{eigsys-fbr}, Remark \\ref{asy-beta} and the following recurrence relations\n\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\ttJ_n^{\\prime}(t) = nJ_n(t) - tJ_{n+1}(t), \\quad\n\t\t\ttH_n^{(1)\\prime}(t) = tH_{n-1}^{(1)}(t) - nH_n^{(1)}(t),\n\t\t\\end{split}\n\t\\end{equation*} \nwe find\n\t\\begin{equation}\\label{FBR-VAL}\n\t\t\\begin{split}\n\t\t\\lambda^{(n)}_{B_R,1} &= \\sqrt{\\frac{8\\pi}{\\omega}}\\frac{inJ_n(t_p) - it_pJ_{n+1}(t_p) - n J_n(t_s)\\sigma_1^{(n)}}{t_p H_n^{(1)\\prime}(t_p) + in H_n^{(1)}(t_s)\\sigma_1^{(n)}} \\\\\n\t\t&= -\\sqrt{\\frac{2\\pi}{\\omega}} \\frac{\\pi  t_p^{2n+2} t_s^{2n}}{2^{2n} (n+1)! n! (t_p^{2n} + t_s^{2n})} \\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\\\\ \n\t\t\\lambda^{(n)}_{B_R,2} &= \\sqrt{\\frac{8\\pi}{\\omega}}\\frac{it_p J_n^{\\prime}(t_p) - n J_n(t_s)\\sigma_2^{(n)}}{t_p H_{n-1}^{(1)}(t_p) - nH_n^{(1)}(t_p) + in H_n^{(1)}(t_s)\\sigma_2^{(n)}} \\\\\n\t\t&= -\\sqrt{\\frac{2\\pi}{\\omega}} \\frac{\\pi   (t_p^{2n} + t_s^{2n})}{2^{2n-2} (n-1)! (n-2)! t_p^2} \\left(1+\\mathcal{O}\\left(\\frac{1}{n}\\right)\\right).\n\t\t\\end{split}\n\t\\end{equation}\nFurther, from (\\ref{5.34}) we get the asymptotics of the eigenvectors of $\\Sigma_n$ as follows\t\n\\begin{equation}\\label{FBR-VEC}\n\t\t\\begin{split}\n\t\t\tX_{n,1} &= (X_{n,1}^{(1)},X_{n,1}^{(2)})^T = \\left(\\frac{t_s^{n}}{\\sqrt{t_p^{2n} + t_s^{2n}}}, \\frac{i t_p^{n}}{\\sqrt{t_p^{2n} + t_s^{2n}}}\\right)^T  \\; \\left(1+\\mathcal{O}\\left(\\frac{1}{n}\\right)\\right) ,\\\\\n\t\t\tX_{n,2} &= (X_{n,2}^{(1)},X_{n,2}^{(2)})^T = \\left(\\frac{ t_p^{n}}{\\sqrt{t_p^{2n} + t_s^{2n}}}, \\frac{-i t_s^{n}}{\\sqrt{t_p^{2n} + t_s^{2n}}}\\right)^T\\;  \\left(1+\\mathcal{O}\\left(\\frac{1}{n}\\right)\\right)  .\n\t\t\\end{split}\n\t\\end{equation}\n\n\n\n\n\n\n\t\\subsection{Proof of Corollaries \\ref{fm_convex_domain} and  \\ref{fm_convex_domainpp}  for testing disks}\\label{sec5.3}\nIn this subsection, we will use the eigensystem $(\\lambda_{B_R, j}^{(n)}, X_{B_R, j}^{(n)})$  for $n\\in {\\mathbb N}, j=1,2$ (see \\eqref{eigsys-fbr}) of the far-field operator $F_{B_R}$ to verify Corollaries \n\\ref{fm_convex_domain} and  \\ref{fm_convex_domainpp} with $\\Omega = B_R$.   Corollary \\ref{fm_convex_domain} can be rephrased as\n\t\\begin{corollary}\\label{fmdiskff}\n\t\tLet $v^\\infty\\in \\left(L^2({\\mathbb S})\\right)^2$ and assume that\n\t\t$\\omega^2$ is not a Dirichlet eigenvalue of $-\\Delta^{\\ast}$ over $B_R$. Then \n\t   \\begin{equation}\n\t\t   I(B_R) = \\sum_{n \\in \\mathbb{Z}}\\sum_{j=1}^2 \\frac{\\left|\\left\\langle v^{\\infty}, X^{(n)}_{B_R, j}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{B_R, j}\\right|} < + \\infty\n\t   \\end{equation}\n\t\t   if and only if $v^{\\infty}$ is the far-field pattern of some Kupradze radiating solution $v^{s}$, where $v^{s}$ satisfies the Navier equation\n\t   \\begin{equation}\\label{Navier-disk}\n\t\t   \\Delta^{\\ast} v^{s} + \\omega^{2} v^{s} = 0 \\qquad \\text{in} \\quad  \\mathbb{R}^2 \\backslash \\overline{B_R},\n\t   \\end{equation}\n\t\twith the boundary data $v^{s} |_{\\partial B_R}\\in \\left(H^{1\/2}(\\partial B_R)\\right)^2.$\n\t\\end{corollary}\n\t\n\t\\begin{proof} Let $v^s$ be a Kupradze radiating solution to (\\ref{Navier-disk}). \nBy the Hodge decomposition \\eqref{Hodgedecom}, we may decompose $v^s$ into its compressional and shear parts by $v^s = {\\rm grad\\,} \\varphi + {\\rm curl\\,} \\psi$, where $\\varphi = \\sum_{n \\in \\mathbb{Z}}  a_n H^{(1)}_n(k_p r)\\text{e}^{in\\theta_x}$ and $\\psi = \\sum_{n \\in \\mathbb{Z}} b_n H^{(1)}_n(k_s r)\\text{e}^{in\\theta_x}$. Straightforward calculations lead to \n\t\\begin{equation}\n\t\t\\begin{split}\\label{vs}\n\t\t\tv^s &= \\sum_{n \\in \\mathbb{Z}} \\frac{\\text{e}^{in\\theta_x}}{r} \\left[\n\t\t\t\t\\left(\n\t\t\t\t\tk_p r H^{(1)\\prime}_n(k_p r) a_n - in H^{(1)}_n(k_s r)b_n\n\t\t\t\t\\right) \\hat{x} + \n\t\t\t\t\\left(\n\t\t\t\t\tin H^{(1)}_n(k_p r)a_n + k_s r H^{(1)\\prime}_n(k_s r)b_n\n\t\t\t\t\\right) \\hat{x}^{\\bot}\\right]\\\\\n\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{\\text{e}^{in\\theta_x}}{r} \n\t\t\t\\begin{pmatrix}\n\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\tk_p r H^{(1)\\prime}_n(k_p r) & - in H^{(1)}_n(k_s r) \\\\\n\t\t\t\tin H^{(1)}_n(k_p r) & k_s r H^{(1)\\prime}_n(k_s r)\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\ta_n \\\\ b_n\n\t\t\t\\end{pmatrix}.\n\t\t\\end{split}\n\t\\end{equation}\n\tBy the asymptotic behaviour of Hankel functions (see \\cite{kress.1998})\n\t\\begin{equation\n\t\t\\begin{split}\n\t\t\tH_{n}^{(1)}(z) &= \\frac{2^{n}(n-1)!}{\\pi iz^{n}}\n\t\t\n\t\t\t\\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\\quad n \\rightarrow +\\infty,\\\\\n\t\t\tH_{n}^{(1)\\prime}(z) &= -\\frac{2^{n} n!}{\\pi iz^{n+1}} \n\t\t\n\t\t\t\\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\\quad n \\rightarrow +\\infty,\n\t\t\\end{split}\n\t\\end{equation}\n\twe have \n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\tv^s(r,\\theta_x) &= \\sum_{n \\in \\mathbb{Z}} \\frac{\\text{e}^{in\\theta_x}}{r} \n\t\t\t\\begin{pmatrix}\n\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\tk_p r \\frac{-2^n n!}{\\pi i k_p^{n+1} r^{n+1} } & - in \\frac{2^n (n-1)!}{\\pi i k_s^nr^n} \\\\\n\t\t\t\tin \\frac{2^n (n-1)!}{\\pi i k_p^nr^n} & k_s r \\frac{-2^n n!}{\\pi i k_s^{n+1} r^{n+1} }\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\ta_n \\\\ b_n\n\t\t\t\\end{pmatrix} \\left(1+\\mathcal{O}(\\frac{1}{n})\\right)\\\\\n\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{\\text{e}^{in\\theta_x}}{r} \\frac{2^n n!}{\\pi r^{n} }\n\t\t\t\\begin{pmatrix}\n\t\t\t\t\\hat{x} & \\hat{x}^{\\bot}\n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\ti k_p^{-n}  & -  k_s^{-n} \\\\\n\t\t\t\tk_p^{-n} & i k_s^{-n} \n\t\t\t\\end{pmatrix}\n\t\t\t\\begin{pmatrix}\n\t\t\t\ta_n \\\\ b_n\n\t\t\t\\end{pmatrix} \\left(1+\\mathcal{O}(\\frac{1}{n})\\right).\n\t\t\\end{split}\n\t\\end{equation}\nThis gives the leading term of the $H^{1\/2}$-norm on $\\partial B_R$ as\n\\begin{equation}\\label{vs_Hhalfnorm}\n\t\t\\begin{split}\n\t\t\t||v^s||^2_{\\left(H^{1\/2}(\\partial B_R)\\right)^2} &= \\sum_{n \\in \\mathbb{Z}} (1+n^2)^{1\/2} \\frac{1}{R^2} \\frac{2^{2n} n! n! }{\\pi^2  R^{2n}} 2 \\left| i k_p^{-n} a_n - k_s^{-n} b_n\\right|^2  \\left(1+\\mathcal{O}(\\frac{1}{n})\\right)\\\\\n\t\t\t&\\sim  \\sum_{n \\in \\mathbb{Z}}  \\frac{1}{R^2} \\frac{2^{2n+1} (n+1)! n! }{\\pi^2 k_p^n k_s^n R^{2n}}  \\left| i \\left(\\frac{k_s}{k_p}\\right)^{n\/2} a_n - \\left(\\frac{k_p}{k_s}\\right)^{n\/2} b_n\\right|^2  \\left(1+\\mathcal{O}(\\frac{1}{n})\\right)\\\\\n\t\t\t&=  \\sum_{n \\in \\mathbb{Z}} \\frac{2^{2n+1} (n+1)! n! }{\\pi^2 R^2 t_p^n t_s^n }  \\left| i \\left(\\frac{t_s}{t_p}\\right)^{n\/2} a_n - \\left(\\frac{t_p}{t_s}\\right)^{n\/2} b_n\\right|^2  \\left(1+\\mathcal{O}(\\frac{1}{n})\\right)\\\\\n\t\t\t&=  \\sum_{n \\in \\mathbb{Z}} C_n^{(1)} \\frac{2^{2n+1} (n+1)! n! }{\\pi^2 R^2 t_p^n t_s^n } \\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\n\t\t\\end{split}\n\t\\end{equation}\n\twhere $$C_n^{(1)} :=  \\left| \\left(\\frac{t_s}{t_p}\\right)^{n\/2} a_n + \\left(\\frac{t_p}{t_s}\\right)^{n\/2}i b_n\\right|^2.$$\nOn the other hand, the far-field pattern of $v^s$ can be calculated as \n\t\\begin{equation}\n\t\tv^{\\infty}(\\hat{x}) = \\sqrt{\\frac{2k_p}{\\pi}}\\text{e}^{i\\frac{\\pi}{4}} \\sum_{n \\in \\mathbb{Z}} i^{-n} a_n \\text{e}^{in\\theta_x} \\hat{x} + \\sqrt{\\frac{2k_s}{\\pi}}\\text{e}^{i\\frac{\\pi}{4}} \\sum_{n \\in \\mathbb{Z}} i^{-n} b_n \\text{e}^{in\\theta_x} \\hat{x}^{\\bot}.\n\t\\end{equation}\n\tWe proceed with the proof by computing the leading term of the  the indicator $I(B_R)$. \nUsing \\eqref{eigsys-fbr} and \\eqref{FBR-VEC},  the inner product over $L^2({\\mathbb S})^2$ can be calculated as\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\t&\\quad \\left|\\left\\langle v^{\\infty},X_{B_R,j}^{(n)}\\right\\rangle_{\\SS} \\right|^2 \\\\\n\t\t\t&= \\left| \\int_{\\SS} \\sqrt{\\frac{2}{\\pi}} k_p \\text{e}^{i\\frac{\\pi}{4}} \\sum_{m \\in \\mathbb{Z}} i^{-m} a_m \\text{e}^{im\\theta_x} X_{n,j}^{(1)}\\text{e}^{-in\\theta_x} + \\sqrt{\\frac{2}{\\pi}}k_s \\text{e}^{i\\frac{\\pi}{4}} \\sum_{m \\in \\mathbb{Z}} i^{-m} b_m \\text{e}^{im\\theta_x} X_{n,j}^{(2)}\\text{e}^{-in\\theta_x} d\\theta_x \\right|^2 \\\\\n\t\t\t&= \\left|  \\sqrt{\\frac{2}{\\pi}}  \\text{e}^{i\\frac{\\pi}{4}} i^{-n} \\int_{\\SS} k_p a_n X_{n,j}^{(1)} + k_s b_n X_{n,j}^{(2)} d\\theta_x \\right|^2\\\\\n\t\t\t&= 8\\pi \\left| k_p a_n X_{n,j}^{(1)} + k_s b_n X_{n,j}^{(2)} \\right|^2,\n\t\t\n\t\t\\end{split}\n\t\\end{equation}\n\tfor $j=1,2$.\nUsing asymptotic behavior shown in (\\ref{FBR-VAL}) and (\\ref{FBR-VEC}), it is easy to check that, as $n \\to \\infty$,\n\\[\n\\left|\\lambda^{(n)}_{B_R,1}\\right| \\sim \\left|\\lambda^{(n)}_{B_R,2} \\right|\\;n^{-4}, \\quad \n\\left|\\left\\langle v^{\\infty},X_{B_R,1}^{(n)}\\right\\rangle_{\\SS} \\right|^2 \\sim \\left|\\left\\langle v^{\\infty},X_{B_R,2}^{(n)}\\right\\rangle_{\\SS} \\right|^2.\n\\] \n\tThus, \n\t\\begin{equation}\n\t\t\\begin{split}\\label{IBR}\n\t\tI(B_R) &= \\sum_{n \\in \\mathbb{Z}} \\left(\\frac{\\left|\\left\\langle v^{\\infty}, X^{(n)}_{B_R,1}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{B_R,1}\\right|} \n\t\t+ \\frac{\\left|\\left\\langle v^{\\infty}, X^{(n)}_{B_R,2}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{B_R,2}\\right|}\\right)\n\t\t\\\\ \n\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{8\\pi \\left| k_p a_n X_{n,1}^{(1)} + k_s b_n X_{n,1}^{(2)} \\right|^2}{\\left|\\lambda^{(n)}_{B_R,1}\\right|}\\left(1+\\mathcal{O}(\\frac{1}{n})\\right) \t\\\\\n\t\t&= \\sum_{n \\in \\mathbb{Z}}  C_{n}^{(2)} \\sqrt{\\frac{\\omega}{2\\pi}} \\frac{2^{2n+3} (n+1)! n!}{t_p^{n}  t_s^{n} R^2 k_p^2 }  \\left(1+\\mathcal{O}(\\frac{1}{n})\\right),\t\\\\\n\t\t\\end{split}\n\t\\end{equation}\n\twhere $$C_{n}^{(2)} :=  \\left| k_p \\left(\\frac{t_s}{t_p}\\right)^{n\/2} a_n + k_s \\left(\\frac{t_p}{t_s}\\right)^{n\/2}i b_n\\right|^2.$$\nNoting that $\\min \\{ k_p^2,k_s^2 \\}C_{n}^{(1)}\\leqslant C_{n}^{(2)} \\leqslant \\max \\{k_p^2,k_s^2 \\}C_{n}^{(1)}$,\n\n\n\n\n\n\n\nwe conclude that the series \\eqref{vs_Hhalfnorm} and \\eqref{IBR} have the same convergence.\n\nSince the boundedness of $||v^s||_{\\left(H^{1\/2}(\\partial B_R)\\right)^2}$ implies that $v^s$ of the form \\eqref{vs} is indeed a radiating solution in $\\mathbb{R}^2 \\backslash \\overline{B_R}$ with the far-field pattern $v^\\infty$. This proves that $I(B_R)<\\infty$ if and only if $v^s$ is a Kupradze radiating solution in $\\mathbb{R}^2 \\backslash \\overline{B_R}$ with the far-field pattern $v^\\infty$ and with the $H^{1\/2}$-boundary data on $\\partial B_R$.  The proof of Corollaries \\ref{fmdiskff} is thus complete.\n\t\\end{proof}\n\nTo prove Corollary \\ref{fm_convex_domainpp} for testing disks, we need to consider spectral systems of the far-field operators\n $F_{B_R}^{(p)}$ and $F_{B_R}^{(s)}$ .  By Definition \\ref{def-ps}, it follows that\n \\begin{equation*}\n \\begin{split}\n( F_{B_R}^{(p)} g_p) (\\hat{x}):=\\text{e}^{-\\frac{i\\pi}{4}}\\sqrt{\\frac{k_p}{\\omega}} \\int_{{\\mathbb S}} \n u^{\\infty}_{B_R,pp}(\\hat{x};d) \\, g_p(d)\\, ds(d),\\quad g_p\\in L_p^2({\\mathbb S}),\\\\\n ( F_{B_R}^{(s)} g_s) (\\hat{x}):=\\text{e}^{-\\frac{i\\pi}{4}}\\sqrt{\\frac{k_s}{\\omega}} \\int_{{\\mathbb S}} \n u^{\\infty}_{B_R,ss}(\\hat{x}; d) \\, g_s(d)\\, ds(d),\\quad g_s\\in L_s^2({\\mathbb S}). \\end{split}\n  \\end{equation*}\nUsing (\\ref{u-inftypp}) and \\eqref{matrix-uinf}, we see \n\t\\begin{equation*}\n\t\tu^{\\infty}_{B_R,pp}(\\hat{x}) = \\sqrt{\\frac{2}{\\pi k_p}} \\text{e}^{i\\frac{\\pi}{4}} i \\sum_{n \\in \\mathbb{Z}} \\Sigma_n(1,1) \\text{e}^{in\\theta},\\quad\n\t\tu^{\\infty}_{B_R,ss}(\\hat{x}) = \\sqrt{\\frac{2}{\\pi k_s}} \\text{e}^{i\\frac{\\pi}{4}} i \\sum_{n \\in \\mathbb{Z}} \\Sigma_n(2,2) \\text{e}^{in\\theta},\n\t\\end{equation*}\n\twhere $\\Sigma_n(i,j)$ dentoes the $(i,j)$-th entry of the matrix $\\Sigma_n$.\nNow we can get the spectral systems of the operators $F_{B_R}^{(p)}$ and $F_{B_R}^{(s)}$:\n\t\\begin{equation}\\label{fbr-eigpp}\n\t\t\\begin{split}\n\t\t\t\\eta_{B_R,p}^{(n)} &= \\sqrt{\\frac{2}{\\pi k_p}} \\text{e}^{i\\frac{\\pi}{4}} i \\Sigma_n(1,1),\\quad\n\t\t\t\\varphi_{B_R,p}^{(n)}(\\hat{x})= \\text{e}^{in\\theta_x},\\\\\n\t\t\t\\eta_{B_R,s}^{(n)} &= \\sqrt{\\frac{2}{\\pi k_s}} \\text{e}^{i\\frac{\\pi}{4}} i \\Sigma_n(2,2),\\quad\n\t\t\t\\varphi_{B_R,s}^{(n)}(\\hat{x}) = \\text{e}^{in\\theta_x}.\n\t\t\\end{split}\n\t\\end{equation}\n\tTaking $\\Omega = B_R$, we can rewrite Corollary \\ref{fm_convex_domainpp} as \n\t\\begin{corollary\n\t\tLet $w^\\infty_{\\alpha\\alpha} \\in L^2_{\\alpha}({\\mathbb S})$ ($\\alpha = p, s$) and assume that\n\t\t$\\omega^2$ is not a Dirichlet eigenvalue of $-\\Delta^{\\ast}$ over $B_R$. Denote by $(\\lambda^{(n)}_{B_R,\\alpha}, \\varphi^{(n)}_{B_R,\\alpha})$ a spectral system of the positive operator $F^{(\\alpha)}_{B_R,\\#} $. Then \n\t   \\begin{equation}\n\t\t   I^{(\\alpha)}(B_R) = \\sum_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle w^\\infty_{\\alpha\\alpha}, \\varphi^{(n)}_{B_R, \\alpha}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{B_R, \\alpha}\\right|} < + \\infty\n\t   \\end{equation}\n\t\t   if and only if $w^\\infty_{\\alpha\\alpha}$ is the far-field pattern of some Sommerfeld radiating solution $w^s_{\\alpha\\alpha}$ which is defined in $\\mathbb{R}^2 \\backslash \\overline{B_R}$ and $w^s_{\\alpha\\alpha} |_{\\partial B_R(z)}\\in H^{-1\/2}(\\partial B_R).$ That is,  $w^s_{\\alpha\\alpha}$ satisfies the following boundary value problem of the Helmholtz equation\n\t   \\begin{equation}\\label{Helmholtz-disk}\n\t\t   \\Delta w^s_{\\alpha\\alpha} + k_{\\alpha}^{2} w^s_{\\alpha\\alpha} = 0 \\  \\text{in} \\  \\mathbb{R}^2 \\backslash \\overline{B_R},\\quad  w^s_{\\alpha\\alpha} |_{\\partial B_R}\\in H^{-1\/2}(\\partial B_R).\n\t   \\end{equation}\n\t\t\n\t\\end{corollary}\n\n\n\t\\begin{proof}\n\tWithout losing generality, we only consider the case of $\\alpha = p$. The case of $\\alpha = s$ can be proceeded in a similar manner.\nSince\n\t\\begin{equation*}\n\t\t\\lambda_{B_R,p}^{(n)} = |\\text{Re}(\\eta_{B_R,p}^{(n)})| + |\\text{Im}(\\eta_{B_R,p}^{(n)})|,\n\t\\end{equation*}\t\nwe deduce from \\eqref{fbr-eigpp} and the definition  $\\Sigma_n:=\\mathcal{H}_n^{-1} \\mathcal{J}_n$ that\t\\begin{equation}\n\t\t\\lambda_{B_R,p}^{(n)} = \\sqrt{\\frac{\\pi}{ k_p}} \\frac{t_p^{2n}}{2^{2n-1} (n-1)! (n-2)! (t_p^2 + t_s^2)}\\left(1+\\mathcal{O}(\\frac{1}{n})\\right), \\ \n\t\t\\varphi_{B_R,p}^{(n)} = \\text{e}^{in\\theta_x}.\n\t\\end{equation}\nBy the Jacobi-Anger expansion (see e.g.,\\cite{kress.1998}), a Sommerfeld radiating solution $w^s_{pp}$ to the Helmholtz equation in $|x|>R$  can be expanded into the series\n    \\begin{equation}\n\t\tw^s_{pp}(x) = \\sum_{n \\in \\mathbb{Z}} D_{n} H_{n}^{(1)}(k_p |x|) \\text{e}^{i n \\theta_x},\\quad |x|>R,\n    \t\\quad x=(|x|,\\theta_x),\n    \\end{equation}\nwith the far-field pattern given by (see \\cite[(3.82)]{kress.1998})\n    \\begin{equation}\n    w^{\\infty}_{pp}(\\hat{x}) = \\sum_{n \\in \\mathbb{Z}} D_{n} C_{n,p} \\text{e}^{in \\theta_x}, \\quad C_{n,p}:=\\sqrt{\\frac{2}{k_p \\pi}} \\text{e}^{-i (\\frac{n\\pi}{2}+\\frac{\\pi}{4})}.\n    \\end{equation}\n\tHence,\n\t\\begin{equation}\\label{IBRP}\n\t\t\\begin{split}\n\t\t\tI^{(p)}(B_R) &= \\sum_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle \\sum_{m \\in \\mathbb{Z}} D_{m} C_{m,p} \\text{e}^{im \\theta_x}, \\varphi^{(n)}_{B_R, p}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{B_R, p}\\right|}\\\\\n\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\frac{\\left| 2\\pi D_{n} C_{n,p}\\right|^{2}}{\\left|\\lambda^{(n)}_{B_R, p}\\right|}\\\\\n\t\t\t&= \\sum_{n \\in \\mathbb{Z}} \\sqrt{\\frac{\\pi}{k_p}} \\frac{2^{2n+2} (n-1)! (n-2)! (t_p^2+t_s^2)}{t_p^{2n}}|D_n|^2 \\left(1+\\mathcal{O}(\\frac{1}{n})\\right).\n\t\t\\end{split}\n\t\\end{equation}\n\tBy the definition of $H^{-1\/2}$-norm on $\\partial B_R$, we get\n\t\\begin{equation}\\label{ws_bound}\n\t\t\\begin{split}\n\t\t   ||w^s_{pp}(x)||^{2}_{H^{-1\/2}(\\partial B_R)} &= \\sum_{n \\in \\mathbb{Z}} (1+n^2)^{-1\/2} |D_n H^{(1)}_n(t_p)|^2\\\\\n\t\t   &=\\sum_{n \\in \\mathbb{Z}} |D_{n}|^{2} \\frac{2^{2n} (n-1)! (n-2)!}{\\pi ^2 t_p^{2n}}\\left(1+\\mathcal{O}(\\frac{1}{n})\\right).\n\t\t\\end{split}\n\t\\end{equation}\n\tObviously, the series \\eqref{IBRP} and \\eqref{ws_bound} have the same convergence. \n\n\tOn the other hand, following the proof of \\cite[Theorem 2.15]{kress.1998}, it is not difficulty to prove that the boundedness of $||w^s_{pp}||_{H^{-1\/2}(\\partial B_R)}$  implies  that $w^s_{pp}$ is a radiating solution in $|x|>R$ with the far-field pattern $w^{\\infty}_{pp}(\\hat{x})$. This proves that $I^{(p)}(B_R)<\\infty$ if and only if $w^s_{pp}$ is a radiating solution to the boundary value problem of the Helmholtz equation \\eqref{Helmholtz-disk}, with the far-field pattern $w^{\\infty}_{pp}(\\hat{x})$. \n\t\\end{proof}\n\n\t\n\t\\section{Imaging schemes with testing disks}\\label{sec6}\n\n\t\nLet $B_R(z)=z+B_R \\coloneqq \\{y\\in {\\mathbb R}^2 : y=z+x, x\\in B_R\\}$ be a rigid disk centered at $z\\in {\\mathbb R}$ with radius $R>0$. By the translation relations (see e.g., (2.13)-(2.16),\\cite{liuxd2019}), we know\n\t\\begin{equation}\n\t\tu_{B_R(z),\\alpha\\beta}^{\\infty}(\\hat{x}) = \\text{e}^{-ik_{\\alpha}z\\cdot \\hat{x}} \\text{e}^{ik_{\\beta}z\\cdot d} u_{B_R,\\alpha\\beta}^{\\infty}(\\hat{x}),\n\t\\end{equation}\n\twhere $\\alpha = p,s$ and $\\beta = p,s$. Define the matrices\n\t\\begin{equation*}\n\t\\begin{split}\n\t\t&U_{B_R(z)}^{\\infty}(\\hat{x}) \\coloneqq\n\t\t\\begin{pmatrix}\n\t\t\tu^{\\infty}_{B_R(z),pp}(\\hat{x}) & u^{\\infty}_{B_R(z),ps}(\\hat{x}) \\\\ \n\t\t\tu^{\\infty}_{B_R(z),sp}(\\hat{x}) & u^{\\infty}_{B_R(z),ss}(\\hat{x})\n\t\t\\end{pmatrix},\\\\\n\t&\tM_{d,z} \\coloneqq \n\t\t\\begin{pmatrix}\n\t\t\t\\text{e}^{-ik_pz\\cdot d} & 0\\\\\n\t\t\t0 & \\text{e}^{-ik_sz\\cdot d}\n\t\t\\end{pmatrix}, \\quad\n\t\tM_{\\hat{x},z} \\coloneqq \n\t\t\\begin{pmatrix}\n\t\t\t\\text{e}^{-ik_pz\\cdot \\hat{x}} & 0\\\\\n\t\t\t0 & \\text{e}^{-ik_sz\\cdot \\hat{x}}\n\t\t\\end{pmatrix}.\n\t\t\\end{split}\n\t\\end{equation*}\n\tThen, it holds that\n\t\\begin{equation}\n\t\tU_{B_R(z)}^{\\infty} = M_{d,z}^{-1}\\; U^\\infty_{B_R} \\;M_{\\hat{x},z}.\n\t\\end{equation}\nUsing the previous relation, we obtain spectral systems for the operators $F_{\\Omega}$, $F_{\\Omega}^{(p)}$ and \t$F_{\\Omega}^{(s)}$ with $\\Omega=B_{R}(z)$ as follows.\n\t\\begin{corollary}\\label{C6.1} The eigenvalues $\\lambda_{B_R(z), j}^{(n)}$ and the associated eigenfunctions $X^{(n)}_{B_R(z), j}$ of the far-field operator $F_{B_R(z)}$ are given by (e.g. (\\ref{eigsys-fbr}))\n\\begin{equation*}\n\\begin{split}\n&\\lambda_{B_R(z), j}^{(n)}=\\lambda_{B_R, j}^{(n)},\\\\\n&X^{(n)}_{B_R(z), j}(\\hat{x})=\\frac{\\left(\\sqrt{k_p}\\text{e}^{-ik_pz\\cdot\\hat{x}}\\hat{x}+\\sqrt{k_s}\\sigma_j^{(n)}\\text{e}^{-ik_sz\\cdot \\hat{x}}\\hat{x}^{\\bot}\\right)}{\\sqrt{1+|\\sigma_j^{(n)}|^2}}\n\\text{e}^{in \\theta_x}\n\\end{split}\t\n\\end{equation*}\t\n\tfor $n\\in {\\mathbb Z}, \\quad j=1,2$.\n\tMoreover, the spectral systems of the operators $F_{B_R(z)}^{(p)}$ and $F_{B_R(z)}^{(s)}$ take the form (e.g. (\\ref{fbr-eigpp}))\n\t\\begin{equation\n\t\t\\begin{split}\n\t\t\t\\eta_{B_R(z),p}^{(n)} &= \\sqrt{\\frac{2}{\\pi k_p}} \\text{e}^{i\\frac{\\pi}{4}} i \\Sigma_n(1,1),\\quad\n\t\t\t\\varphi_{B_R(z),p}^{(n)}(\\hat{x})= \\text{e}^{in\\theta_x}\\text{e}^{-ik_pz\\cdot\\hat{x}},\\\\\n\t\t\t\\eta_{B_R(z),s}^{(n)} &= \\sqrt{\\frac{2}{\\pi k_s}} \\text{e}^{i\\frac{\\pi}{4}} i \\Sigma_n(2,2),\\quad\n\t\t\t\\varphi_{B_R(z),s}^{(n)}(\\hat{x}) = \\text{e}^{in\\theta_x}\\text{e}^{-ik_sz\\cdot\\hat{x}}.\n\t\t\\end{split}\n\t\\end{equation}\\end{corollary}\nFurthermore, taking $\\Omega = B_R(z)$, we can rewrite results of Theorems \\ref{One-FM} and  \\ref{One-FMpp} as\n\t\\begin{theorem}\\label{One-FMdisk}\n\t\tDefine\t\n\t\t\\begin{equation}\\label{svd-disk}\n\t\t\\begin{split}\n\t\t\t&W(B_R(z)) := \\sum_{n \\in \\mathbb{Z}}\\sum_{j=1}^2 \\frac{\\left|\\left\\langle u_D^{\\infty}, X^{(n)}_{B_R(z), j}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{B_R(z), j}\\right|},\\\\\n\t\t\t&W^{(\\alpha)}(B_R(z)) := \\sum_{n \\in \\mathbb{Z}} \\frac{\\left|\\left\\langle u_{D,\\alpha \\alpha}^{\\infty}, \\varphi^{(n)}_{B_R(z),\\alpha}\\right\\rangle_{\\mathbb{S}}\\right|^{2}}{\\left|\\lambda^{(n)}_{B_R(z),\\alpha}\\right|},\\quad \\alpha=p, s\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\twhere \n\t\t\\[\n\t\t\\lambda^{(n)}_{B_R(z),\\alpha}\n\t\t= \\left| \\text{\\rm Re}\\left(\\eta_{B_R(z),\\alpha}^{(n)}\\right)\\right| +\\left |\\text{\\rm Im}\\left(\\eta_{B_R(z),\\alpha}^{(n)}\\right)\\right|\t.\t\t\t\\]\n\t\tThen $W(B_R(z))<\\infty$\tif and only if $D \\subseteq B_R(z)$ and the same conclusion applies to $W^{(\\alpha)}(B_R(z))$. \n\t\\end{theorem}\n\n\n\n\n\t\t\t\n\n\n\n\t\nFinally, we describe our imaging scheme for solving the inverse problems {\\bf IP-F},\n{\\bf IP-P} and {\\bf IP-S} stated at the end of Section \\ref{sec:2}.\nLet $D$ be a convex rigid polygon to be recovered from far-field data. The procedure consists of the following steps:\n\t\\begin{itemize}\n\t\t\\item Suppose that $B_R \\supset D$ for some $R>0$ and collect the measurement data $u_D^{\\infty}(\\hat{x})$,  $u_{D,pp}^{\\infty}(\\hat{x})$ or \n\t\t$u_{D,ss}^{\\infty}(\\hat{x})$ for all $\\hat{x} \\in \\mathbb{S}$. Let $Q\\supset D$ be our search\/computational region for imaging $D$;\n\t\t\\item Choose sampling centers $z_{n} \\in \\Gamma_{R}:= \\{x:\\ |x|=R\\}$ for $n = 1,2,\\cdots,N$ and\n\t\tchoose sampling radii $h_m\\in(0,2R)$ ($m=1,2,\\cdots, M$) to get the spectral systems for the operators $F_{\\Omega}$, $F_{\\Omega}^{(p)}$ and \t$F_{\\Omega}^{(s)}$ with $\\Omega=B_{h_m}(z_n)$, $n=1,2,\\cdots, N$, $m=1,2,\\cdots, M$ (see \\eqref{eigsys-fbr} and Corollary \\ref{C6.1});\n\t\t\\item For each $z_n\\in \\Gamma_R$, define the function over the grid points $x\\in Q$ satsfying $h_{m+1}>|x-z_n|\\geqslant h_m$ for some $m=1,2, \\cdots, M$ by (see \\eqref{svd-disk}):\n\t\t\\begin{equation*}\n\t\t\\mathcal{I}_{n}(x)=\\left\\{\\begin{array}{lll}\n\t\t&[W(B_{h_m}(z_n))]^{-1}\\qquad\\mbox{for the inverse problem {\\bf IP-F}};\\\\\n\t\t&[W^{(p)}(B_{h_m}(z_n))]^{-1}\\quad\\mbox{for the inverse problem {\\bf IP-P}};\\\\\t\t&[W^{(s)}(B_{h_m}(z_n))]^{-1}\\quad\\mbox{for the inverse problem {\\bf IP-S}};\t\t\\end{array}\\right.  \n\t\t\t\t\\end{equation*}\n\t\t\n\t\\item The imaging function for recovering $D$ is defined as $\\mathcal{I}(x)= \\sum_{n=1}^{N_z}\\mathcal{I}_{n}(x)$,  where $x\\in Q$ are the grid points. This can be considered as imaging function over $Q$ if the grids are sufficiently fine.\n\t\\end{itemize}\nWe expect the values of the indicator function $\\mathcal{I}$ for grid points $x\\in D$ should be larger than those for $x\\in Q\\backslash\\overline{D}$, because \n\\begin{equation*}\n\\begin{split}\n&[W(B_{h_m}(z_n))]^{-1}=0\\quad\\mbox{if}\\quad h_m\\leq\\max_{y\\in \\partial D}|z_n-y|;\\\\\n&[W(B_{h_m}(z_n))]^{-1}<\\infty\\quad\\mbox{if}\\quad h_m>\\max_{y\\in \\partial D}|z_n-y|;\\end{split}\n\\end{equation*}\nand the same indicating behavior applies to $[W^{(\\alpha)}(B_{h_m}(z_n))]^{-1}$, $\\alpha=p,s$.\t\n\\begin{remark}\nIn implementing the above scheme, the spectral data appeared in Theorem \\ref{One-FMdisk} are all given explicitly by Corollary \\ref{C6.1}. For each sampling disk $B_{h_m}(z_n)$ with $n=1,2,\\cdots N, m=1,2,\\cdots, M$,  they can be easily calculated and stored off-line before the inversion process. \nThis is just the advantageous of using testing disks instead of other testing scatterers. \n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\t\\section{Acknowledgements}\nThis work was supported by NSFC 12071236 and NSAF U1930402.\n\n\n\n\n\t\n\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:1}\nDespite the tremendous efforts that have been devoted to the quantum aspects of gravity, the shared conclusion has been simply that \\textit{there is no viable description of quantum gravity} \\cite{quantum-gravity}. On the other hand, electroweak and strong interactions of quarks and leptons are simply understood in the context of gauge field theory and are successfully encoded in the standard model (SM) of particle physics that has been completed thanks to the discovery of the Higgs boson \\cite{LHC}. The gap existing between the physics of gravity and that of the SM has led some people to think about gravity as an emergent or an\\enquote{induced} phenomenon that does not necessitate any quantum description, but rather, it emerges from manifestations of quantum matter fields \\cite{sakharov,adler1,*adler2,*zee}. One of the attempts that gained much attentions has been the idea of induced gravity firstly proposed by Sakharov \\cite{sakharov}. It is argued there that gravity as a classical force, could arise from quantum fields of elementary particles when treated in a \\enquote{classical} curved spacetime. \n\nIn induced gravity models, we generally suggest the existence of a geometric background; a \\enquote{Lorentzian manifold} on which one proceeds to construct a quantum field theory. In this setup, the geometry is considered classical, whereas matter fields are quantized. This construction automatically generates Einstein's general relativity (GR) and higher curvature gravities.\n\nThe framework is generally based on the one-loop effective action of a classical theory of a massive scalar $\\phi$ coupled nonminimally to the spacetime scalar curvature $R\\left(g\\right)$ as\n\\begin{eqnarray}\n\\label{metric-induced gravity action}\nS[g,\\phi]=\\int d^{4}x \\sqrt{-g}\n\\left[\\frac{\\xi \\phi^{2}}{2}R(g)-\\frac{1}{2}(\\partial \\phi)^{2}-V(\\phi) \\right]\n\\end{eqnarray}\nwhere $g_{\\mu\\nu}$ refers to the metric tensor of the manifold and $\\xi$ is a constant dimensionless parameter.\nHistorically the model above which is generically called an induced gravity model is a particular type of the so called scalar-tensor theory proposed originally by Jordan \\cite{jordan} which stands on embedding the four-dimensional spacetime in a five-dimensional flat space and allows one to describe a variable gravitational constant \\cite{maeda}. More influential are Brans-Dicke scalar-tensor gravities \\cite{BD} which are also based on the same feature; the nonminimal coupling dynamics (see Ref.\\cite{maeda} for both historical and technical details.) Among various and several examples, some earlier applications of scalar-tensor theories have included gravitational waves \\cite{wagoner}, and later black hole structures \\cite{bronnikov} as well as attempts to explain the accelerating expansion of the universe \\cite{boisseau}.     \nAt the classical level, models such as (\\ref{metric-induced gravity action}) have been used as an attempt to incorporate the spontaneous symmetry breaking in a curved background that leads to the scale of gravity \\cite{adler1}. Furthermore, they have been considered as possible gravitational models for the early universe, particularly to drive cosmological inflation \\cite{*early1,*early2,*early3,*early4}. \n\nThe essential ingredient in Sakharov's induced gravity is the following one-loop effective action obtained from (\\ref{metric-induced gravity action}) by integrating out the scalar degrees of freedom (the general case would include fermions and gauge bosons) \\cite{sakharov,adler1}\n\\begin{eqnarray}\n\\label{metric one-loop}\n\\Delta S = \\frac{i}{2}\\text{Tr ln}\\left[\\Box_{g}+V^{\\prime\\prime} \n+\\xi R(g)\n\\right].\n\\end{eqnarray}\n\nAlthough it leads correctly to Einstein's general relativity and generalized theories of gravity, this ancient induced gravity may not reflect a correct emergence of gravity based on its \\enquote{metrical} structure. In fact, classical gravity described by Einstein's general relativity is a theory of the spacetime metric. The latter plays the role of the gravitational field which in turn appears as curvature effects on rods and clocks. These effects are encoded in the metric tensor, and then it is this \\enquote{metrical elasticity} which is the origin of gravity at large scales. At that end, a correct theory of induced gravity must be able to generate this metrical elasticity of space. In the standard induced gravity, this metric structure is already postulated as a Lorentzian manifold, thus generation of Einstein-Hilbert action may not mean generation of \\enquote{metrical elasticity}.\n\nThe question now, is it possible to generate theories of gravity with their metrical structure? If yes, do they arise \\enquote{classically} or at the quantum level like standard induced gravity where matter is quantized?\n\nThe aim of this paper is to show that such a theory might be possible in the framework of \\enquote{affine} spacetime, where the metric tensor is completely absent in the beginning \\cite{azri-induced}. Our setup will be based on an affine action in which an affine connection, introduced as fundamental quantity, is coupled to scalar fields through curvature. We will propose to proceed in two possible ways:\n\\begin{enumerate}[A.]\n\\item The first step that we follow in Section \\ref{sec:2} is to expand our pure affine action around a background up to quadratic terms of the field. Stationary of the action against variation with respect to the affine connection leads to dynamical equation from which emerges the metric tensor with a constant of proportionality that will be related to the gravitational constant. As a result, the action of the background field is reduced to Einstein-Hilbert action where any constant piece that appears \\textit{a priori} in the potential, which correspond to a possible cosmological constant, is completely absorbed in the definition of this gravitational sector. On the other hand, the terms quadratic in the field will bring higher order curvature terms that correspond to extended gravity models. This terminates the classical part of our approach to induced gravity. \n\\item\nThe second part will be devoted to constructing the one-loop effective action of the theory. This maybe considered as the first essay towards effective actions in affine space. It is the emergence of the metric structure in (A) that allows us to calculate this quantity. In fact, the volume elements in affine space which are necessary in performing the integrals are reduced to metric volume elements thanks to step (A). We then adopt an ultraviolet (UV) cutoff to regularize the integrals which in turn lead to an induced gravitational and cosmological constants. As in Sakharov's induced gravity, the experimental value of Newton's constant requires a UV boundary of the order of the Planck mass. This, unfortunately, pushes the induced cosmological constant to a very large value due to the presence of the quartic and quadratic UV divergent terms.  \n\\end{enumerate}\nWith the last step, we have a complete approach to induced gravity where not only the gravitational parameters (Newton and cosmological constants) but interestingly, the metric tensor also gains an emergent character.\nA summary of these results will be given in Section \\ref{sec:3}.\n\n\\section{Generating metric gravity}\n\\label{sec:2}\nAs we have mentioned above, our aim is to construct a model of induced gravity with a crucial feature; the induction of the spacetime metric. This suggests a primary \\enquote{metric-less} classical action from which one proceed to extract the one-loop effective action. Thus, we begin with a very simple spacetime which does not recognize a metrical structure where angles and distances measurements take place. However, different events in different points are studied and compared only through parallel displacements of vectors and tensors \\cite{azri-induced,azri-affine,demir-eddington,*azri-immersed,*azri-separate,*poplawski,*cota,*oscar,*oscar2}. \nTo that end, this spacetime arena is endowed with only an affine connection $\\Gamma$ and its associated curvature. This simple structure offers a viable calculus of variation by coupling scalar fields to the affine connections \\cite{kijowski1,*kijowski2}. Indeed, a viable theory of gravity must be described by a covariant field equations that arise from an action principle.\nHere we want to construct an affine action that enables us to induce the scale of gravity in the philosophy of GR counterpart (\\ref{metric-induced gravity action}), i.e, when $ \\xi \\phi^{2}$ $\\rightarrow $ $M_{Pl}^{2} $ (for constant fields.) With this, the general coordinate transformations of affine spacetime suggest then a combination of the Ricci tensor of the affine connection $R_{\\mu\\nu}(\\Gamma)$ and scalar field kinetic structure $\\nabla_{\\mu}\\phi\\nabla_{\\nu}\\phi$ as well as a scalar potential $V(\\phi)$.\nDimensional analysis implies that this combination could simply come out in the following form \n\\begin{eqnarray}\n\\label{action0}\nS \\left[\\phi \\right]= 2\\int d^{4}x \\frac{\\sqrt{\\left| \\right| \\xi \\phi^2 R_{\\mu\\nu}\\left(\\Gamma\\right)- \\nabla_{\\mu}\\phi\\nabla_{\\nu}\\phi \\left| \\right|}}{V(\\phi)},\n\\end{eqnarray}\nwhich indeed leads to GR with an induced gravity scale for constant fields \\cite{azri-induced} (see also Ref.\\cite{azri-review} on how to construct pure affine actions.)\n\nWe notice here that we take only the symmetric part of the Ricci Tensor of the symmetric affine connection.\n\nThis action has been introduced for the first time by the author and his collaborator as an attempt to a new approach to induced gravity in the philosophy of spontaneous symmetry breaking \\cite{azri-induced}. It is worth enlightening briefly the main results of the mentioned work. First of all, this action runs along two important and explicit properties:  \n\\begin{enumerate}[(a)]\n\\item\nBoth geometry and scalar field terms define the invariant volume measure, i.e, the square root of the determinant. The scalar field enters this measure by its derivative (kinetic part) in a tensorial form. The property of \\enquote{contraction} is absent at this stage since there is no notion of metric tensor.\n\\item \nThe potential energy enters the action separately in division, and then the action is well defined only for nonzero potential energy, $V\\left(\\phi \\right)\\neq 0$.\n\\end{enumerate}\n\nThe second property reflects the viability of the affine models in studying the early universe where the scalar field $\\phi$ requires a nonzero potential enegry to get all the phase of inflation done \\cite{azri-induced,azri-affine,guth,*linde1,*albrecht,*linde2,*higgs-inflation,*bauer1,*bauer2}. The crucial importance of this property will become clear later when deriving the gravitational actions based on the model given in (\\ref{action0}). It has been shown that (affine) gravity is induced via spontaneous symmetry breaking where the gravity scale arises from the constant vacuum expectation value of a heavy scalar, and the metric tensor is generated thanks to the nonzero vacuum energy left after symmetry breaking \\citep{azri-induced}.    \n \nReturning to the present work, the first step towards the emergence of metric gravity is to expand the field $\\phi$ around a \\enquote{constant} background $\\phi_{c}$ as\n\\begin{eqnarray}\n\\label{field expansion}\n\\phi=\\phi_{c}+\\varphi.\n\\end{eqnarray}\nThis has been taken only for the sake of simplicity and it could be trivially generalized to the case where the background is not constant.\n \nAn important remark here, is that the field must not have a zero potential at this background, $V\\left(\\phi_{c}\\right) \\neq 0$. This is an essential postulate for the upcoming results.\n\nTo that end, and up to second order in $\\varphi$, the action (\\ref{action0}) takes the following form\n\\begin{eqnarray}\n\\label{expanded action}\nS\\left[\\varphi+\\phi_{c}\\right]=\nS\\left[\\phi_{c}\\right]+I_{1}[\\varphi]+I_{2}[\\varphi\\,\\partial^{2}\\varphi]+I_{3}[\\varphi^{2}]+\\dots \\nonumber \\\\\n\\end{eqnarray}\nwhere the first term in the right hand side is nothing but action (\\ref{action0}) evaluated at the background $\\phi_{c}$, and the last three terms are respectively given as follows \n\\begin{widetext}\n\\begin{eqnarray}\n\\label{i1}\n&&I_{1}[\\varphi]=\n2\\int d^{4}x \n\\frac{\\sqrt{||K(\\phi_{c})||}}{V^{2}(\\phi_{c})}\\Big[\\xi \\phi_{c} V(\\phi_{c})\n(K^{-1})^{\\alpha\\beta}R_{\\alpha\\beta}-V^{\\prime}(\\phi_{c})\\Big]\\varphi\n\\\\&&\nI_{2}[\\varphi\\,\\partial^{2}\\varphi]=\\int d^{4}x\\,\\varphi\\, \\partial_{\\alpha}\n\\left( \\frac{\\sqrt{||K(\\phi_{c})||}}{V(\\phi_{c})} (K^{-1})^{\\alpha\\beta}\\partial_{\\beta}\\varphi\n\\right),\n\\label{i2}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{i3}\nI_{3}[\\varphi^{2}]&&=\n\\int d^{4}x \n\\frac{\\sqrt{||K(\\phi_{c})||}}{V^{2}(\\phi_{c})}\n\\left[ \\left(\\xi V(\\phi_{c})-2\\xi \\phi_{c}V^{\\prime}(\\phi_{c}) \\right)\n(K^{-1})^{\\alpha\\beta}R_{\\alpha\\beta}-2\\xi\\phi_{c}^{2}V(\\phi_{c})\n(K^{-1})^{\\alpha\\beta}\n(K^{-1})^{\\nu\\lambda}R_{\\beta\\nu}R_{\\lambda\\alpha}\n \\right] \\varphi^{2} \\nonumber \\\\&&\n+\\int d^{4}x \n\\frac{\\sqrt{||K(\\phi_{c})||}}{V^{2}(\\phi_{c})} \n\\left[\\frac{2V^{\\prime 2}(\\phi_{c})}{V(\\phi_{c})}-V^{\\prime \\prime}(\\phi_{c})\n+\\xi^{2}\\phi_{c}^{2}V(\\phi_{c})\\left((K^{-1})^{\\alpha\\beta}R_{\\alpha\\beta}\\right)^{2} \\right]\n\\varphi^{2}\n\\end{eqnarray}\n\\end{widetext}\nwhere for simplicity we have introduced the tensor $K_{\\mu\\nu}$ which is given as follows\n\\begin{eqnarray}\n\\label{tensor k}\nK_{\\mu\\nu}\\left(\\phi\\right)=\\xi \\phi^2 R_{\\mu\\nu}\\left(\\Gamma\\right)- \\nabla_{\\mu}\\phi\\nabla_{\\nu}\\phi.\n\\end{eqnarray}\nFinally we have an expansion of action (\\ref{action0}) up to quadratic terms of the scalar field where all the geometric parts (the Ricci tensor and its inverse) that appear in this expansion are given only in terms of the affine connection.\n\nOur next step is to extract the new feature behind this expansion which will be certainly the \\textit{induction} of metric gravity (GR and its extensions). In the subsequent section we show how Einstein-Hilbert action arises after the \\textit{generation} of the metric tensor dynamically. This classical setup will be of a great importance since it helps, later on, in performing the one-loop effective action which is considered as the main part in inducing gravity \\`a la Sakharov.  \n \n\\subsection{Classical setup: Inducing metric structure}\nFirst of all, let us simplify the expression of the expansion (\\ref{expanded action}) by using every possible equation of motion. \n\nHere, the term (\\ref{i1}) which is linear in $\\varphi$ vanishes by use of the equation of motion of the background\n\\begin{eqnarray}\n\\frac{\\delta S[\\phi_{c}]}{\\delta \\phi_{c}}=0,\n\\end{eqnarray}\nwhich is given explicitly as\n\\begin{eqnarray}\n\\label{equation of motion of phi}\n\\frac{\\sqrt{||K(\\phi_{c})||}}{V^{2}(\\phi_{c})}\\left[\\xi \\phi_{c} V(\\phi_{c})\n(K^{-1})^{\\alpha\\beta}R_{\\alpha\\beta}-V^{\\prime}(\\phi_{c})\\right]=0\n\\end{eqnarray}\nThis equation of motion can be simplified using the geometrical equations which arises from varying the background action with respect to the affine connection $\\Gamma$. In fact, stationary of $S[\\phi_{c}]$ against this variation \n\\begin{eqnarray}\n\\frac{\\delta S[\\phi_{c}]}{\\delta \\Gamma^{\\lambda}_{\\,\\mu\\nu}}=0,\n\\end{eqnarray}\nwould simply lead to the following constraint\n\\begin{eqnarray}\n\\label{dynamical eq}\n\\nabla_{\\mu}\\left(\\xi \\phi_{c}^{2} \\frac{\\sqrt{\\left| \\right| K\\left(\\phi_{c}\\right) \\left| \\right|}}{V(\\phi_{c})}\n\\left( K^{-1}\\right)^{\\alpha\\beta} \\right)=0.\n\\end{eqnarray}\nThis implies a natural \\enquote{induction} of an invertible (rank-two) tensor $g_{\\mu\\nu}$ such that\n\\begin{eqnarray}\n\\label{metric}\n\\sqrt{\\left| \\right| g \\left| \\right|}\\left(g^{-1} \\right)^{\\alpha\\beta}=\\frac{\\xi \\phi_{c}^{2}}{M^{2}} \\frac{\\sqrt{\\left| \\right| K\\left(\\phi_{c}\\right) \\left| \\right|}}{V(\\phi_{c})}\n\\left( K^{-1}\\right)^{\\alpha\\beta},\n\\end{eqnarray}\nwhere $M$ is a constant of a mass dimension.\n\nThe induced tensor clearly satisfies $\\nabla_{\\mu} g_{\\alpha\\beta}=0$ which appears as a metricity equation, and then it plays the role of the metric tensor. The affine connection $\\Gamma$ in turn, which defines the action (\\ref{action0}) is automatically reduced to Levi-Civita of this metric. \n\nWith this induced metrical structure described by the metric tensor (\\ref{metric}), the equation of motion of the background (\\ref{equation of motion of phi}) takes the form\n\\begin{eqnarray}\n\\xi \\phi_{c} R(g)-V^{\\prime}(\\phi_{c})+\\Psi(\\phi_{c})=0,\n\\end{eqnarray}\nwhere $R(g)$ is now the Ricci scalar of the metric tensor and the last term is given by\n\\begin{eqnarray}\n\\Psi(\\phi_{c})=\\left(1- \\frac{M^{2}}{\\xi\\phi_{c}^{2}}\\right)V^{\\prime}(\\phi_{c}).\n\\end{eqnarray}\nWe mention again that the background field has been taken constant, the case that justifies the absence of the d'Alembert operator in the previous equation \\citep{azri-induced}. At the background, the equality (\\ref{metric}) is nothing but the gravitational field equations with a cosmological constant $V_{0}=V(\\phi_{c})$ or explicitly\n\\begin{eqnarray}\n\\label{einstein equation with cc}\n\\xi\\phi_{c}^{2}R_{\\mu\\nu}(g)=g_{\\mu\\nu}V_{0}\\left(\\frac{M^{2}}{\\xi\\phi_{c}^{2}} \\right).\n\\end{eqnarray}\nConsistency with Einstein's field equations implies that\n\\begin{eqnarray}\nM^{2}=\\xi\\phi_{c}^{2}\\equiv (8\\pi G_{N})^{-1}.\n\\end{eqnarray}\nNow the remaining terms in the expansion (\\ref{i2}), (\\ref{i3}) as well as $S\\left[\\phi_{c}\\right]$ are reduced to quantities that depend on the induced metric and they lead finally to the gravitational actions  \n\\begin{eqnarray}\n\\label{induced action}\nS\\left[\\varphi+\\phi_{c}\\right]&&= \\int d^{4}x \\sqrt{\\left| \\right| g \\left| \\right|}\\,\n\\frac{R\\left(g\\right)}{16\\pi G_{N}}\n\\nonumber \\\\ &&\n+\\int d^{4}x \n\\sqrt{\\left| \\right| g \\left| \\right|}\\, \\, \\varphi\\, \\Big[\\Box_{g} +c_{1}+c_{2} R\\left(g\\right)\\Big]\\varphi \\nonumber \\\\ &&\n+ \\int d^{4}x \n\\sqrt{\\left| \\right| g \\left| \\right|}\\, \\, \\varphi^{2}\\, \\Big[\nc_{3} R^{2}\\left(g\\right)+\nc_{4} R^{\\mu\\nu}R_{\\mu\\nu} \\Big] \\nonumber \\\\\n\\end{eqnarray}\nwhere the constants $c_{i}$ are listed as follows\n\\begin{eqnarray}\n\\label{c1}\n&&c_{1}=\\frac{2V^{\\prime 2}(\\phi_{c})}{V(\\phi_{c})}-V^{\\prime \\prime}(\\phi_{c})\n\\\\\n&&c_{2}=\\xi - \\frac{2\\xi \\phi_{c} V^{\\prime}(\\phi_{c})}{V(\\phi_{c})} \\\\\n&&c_{3}=\\frac{\\xi\\phi^{2}_{c}}{V(\\phi_{c})} \\\\\n&&c_{4}=-\\frac{2\\xi\\phi^{2}_{c}}{V(\\phi_{c})}.\n\\end{eqnarray}\nIt is worth noticing that in order to get the Ricci scalar in the first term of (\\ref{induced action}), we have used equation (\\ref{metric}) in its tensor form where the potential $V(\\phi_{c})$ which appears in $S[\\phi_{c}]$ is then written in terms of the Ricci scalar\\footnote{Here, the explicit relation between $V_{0}$ and $R(g)$ is obtained by tracing equation (\\ref{einstein equation with cc}). The author thanks D. Demir for drawing his attention to this step.}. \n\nThe induced actions (\\ref{induced action}) are written in terms of the metric tensor and then they describe metrical gravitational theories. These metrical theories are originated from the simple and primarily action (\\ref{action0}). We thus enumerate the main results found above as follows:\n\\begin{enumerate}[(i)]\n\\item Einstein-Hilbert action which describes GR is \\textit{generated} from the (constant) background field via the action $S\\left[\\phi_{c}\\right]$. This is nothing but Eddington (affine) action written in an associated metrical form \\cite{eddington,*schrodinger}. A nonzero potential is a very crucial condition here. Here, we notice the absence of any constant term that could correspond to a \\enquote{classical} cosmological constant. The reason is that at the background field, the constant potential $V_{0}=V(\\phi_{c})$ is \\enquote{completely} transformed to the gravitational sector which is formed by the Planck mass and the Ricci scalar. Thus, the affine to GR transition is followed by the absorption of the cosmological constant.     \n\\item\nThis pure affine model (\\ref{action0}) generates also higher order curvature terms which are seen in the last integral of (\\ref{induced action}). Unlike metric induced gravity, these terms appear here in the action without calculating the effective action. In metric induced gravity the higher order curvature terms arise only after adopting an explicit UV cutoff and regularizing the action. The one-loop effective action will be our object of interest in the subsequent Section. \n\\end{enumerate}\n\\subsection{Quantum corrections: Gravity \\`a la Sakharov}\n\nUp to now, like geometry, we have treated the matter fields (scalars here) as pure classical fields. In what follows, we will define the quantum matter fields on the classical background (affine) geometry. The aim of this part of work is to consider the effects of quantum matter fields on this geometry which is endowed with only an affine connection and its curvature.\n\nThe object of interest here is the effective action $\\Delta S[\\phi]$ for (\\ref{action0}) by integrating out the scalar fields. The one-loop contribution to this effective action is generally given by \\cite{peskin,*davies,*buch}\n\\begin{eqnarray}\n\\label{affine one-loop}\ne^{i\\Delta S[\\phi]} \\equiv \n\\int \\mathcal{D}\\varphi \\exp \\Big[ \\frac{i}{2}\\varphi \\left(\\frac{\\delta^{2}S[\\phi]}{\\delta\\phi\\delta\\phi} \\right)\\varphi \\Big].\n\\end{eqnarray}\nAs one may easily show, the integrand in this expression is given by the sum of the two terms (\\ref{i2}) and (\\ref{i3}) quadratic in the field. Thus, the expression (\\ref{affine one-loop}) is a function of the affine connection and its associated curvature and it can be considered as the effective action for the classical \\textit{affine connection} $\\Gamma_{\\,\\mu\\nu}^{\\lambda}$ which appears (in terms of Feynman diagrams) in the external legs, whereas the scalar fields inside the loops. However, it is worth emphasising here that, as we shall see below, this method will be based on a transition from pure-affine to pure-metric action where the metric tensor gains an induced character from the background action. Thus, one has to pay attention to that and not confuse this case with the general classical cases in Ref.\\cite{azri-affine,azri-induced} where the metric tensor does not arise from ''only'' the background but satisfies a complete Einstein equations of motion with scalar fields\\footnote{By this we mean that the present method of induced gravity has nothing to do with any quantum corrections of the pure-affine inflationary models given in Ref.\\cite{azri-affine,azri-induced,azri-review}.}.  \n\nAs we have mentioned in the previous part of this section, the geometric side of the equations of motion, or the equations arising from variation with respect to the affine connection leads to dynamical constraint (\\ref{dynamical eq}) which in turn necessitates the existence of an emergent metric field (\\ref{metric}). This step adds a new feature to induced gravity in general which is the classical transition\n\\begin{center}\n\\textit{Affine spacetime} $\\xrightarrow{\\text{Eq}.(\\ref{dynamical eq})}$ \\textit{Metric spacetime}.\n\\end{center} \nThe important utility of this transition in evaluating the effective action, is the fact that spacetime affine measure is reduced to spacetime metric measure, or  \n\\begin{eqnarray}\n\\frac{\\xi^{2}\\phi_{c}^{4}}{M^{4}}\\frac{\\sqrt{||K(\\phi_{c})||}}{V^{2}(\\phi_{c})}\n\\xrightarrow{\\text{Eq}.(\\ref{metric})} \\sqrt{||g||}.\n\\end{eqnarray}\nTo that end, in our approach, gravity will be induced from the one-loop effective action (\\ref{affine one-loop}) which takes the form\n\\begin{eqnarray}\ne^{i\\Delta S[\\phi]} \\rightarrow\n\\int \\mathcal{D}\\varphi \\exp \\Big\\lbrace  i\n\\int d^{4}x \\sqrt{||g||}\\,\n\\varphi \\Big[\\Box_{g} +\\mathcal{H}(g) \\Big]\\varphi \\Big\\rbrace \\nonumber \\\\\n\\end{eqnarray}\nwhere the operator $\\mathcal{H}(g)$ is given by\n\\begin{eqnarray}\n\\mathcal{H}(g)=c_{1}+c_{2} R\\left(g\\right) \n+c_{3}R^{2}\\left(g\\right)+c_{4}R^{\\mu\\nu}R_{\\mu\\nu}.\n\\end{eqnarray}\nThis simply leads to\\footnote{The final result includes a factor two in front of the operator $\\Box_{g} +\\mathcal{H}(g)$ in order to have a correct Gaussian integral. However, we have ignored this factor since the quantity of interest will be difference\nin the one-loop contribution to the effective actions of two metrics using the familiar formula of $\\ln(a\/b)$ (see References \\cite{visser,extended-sakharov} for details.)}\n\\begin{eqnarray}\n\\label{final one-loop}\n\\Delta S \\rightarrow \\frac{i}{2}\\,\\text{Tr}\\ln \\left[\\Box_{g} +\\mathcal{H}(g)\\right]. \n\\end{eqnarray}\nThis is the final expression of the one-loop effective action that arises from pure affine action (\\ref{action0}). There are crucial differences from the one-loop effective action (\\ref{metric one-loop}) in metric theory which could be seen if one considers a simple massive scalar field (see discussion below).\n\nA common fact however is that those expressions diverge when performing the integration. The detailed analysis behind the evaluation of the integrands (\\ref{metric one-loop}) and (\\ref{final one-loop}) put it beyond the scope of this paper. However, we mention here that we follow the Heat Kernel method where the heat kernel is expressed in terms of the Seeley-DeWitt expansion, and the integrals are regulared by adopting a UV cutoff \\cite{davies,buch}. \n\nBelow, we will not be interested in the higher order (curvature) terms but only in the divergences which are proportional to the Ricci scalar $R$ and that corresponding to a volume element. Detailed calculation shows that those two terms \\textit{induce} both Newton's constant $G_{\\text{ind}}$ and a cosmological constant $V_{0}^{\\text{ind}}$ respectively  \n\\begin{eqnarray}\n\\label{induced-G}\n\\frac{1}{G_{\\text{ind}}}=&&\\left(\\frac{1}{6}-c_{2} \\right)\\frac{\\Lambda_{\\text{UV}}^{2}}{2\\pi} \\nonumber \\\\\n&&-\\left(\\frac{1}{6}-c_{2} \\right)\\frac{c_{1}}{\\pi}\n\\ln\\left(\\frac{\\Lambda_{\\text{UV}}}{\\mu} \\right)\\nonumber \\\\\n&&+\\text{UV-finite}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{induced-cc}\nV_{0}^{\\text{ind}}=&&\n-\\frac{\\Lambda_{\\text{UV}}^{4}}{128\\pi^{2}} \\nonumber \\\\\n&&+c_{1}\\frac{\\Lambda^{2}_{\\text{UV}}}{64\\pi^{2}}-\\frac{c_{1}^{2}}{64\\pi^{2}}\n\\ln\\left(\\frac{\\Lambda_{\\text{UV}}}{\\mu} \\right)\\nonumber \\\\\n&&+\\text{UV-finite},\n\\end{eqnarray}\nwhere the parameter $\\mu$ is an infrared cutoff.\n\nThese expressions show the main idea behind induced gravity which lies in determining the gravitational couplings from the particle spectrum and the UV cutoff. The latter, however, will be fixed by the numerical values provided by experiments and it will certainly be related to the upper scale for the valid effective theory. For the previous setup to make physical sense, the UV boundary must be of the order of the Planck mass $\\Lambda_{\\text{UV}} \\simeq M_{Pl}$. Thus, the induced gravitational constant reads \n\\begin{eqnarray}\n\\frac{1}{G_{\\text{ind}}} \\simeq M_{Pl}^{2}.\n\\end{eqnarray}\nThis has a direct but sever implication on the value of the cosmological constant. While its observational value is estimated to be of the order of the neutrino mass density, i.e, $(10^{-3} \\text{eV})^{4}$ \\cite{planck}, the value of the induced quantity (\\ref{induced-cc}) manifests as \n\\begin{eqnarray}\n|V_{0}^{\\text{ind}}| \\simeq M_{Pl}^{4} \\simeq (10^{19}\\text{GeV})^{4}.\n\\end{eqnarray}\nThe discrepancy between the observational and theoretical estimations of the value of the cosmological constant is not restricted to the present model of induced gravity but it holds in most of gravity models albeit with quantum matter fields. This is simply the origin of the celebrated cosmological constant problem \\cite{ccp1,*ccp2}. \n\\begin{table*}[t]\n\\caption{\\label{table1}%\nThe one-loop induced parameters in standard (metric) induced gravity versus those of the present (affine) approach for quadratic potential. In both approaches, the gravitational and the cosmological constants are dominated by power-law UV terms, however, in those terms the conformal value $\\xi=1\/6$ of metric gravity is no longer preserved in our approach. This could be a consequence of the absence of (metric) conformal transformation in affine gravity \\cite{azri-review}. The crucial differences between the approaches lie in the induced higher order gravity where the couplings are power-law UV sensitive in the present model. This is an expected result since higher order gravity emerges classically in (\\ref{induced action}) before performing the effective action. }\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\n\\textrm{Induced constants}&\n\\textrm{Metric induced gravity}&\n\\textrm{Affine induced gravity}\\\\\n\\colrule\n\\\\\n$V_{0}^{\\text{ind}}$ & $-\\frac{1}{64\\pi^{2}}\\left[\\frac{\\Lambda^{4}}{2}-m^{2}\\Lambda^{2}-m^{4}\\ln(\\Lambda\/\\mu) \\right]$ & $-\\frac{1}{64\\pi^{2}}[\\frac{\\Lambda^{4}}{2}-3m^{2}\\Lambda^{2}+9m^{4}\\ln(\\Lambda\/\\mu) ]$\\\\ \n\\,&\\,&\\, \\\\\n$1\/G_{\\text{ind}}$ & $\\frac{1}{\\pi}(\\frac{1}{6}-\\xi)\\left[\\frac{\\Lambda^{2}}{2}-m^{2}\\ln(\\Lambda\/\\mu)\\right]$ & $\\frac{1}{\\pi}(\\frac{1}{6}+3\\xi)\\left[\\frac{\\Lambda^{2}}{2}-3m^{2}\\ln(\\Lambda\/\\mu)\\right]$\\\\\n\\,&\\,&\\, \\\\\n$c_{_{R}}$ & $\\frac{1}{32\\pi^{2}}\\left(\\frac{1}{36}-\\frac{\\xi}{3}+\\xi^{2}\\right)\\ln(\\Lambda\/\\mu)$ & $\\frac{1}{32\\pi^{2}}\\left[-2\\xi^{2}\\frac{\\Lambda^{2}}{m^{2}}+ \\left(\\frac{1}{36}+13\\xi+9\\xi^{2}\\right)\\ln(\\Lambda\/\\mu)\\right]$  \\\\\n\\,&\\,&\\, \\\\\n$\\bar{c}_{_{R}}$ &$-\\frac{1}{2880\\pi^{2}}\\ln(\\Lambda\/\\mu)$ & $\\frac{1}{32\\pi^{2}}\\left[4\\xi\\frac{\\Lambda^{2}}{m^{2}}- \\left(\\frac{1}{90}+24\\xi\\right)\\ln(\\Lambda\/\\mu)\\right]$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\nLast but not least, we summarize in Table \\ref{table1} the UV coefficients of the induced gravitational and cosmological constants, as well as the higher order curvature terms, for the simple example of a massive scalar field where $V(\\phi)=m^{2}\\phi^{2}\/2$ using the following standard writings\n\\begin{eqnarray}\n\\label{example}\n\\Delta S \\supset\n\\int d^{4}x \\sqrt{||g||}\\,  \\Big[\\frac{R(g)}{16\\pi G_{\\text{ind}}}\n-2V_{0}^{\\text{ind}}\n+c_{_{R}}R^{2} \\nonumber \\\\\n+\\bar{c}_{_{R}}R_{\\mu\\nu}R^{\\mu\\nu}+\\dots\n\\Big].\n\\end{eqnarray}\nHere, since curvature terms like those at the end of the last expression are also generated classically in (\\ref{induced action}), then the related coefficients will be certainly sensitive to UV power-law. This is no longer the case of metric induced gravity.\n \nThe discussion above shows how one could realize an induced gravity model from a pure affine spacetime free of not only the gravitational constant which provides the measure of weights, but also the metric tensor which plays the role of the gravitational field.  The present setup is able to induce both quantities as well as a vacuum energy from the one-loop (affine) effective action. \n\nWe conclude by mentioning that the present approach may need to be extended by illuminating the following points:\n\\begin{enumerate}[i.]\n\\item First of all, the model described by action (\\ref{action0}) is based mainly on scalar degrees of freedom and it could be easily generalized to multi-scalar fields \\cite{azri-review, azri-thesis}. However, the fact remains that fermionic fields are not yet viably accommodated in affine space. This could be an obstacle if the gravitational couplings have to be induced from \\enquote{all} the SM particles spectrum. In this case, extending (\\ref{action0}) to include the SM matter fields in a unified picture would be necessary.\n\\item Second, unlike the case of induced gravity models based on their prior metric structure, in the present approach, we were able to generate higher order gravity terms only classically. Although these terms receive a UV power laws corrections (see Table \\ref{table1}), one has to explore the possible and considerable new features of these terms, such as deriving cosmological inflation.        \n\\end{enumerate}   \n\n\\section{Summary}\n\\label{sec:3}\n\nThe physics of gravity at the microscopic scale remains one of the puzzles on which debates have never been settled down. Since the standard model of particle physics does not accommodate this interaction, people turned to thinking about its origin as a non-fundamental force. Gravity is induced or emerged from the micro-physical phenomena is the common framework for most of the proposed models \\cite{sakharov,extended-sakharov1}.\n\nInduced gravity, particularly in Sakharov's approach, aims at describing classical gravity by the quantum fields of matter. The latter need to be coupled to spacetime curvature with no signature of Newton's constant that translates the presence of gravity. At this stage, the \\enquote{classical} spacetime geometry does not show to have any specific dynamics. Nevertheless, its response to quantum matter fields ends up with an induction of the gravitational constant via the one-loop effective action. In Sakharov's approach, the spacetime gains a Riemannian geometry from the scratch, while in other (almost) similar models, it has been taken arbitrary \\cite{extended-sakharov}. However, in all these models, the main postulate is that spacetime is \\enquote{endowed} with a metric tensor.    \n\nIn this paper we have argued that gravity, if induced, must arise with the metrical structure of spacetime. The latter property may not be imposed from the scratch. In fact, it has been shown that matter fields may not require the metric tensor to couple to curvature of spacetime \\cite{azri-induced,azri-affine,kijowski1,kijowski2}. It turned out that the metric tensor is generated dynamically from an affine variational principles. This feature has been at the heart of our approach presented in this paper where we have proposed a pure affine action in which a scalar field is coupled to the affine connection. Unlike the models that we have referred to previously, our approach is founded on two main roads, where in the first, we have generated the metric structure dynamically. Then we have proceeded to calculating the one-loop effective action and obtained the regularized induced parameters; Newton's constant and the cosmological constant, providing that the ultraviolet cutoff is comparable to the Planck mass.\n\nInduced gravity does not only provide us with an origin to the gravitational couplings from the particle spectrum, but it may also have an interesting effects on the standard model of particle physics particularly in counteracting the ultraviolet sensitivity of the latter \\cite{demir-uv0,*demir-uv1,*demir-uv2,*demir-uv3,*demir-uv4}. The present approach to induced gravity may reveal some important and new features of the physics of standard model when incorporating gravity. This will be explored in a possible future work.      \n\n\\section*{acknowledgments}\nThe author thanks Durmu\\c{s} Demir and Tomi Koivisto for their constructive comments. Supports (in part) by T\\\"{U}B\\.{I}TAK grant 117F111 are acknowledged.\n\n\n\n\n\\bibliographystyle{apsrev4-1.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\\iffalse \nHistorical developments of the views of Condorcet and de\nBorda on the best method for elections has led to two main types of\nsingle-winner voting rules which are now known as Condorcet-consistent\nvoting rules and scoring (or positional) voting rules, respectively\n\\cite{bra-fis:b:polsci:voting-procedures}.  Nowadays, the scoring\nrules are perhaps the best understood voting rules among single-winner\nelection systems.\n\nAny voting rule has two complementary characterizations: algorithmic\nand axiomatic. These two are equally important: knowing the algorithm\none will be unaware of the normative properties of the rule and\nknowing an axiomatic characterization one would have no clue how to\nexecute the rule.\nThe study of axiomatic properties of voting rules has been initiated\nby Arrow~\\cite{arr:b:polsci:social-choice}, and in a somehow more\nnarrow framework, by May~\\cite{mayAxiomatic1952}.  Axiomatic\ncharacterizations of the class of single-winner scoring rules were\nprovided by G{\\\"a}rdenfors~\\cite{gardenfors73:scoring-rules},\nSmith~\\cite{smi:j:scoring-rules}, and\nYoung~\\cite{you:j:scoring-functions}, and in a more general setting,\nby Myerson~\\cite{Myerson1995} and Pivato~\\cite{pivato2013variable}.\nAxiomatic characterizations of particular most prominent rules have\nbeen also achieved. These include characterizations of\nthe Borda rule~\\cite{youngBorda, hansson76, fishburnBorda,\n  smi:j:scoring-rules}, and of the Plurality~\\cite{RichelsonPlurality,\n  Ching1996298} and Antiplurality rules~\\cite{Barbera198249} (see also\nthe surveys of Chebotarev and Shamis~\\cite{che-sha:j:scoring-rules}\nand of Merlin~\\cite{merlinAxiomatic}).  \n\\fi\n\nAxiomatic studies of multiwinner voting rules\ngo back to Felsenthal and Maoz~\\cite{fel-mao:j:norms} and\nDebord~\\cite{deb:j:k-borda}, but a systematic work on the topic has began only\nrecently and on several different fronts.  New results appear within social choice theory,\ncomputer science, artificial intelligence, and a number of other\nfields (see the work of Faliszewski et al.~\\cite{fal-sko-sli-tal:b:multiwinner-trends} for more\ndetails on the history as well as recent progress).\nThe reason for this explosion of interest from a number of research\ncommunities is the wide range of applications of\nmultiwinner %\nvoting rules on the one hand, and the corresponding richness and\ndiversity of the spectrum of those rules on the other.  Typically,\nsocial-choice theorists study normative properties of various\nmultiwinner rules, computer scientists investigate feasibility of computing the\nelection results, and researchers working within artificial\nintelligence use multiwinner elections as a versatile tool (e.g.,\nuseful in genetic\nalgorithms~\\cite{fal-saw-sch-smo:j:multiwinner-genetic}, for ranking\nsearch results~\\cite{sko-lac-bri-pet-elk:c:proportional-rankings}, or\nfor providing personalized\nrecommendations~\\cite{bou-lu:c:chamberlin-courant}).  Yet, there is a\ngrowing interplay between these areas and an increased need for a new\nlevel of comprehension of results obtained in all of\nthem. %\nIn this paper we partially address this need by linking syntactic\nfeatures of certain families of committee scoring rules with their\nnormative properties.  The syntactic features of the rules are useful,\ne.g., for establishing their computational\nproperties~\\cite{sko-fal-lan:c:collective,fal-sko-sli-tal:c:top-k-counting},\nor for viewing those rules as achieving certain optimization goals (which\nallows one to consider these rules as tools for certain tasks from\nartificial intelligence and operation research).\nThe normative properties, on the other hand, are useful for\nunderstanding the `behavior' of these rules and the settings for which\nthey may be appropriate.\n\n\nThe model of multiwinner elections studied in this paper is as\nfollows. We are given a set of candidates, a collection of\nvoters---each with a preference order in which the candidates are ranked from the best\nto the worst---and an integer~$k$. %\nA multiwinner rule \nmaps this input to a subset of $k$ candidates (i.e., a committee; we discuss tie-breaking later)\nthat, in some sense, best reflects the voters' preferences. \nFor example, the Single Non-Transferable Vote rule (the SNTV rule)\nchooses $k$ candidates that are top-ranked most frequently, whereas\nthe Bloc rule selects $k$ candidates that are ranked most frequently among top~$k$\npositions  (equivalently, under Bloc each voter names\nmembers of his or her favorite committee, and those that are mentioned\nmost often are selected).\nNaturally, there are many other multiwinner rules to choose from, \ndefined in various ways.\n\n\n\n\n\nIn this paper we\nfocus on the class of committee scoring rules, \nintroduced by Elkind et\nal.~\\cite{elk-fal-sko-sli:j:multiwinner-properties} as multiwinner\ngeneralizations of classic positional scoring rules. The main idea of\ncommittee scoring rules is essentially the same as in the\nsingle-winner case: Each voter gives each committee a score based on\nthe positions of members of this committee in the voter's ranking, \nscores from individual voters are aggregated into the societal scores\nof the committees, and the committee(s) with the highest score wins.\nCommittee scoring rules appear to form a remarkably rich class that\nincludes both very simple rules, such as SNTV and Bloc, and rather\nsophisticated ones, such as the rule of Chamberlin and\nCourant~\\cite{cha-cou:j:cc} or variants of the Proportional Approval\nVoting rule~\\cite{kil-handbook}. As these rules tend to be very different in nature,\nthey are suitable for different purposes,\nsuch as selecting a diverse committee, selecting a committee that\nproportionally represents the electorate, or selecting a committee\nconsisting of $k$ individually best candidates.\nThis richness is the main strength of the class of committee scoring\nrules, but to choose rules for given settings wisely, it is important\nto understand the internal structure of the class.\nUnderstanding this structure  is the main goal of the current paper.\n\n\n\n\n\n\n\n\nSo far, researchers have identified the following subclasses of\ncommittee scoring rules (we provide their formal definitions in\nSections~\\ref{sec:prelim} and~\\ref{sec:classification}; here we give\nintuitions only).  (Weakly) separable rules, introduced by Elkind et\nal.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}, are those rules\nwhere we compute a separate score for each candidate (using a\nsingle-winner scoring rule) and then pick $k$ candidates with the top\nscores (for example, using Plurality scores leads to\nSNTV).\\footnote{If the underlying single-winner scoring rule does not\n  depend on the size of the committee (as in the case of SNTV) then\n  the rule is referred to as separable. If there is such dependence\n  (as in the case of Bloc), then the rule is weakly separable.}\nRepresentation-focused rules, also introduced by Elkind et\nal.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}, are similar in\nspirit to the Chamberlin--Courant rule, whose aim is to ensure that in the\nelected committee each voter's most preferred committee member (his or\nher representative) is ranked as high as possible. On the other\nhand, top-$k$-counting rules, introduced by Faliszewski et\nal.~\\cite{fal-sko-sli-tal:c:top-k-counting}, capture rules where each\nvoter evaluates the quality of a committee by the number of members of that committee \nthat he or she ranks among the top $k$ ones;\nBloc is a\nprime example of a top-$k$-counting rule.  \nFinally, the class of OWA-based rules---introduced by Skowron et\nal.~\\cite{sko-fal-lan:c:collective}, also studied in the approval-based\nelection\nmodel~\\cite{azi-bri-con-elk-fre-wal:j:justified-representation,azi-gas-gud-mac-mat-wal:c:approval-multiwinner,lac-sko:t:approval-thiele}---contains\nall the previously mentioned classes. Under these rules a voter\ncalculates the score of a committee as\nthe \\emph{ordered weighted average} (OWA) of the scores of\nthe candidates in that committee.\\footnote{See the\n  original work of Yager~\\cite{yag:j:owa} for a general discussion of\n  OWAs, and, e.g., the works of Kacprzyk et\n  al.~\\cite{kac-nur-zad:b:owa-social-choice} or Goldsmith et\n  al.~\\cite{gol-lan-mat-per:c:rank-dependent} for their other\n  applications in voting.}\nIn this paper we also introduce the class of \\emph{decomposable}\ncommittee scoring rules that strictly contains all the OWA-based ones,\nhas interesting applications, and appears to be easier to work with axiomatically.\n\nAll these classes have been defined purely in terms of the syntactic\nfeatures of the functions used to calculate the scores of the\ncommittees.\\footnote{%\n  A notable exception is the class of top-$k$-counting\n  rules which were discovered while characterizing those committee scoring\n  rules that satisfy the fixed-majority\n  property~\\cite{fal-sko-sli-tal:c:top-k-counting}.}  These syntactic\nfeatures are important if, for example, one wants to assess\nsome computational properties of the rules (e.g., it is known that weakly\nseparable rules are polynomial-time\ncomputable~\\cite{elk-fal-sko-sli:j:multiwinner-properties}, that\nrepresentation-focused rules tend to be ${\\mathrm{NP}}$-hard to\ncompute~\\cite{pro-ros-zoh:j:proportional-representation,bou-lu:c:chamberlin-courant,sko-fal-lan:c:collective},\nand that the structure of the functions used within OWA-based rules\naffects the ability to compute their results\napproximately~\\cite{sko-fal-lan:c:collective}). Such syntactic features are also\nessential when we view committee scoring rules as specifying\noptimization goals for particular applications\n(for example, since under the Chamberlin--Courant rule each voter's\nscore depends solely on his or her representative in the elected\ncommittee, this rule is particularly suitable in the context of\ndeliberative democracy~\\cite{cha-cou:j:cc}, for targeted\nadvertising~\\cite{bou-lu:c:chamberlin-courant,bou-lu:c:value-directed-cc},\nor for certain facility location\nproblems~\\cite{far-hek:b:facility-location}).  Nonetheless, these\nsyntactic features do not tell us much about the behavior of\nthe rules.\n\n\nOur first result reinforces the syntactic hierarchy of committee\nscoring rules. We show that the class of committee scoring rules strictly contains\nthe class of decomposable rules, which, in turn, strictly contains the class of\nOWA-based ones, and that the class of OWA-based rules strictly\ncontains the classes of (weakly) separable rules,\nrepresentation-focused rules, and top-$k$-counting rules. For each\npair of the latter three classes, we show that their intersection\ncontains exactly one, previously-known, non-trivial voting rule.  See\nFigure~\\ref{fig:hierarchy} for a visualization of the syntactic\nhierarchy of committee scoring rules.\n\n\n\nOur second, and the main, result establishes a link between several levels\nof the syntactic hierarchy and respective normative properties. In\nother words, we establish axiomatic characterizations of some of the\nstudied subclasses of committee scoring rules.\nUntil now, the only result of this form,\nwhich is due to Faliszewski et\nal.~\\cite{fal-sko-sli-tal:c:top-k-counting}, was a characterization of\nfixed-majority consistent committee scoring rules as those\ntop-$k$-counting rules whose scoring functions satisfy (a relaxed\nvariant of) the convexity property.\nHere, our main result is that many of the syntactic properties of our\nrules nicely correspond to certain types of\nmonotonicity. Specifically, we focus on the \\emph{committee\n  enlargement monotonicity}\\footnote{This notion is also known as\n  \\emph{committee\n    monotonicity}~\\cite{elk-fal-sko-sli:j:multiwinner-properties} and\n  \\emph{enlargement\n    consistency}~\\cite{bar-coe:j:non-controversial-k-names}. We chose\n  a name that is more informative than the former, but which is not\n  tied to the realm of resolute rules, as the latter.  In the\n  literature on apportionment rules, a related property is often called\n  \\emph{house monotonicity}~\\cite{Puke14a,\n    bal-you:b:polsci:representation}.} property, which requires that\nif we increase the size of the committee sought in the election, then the new\nwinning committee should be a superset of the old winning committee,\nand on variants of the \\emph{non-crossing monotonicity} property,\nwhich requires that if we shift forward some members of a\nwinning committee within any vote in a way that does not affect the\npositions of the remaining members of this committee, then this\ncommittee should still win.\nWe show that committee enlargement monotonicity characterizes exactly\nthe class of separable rules among committee scoring rules, and that\nnon-crossing monotonicity characterizes the class of weakly separable\nones.  Then we introduce top-member monotonicity (a variant of\nnon-crossing monotonicity restricted within each vote to shifting only\nthe highest-ranked member of the winning committee) and show that\ntogether with narrow-top consistency (which requires that if there are\nat most $k$ candidates that are ever ranked in the top position within\na vote, then these candidates should belong to the winning committee)\nit characterizes the class of representation-focused rules.  Finally,\nwe show that if a committee rule is prefix-monotone (i.e., satisfies a\nyet another restricted variant of non-crossing monotonicity) then it\nmust be decomposable.\n\n\n\n\n\n\n\n\n\n\n\n\n\n  \n\n\n\nThe paper is organized as follows. In Section~\\ref{sec:prelim} we\ndescribe the model of multiwinner elections, define the class of\ncommittee scoring rules, provide their basic properties, and show\nseveral examples of committee scoring rules.\nSection~\\ref{sec:classification} is devoted to structural properties\nof classes of committee scoring rules. Here we build the\nhierarchy of the classes and show results regarding\ncontainments and intersections among these classes.\nIn Section~\\ref{sec:axiomatic_properties} we switch to axiomatic\nproperties of the rules in the classes of the hierarchy and give\nseveral axiomatic characterizations of those classes. \nFinally, we discuss related work in Section~\\ref{sec:related} and\nconclude in Section~\\ref{sec:concl}.\n\n\n\n\n\n\\section{Multiwinner Elections and Committee Scoring Rules}\n\\label{sec:prelim}\nIn this section we set the stage for the discussions provided\nthroughout the rest of the paper by providing preliminary definitions\nas well as introducing the class of committee scoring rules.  For each\npositive integer $t$, we write $[t]$ to denote $\\{1, \\ldots, t\\}$. By\n$\\mathbb{R}_{+}$ we mean the set of nonnegative real numbers.\n\n\n\\subsection{Preliminaries}\n\nAn election is a pair $E = (C,V)$, where $C = \\{c_1, \\ldots, c_m\\}$ is\na set of candidates and $V = (v_1, \\ldots, v_n)$ is a collection of\nvoters. Each voter $v_i$ has a \\emph{preference order} $\\succ_i$,\nexpressing his or her ranking of the candidates, from the most\ndesirable one to the least desirable one.  Given a voter $v$ and a\ncandidate $c$, by ${{{\\mathrm{pos}}}}_v(c)$ we mean the position of $c$ in $v$'s\npreference order (the top-ranked candidate has position $1$, the next\none has position $2$, and so on).\n\n\nA \\emph{multiwinner voting rule} is a function $\\mathcal{R}$ that given an\nelection $E = (C,V)$ and a committee size $k$, $1 \\leq k \\leq |C|$,\nreturns a family $\\mathcal{R}(E,k)$ of size-$k$ subsets of $C$, i.e., the\nset of committees that tie as winners of the election (we use the\nnonunique-winner model or, in other words, we assume that multiwinner\nrules are irresolute).  We provide a few concrete examples of multiwinner rules in Section~\\ref{sec:rules-examples}.\n\n\n\n\nMost of the multiwinner rules that we study are based on single-winner\nscoring functions.  A~\\emph{single-winner scoring function} for $m$\ncandidates is a nonincreasing function $\\gamma \\colon [m] \\rightarrow\n\\mathbb{R}_{+}$ that assigns a score value to each position in a\npreference order.\nGiven a preference order $\\succ_i$ and a candidate $c$, by the\n$\\gamma$-score of $c$ (given by voter $v_i$) we mean the value\n$\\gamma({{{\\mathrm{pos}}}}_{v_i}(c))$.  \nThe two most commonly used %\nscoring functions are the Borda scoring function,\n\\begin{align*}\n\\beta_m(i) = m-i,\n\\end{align*}\nand the $t$-approval scoring function,\n\\begin{align*}\n\\alpha_t(i) = \n  \\begin{cases}\n    1       & \\quad \\text{if~} i \\leq t \\\\\n    0       & \\quad \\text{otherwise.}\n  \\end{cases}\n\\end{align*}\nIn particular, $\\alpha_1$ is known as the Plurality scoring function.\n\n\n\n\nCommittee scoring functions generalize single-winner scoring functions\nto the multiwinner setting in a natural way, by assigning scores to\nthe positions of the whole committees. Formally, given a vote $v$ and\na committee $S$ of size $k$, %\nthe \\emph{committee position} of $S$ in $v$, denoted ${{{\\mathrm{pos}}}}_v(S)$, is a\nsequence $(i_1, \\ldots, i_k)$ that results from sorting the set $\\{\n{{{\\mathrm{pos}}}}_v(c) \\mid c \\in S\\}$ in the increasing order.  We write $[m]_k$\nto denote the set of all such length-$k$ increasing sequences of\nnumbers from $[m]$ (in other words, we write $[m]_k$ to denote the set\nof all possible committee positions for the case of $m$ candidates and\ncommittees of size $k$).  Given two committee positions from $[m]_k$,\n$I = (i_1, \\ldots, i_k)$ and $J = (j_1, \\ldots, j_k)$, we say that $I$\n\\emph{weakly dominates} $J$, $I \\succeq J$, if for each $t \\in [k]$,\nit holds that $i_t \\leq j_t$ (we say that $I$ \\emph{dominates} $J$,\ndenoted $I \\succ J$, if at least one of these inequalities is\nstrict\\footnote{In previous papers on committee scoring rules, the\n  notions of weak dominance and dominance were conflated. We believe\n  that giving them clear, separate meanings will help in providing\n  more crisp arguments and discussions.}).  Below we define committee\nscoring functions formally.\n\n\n\\begin{definition}[Elkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}]\n  A committee scoring function for $m$ candidates and a committee size\n  $k$ is a function $f_{m,k} \\colon [m]_k \\rightarrow \\mathbb{R}_{+}$ such\n  that for each two sequences $I,J \\in [m]_k$, if $I$ weakly dominates $J$\n  then $h(I) \\geq h(J)$.\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\n\nLet $f=(f_{m,k})_{k\\le m}$ be a family of committee scoring functions,\nwhere %\neach $f_{m,k}$ is a function for $m$ candidates and committees of size\n$k$. Given an election $E = (C, V)$ with $m$ candidates and a committee~$S$ of\nsize $k$, we define the $f_{m,k}$-score of $S$ to be:\n\\begin{align*}\nf_{m,k}\\hbox{-}{{\\mathrm{score}}}_E(S) = \\sum_{v_i \\in V}f_{m,k}({{{\\mathrm{pos}}}}_{v_i}(S)) \\text{.}\n\\end{align*}\nWhen $f$ is clear from the context, we often speak of the score of a\ncommittee instead of its $f$-score.\nGiven the above notation, we are ready to define committee scoring rules\nformally.\n\n\\begin{definition}\n  Let $f=(f_{m,k})_{k\\le m}$ be a family of committee scoring\n  functions (with one function for each $m$ and $k$, $k \\leq\n  m$). Committee scoring rule $\\mathcal{R}_f$ is a multiwinner voting rule\n  that given an election $E = (C,V)$ and committee size $k$, outputs\n  all size-$k$ committees with the highest $f_{|C|,k}$-score.\n\\end{definition}\n\nWe say that a committee scoring rule $\\mathcal{R}_f$ is \\emph{degenerate} if\nthere is a number of candidates $m$ and a committee size $k$ such that\n$f_{m,k}$ is a constant function. As a consequence, a degenerate rule returns all\nsize-$k$ committees for every election with $m$ candidates.\nThe \\emph{trivial committee scoring} rule is a degenerate rule that\nreturns the set of all size-$k$ committees for all elections and all\nsizes $k$ (naturally, it is defined by a family of constant\nfunctions).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Examples of Committee Scoring Rules}\n\\label{sec:rules-examples}\n\nMany well-known multiwinner rules are, in fact, committee scoring\nrules; below we provide several such examples. For each of the\nrules we provide the family of committee scoring functions used in its\ndefinition, discuss these functions intuitively, and mention some\napplications.\n\n\n\\begin{description}\n\\item[SNTV, Bloc, and $\\boldsymbol{k}$-Borda.] These three rules use the following committee\n  scoring functions:\n  \\begin{align*}\n  f^{{{\\mathrm{SNTV}}}}_{m,k}(\\row ik) &= \\textstyle\\sum_{t=1}^{k}\\alpha_1(i_t) = \\alpha_1(i_1) \\text{,} \\\\\n  f^{{{\\mathrm{Bloc}}}}_{m,k}(\\row ik) &= \\textstyle\\sum_{t=1}^{k}\\alpha_k(i_t) \\text{, and} \\\\\n  f^{{{k\\hbox{-}\\mathrm{Borda}}}}_{m,k}(\\row ik) &= \\textstyle\\sum_{t=1}^k\\beta_m(i_t) \\text{.}\n  \\end{align*}\n  That is, under the SNTV rule we choose $k$ candidates with the\n  highest Plurality scores, under Bloc we choose $k$ candidates with\n  the highest $k$-Approval scores, and under $k$-Borda we choose~$k$\n  candidates with the highest Borda scores.\n  On the intuitive level, under SNTV each voter names his or her\n  favorite committee member, \n  under Bloc each voter names all the $k$ members of his or her\n  favorite committee, and under $k$-Borda each voter ranks all the\n  candidates and assigns them scores in a way which corresponds\n  linearly to their position in the ranking. SNTV and Bloc are\n  sometimes used in political elections (with the former used, e.g.,\n  in the parliamentary elections in Puerto Rico, and with the latter\n  often used for various local elections in many countries). $k$-Borda\n  and other rules based on similar scoring schemes are often used to\n  determine finalists of competitions (e.g., the finalists of the\n  Eurovision Song Contest are selected using a system very close to\n  $k$-Borda).\n\n\\item[The Chamberlin--Courant rule.] Under the Chamberlin--Courant rule\n  (the $\\beta$-CC rule), the score that a voter $v$ assigns to a\n  committee $S$ depends only on how $v$ ranks his or her favorite\n  member of~$S$ (referred to as $v$'s \\emph{representative} in~$S$). The\n  Chamberlin--Courant rule seeks committees in which each voter ranks\n  his or her representative as high as possible. Formally, the rule\n  uses functions:\n  \\begin{align*}\n    f^{{{\\beta\\hbox{-}\\mathrm{CC}}}}_{m,k}(\\row ik) = \\beta_m(i_1) \\text{.}\n  \\end{align*}\n  This is the variant of the rule originally proposed by Chamberlin\n  and Courant~\\cite{cha-cou:j:cc}, but, subsequently, other authors\n  %\n  %\n  %\n  (e.g., Procaccia et\n  al.~\\cite{pro-ros-zoh:j:proportional-representation}, Betzler at\n  al.~\\cite{bet-sli-uhl:j:mon-cc}, and Faliszewski et\n  al.~\\cite{fal-sli-sta-tal:c:cc-mon-clustering}) considered other\n  ones, based on other single-winner scoring functions. In particular,\n  we will be interested in the $k$-Approval Chamberlin--Courant rule,\n  ($\\alpha_k$-CC) which is defined through functions:\n  \\begin{align*}\n  f^{{{\\alpha_k\\hbox{-}\\mathrm{CC}}}}_{m,k}(\\row ik) = \\alpha_k(i_1) \\text{.}\n  \\end{align*}\n  Intuitively, both variants of the Chamberlin--Courant rule seek\n  committees of diverse candidates that ``cover'' as broad a spectrum\n  of voters' views as\n  possible. %\n  Lu and Boutilier~\\cite{bou-lu:c:chamberlin-courant} considered the\n  rule in the context of recommendation systems.\n\n\n\\item[The PAV rule.] The Proportional Approval Voting rule (the PAV\n  rule) was originally defined by Thiele~\\cite{Thie95a} in the\n  approval setting (where instead of ranking the candidates, the\n  voters indicate which ones they accept as committee members; for\n  recent discussions of the rule see the overview of\n  Kilgour~\\cite{kil-handbook} and the works of Aziz et\n  al.~\\cite{azi-bri-con-elk-fre-wal:j:justified-representation} and\n  Lackner and Skowron~\\cite{lac-sko:t:approval-thiele}). We model it\n  as a committee scoring rule $\\alpha_t$-PAV, where $t$ is a\n  parameter, defined using scoring functions of the form:\n  \\begin{align*}\n    f^{{{\\alpha_t\\hbox{-}\\mathrm{PAV}}}}_{m,k}(\\row ik) = \\alpha_t(i_1) +\n    \\textstyle\\frac{1}{2}\\alpha_t(i_2) +\n    \\frac{1}{3}\\alpha_t(i_3) +\n    \\cdots + \n    \\frac{1}{k}\\alpha_t(i_k)\n    \\text{.}\n  \\end{align*}\n  PAV is particularly well-suited for electing parliaments. Indeed,\n  Brill et al.~\\cite{bri-las-sko:c:apportionment} have shown that it\n  generalizes the d'Hondt apportionment method, which is used for this\n  purpose in many countries (e.g., in France and Poland).\n  A number of recent works~\\cite{azi-bri-con-elk-fre-wal:j:justified-representation,bri-las-sko:c:apportionment,lac-sko:t:approval-thiele,azi-elk-hua-lac-san-sko:c:ejr-poly}\n  explain why the harmonic sequence used within the PAV scoring\n  function ensures that the elected committee represents the voters\n  proportionally.\n\\end{description}\n\n\n\n\n\n\nNaturally, there are many other committee scoring rules, and we will\ndiscuss some of them throughout the paper. Nonetheless, the above few\nsuffice to illustrate our main points.  There is also a number of\nother multiwinner rules that are not committee scoring rules, such as\nSTV (see, e.g., the work of Tideman and\nRichardson~\\cite{tid-ric:j:stv}), Monroe~\\cite{mon:j:monroe}, Minimax\nApproval Voting~\\cite{bra-kil-san:j:minimax}, or rules which are\nstable in the sense of Gehrlein~\\cite{geh:j:condorcet-committee}. We\ndo not discuss them in this paper, but we provide some literature\npointers in Section~\\ref{sec:related}.\n\n\n\\subsection{Basic Features of Committee Scoring Rules}\n\\label{sec:basic}\n\nThe class of committee scoring rules is very rich and there are only a\nfew basic properties shared by all the rules in this class. Below we\ndiscuss several such properties that will be useful throughout this\npaper.\n\n\nFrom our point of view, the most important common feature of committee\nscoring rules is that they are uniquely defined by their scoring\nfunctions (up to linear transformations). Formally, we have the\nfollowing lemma (we provide the proof in the appendix).\n\n\\newcommand{\\lemuni}{Let $\\mathcal{R}_f$ and $\\mathcal{R}_g$ be two committee\n  scoring rules defined by committee scoring functions $f =\n  (f_{m,k})_{k \\leq m}$ and $g = (g_{m,k})_{k \\leq m}$,\n  respectively. If $\\mathcal{R}_f = \\mathcal{R}_g$ then for each $m$ and~$k$, $k\n  \\leq m$, there are two values, $a_{m,k} \\in \\mathbb{R}_{+}$ and $b_{m,k}\n  \\in \\mathbb{R}$, such that for each $I \\in [m]_k$ we have that\n  $f_{m,k}(I) = a_{m,k} \\cdot g_{m,k}(I) + b_{m,k}$.}\n\n\n\\begin{lemma}\\label{lem:unique}\n  Let $\\mathcal{R}_f$ and $\\mathcal{R}_g$ be two committee scoring rules defined\n  by committee scoring functions $f = (f_{m,k})_{k \\leq m}$ and $g =\n  (g_{m,k})_{k \\leq m}$, respectively. If $\\mathcal{R}_f = \\mathcal{R}_g$ then for\n  each $m$ and $k$, $k \\leq m$, there are two values, $a_{m,k} \\in\n  \\mathbb{R}_{+}$ and $b_{m,k} \\in \\mathbb{R}$, such that for each $I \\in\n  [m]_k$ we have that $f_{m,k}(I) = a_{m,k} \\cdot g_{m,k}(I) + b_{m,k}$.\n\\end{lemma}\n\nDue to Lemma~\\ref{lem:unique}, to show that two committee scoring\nrules are distinct it suffices to show that their scoring functions\nare not linearly related. In particular, this will be very useful when\nwe will be showing that certain rules cannot be represented using\nscoring functions of a given form.\n\n\n\nThe second common feature of committee scoring rules is nonimposition,\nwhich requires that for every committee there is some election where\nit wins uniquely.  Formally, we have the following definition.\n\n\\begin{definition}\\label{def:nonimposition}\n  Let $\\mathcal{R}$ be a multiwinner rule. We say that $\\mathcal{R}$ has the\n  nonimposition property if for each candidate set $C$ and each subset\n  $W$ of $C$, there is an election $E = (C,V)$ such that $\\mathcal{R}(E,\n  |W|) = \\{W\\}$.\n\\end{definition}\n\nNonimposition is such a basic property that it is hardly surprising\nthat all non-degenerate committee scoring rules satisfy it. We prove\nthe next lemma in the appendix.\n\n\\begin{lemma}\\label{lem:nonimposition}\n  Let $\\mathcal{R}_f$ be a committee scoring rule defined by a family of\n  committee scoring functions $f=(f_{m,k})_{k\\le m}$.  $\\mathcal{R}_f$\n  satisfies the nonimposition property if and only if every committee\n  scoring function in $f$ is nontrivial.\n\\end{lemma}\n\nWhile at first sight nonimposition and Lemma~\\ref{lem:nonimposition}\nseem hardly exciting, in fact they are sufficient to illustrate\nintriguing differences between single-winner voting rules and their\nmultiwinner counterparts. For example, one can verify that all\nnontrivial single-winner scoring rules satisfy the following extended\nvariant of the nonimposition property: For every candidate set $C$ and\nits subset $S$, there is an election $E = (C,V)$ where exactly the\ncandidates from $S$ tie as winners. Analogous result does not hold for\ncommittee scoring rules, even for the case of two committees (in which\ncase it could be dubbed as $2$-nonimposition; the example below is due\nto Lackner and Skowron~\\cite{lac-sko:t:approval-thiele}).\n\n\\begin{example}\n  Let us fix some committee size $k$ and a set $C$ containing at least\n  $2k$ candidates.  Consider two disjoint committees $W_1$ and\n  $W_2$. Let $E$ be an arbitrary election where $W_1$ and $W_2$ are\n  tied as winners according to Bloc (such elections exist). We note\n  that each candidate in $W_1$ has exactly the same $k$-Approval score\n  as each candidate in $W_2$ (otherwise at least one of these\n  committees would not be winning).  Consequently, every size-$k$\n  committee $W$ such that $W \\subseteq W_1 \\cup W_2$ is also winning\n  in $E$, so $W_1$ and $W_2$ are not the two unique winning\n  committees.\n\\end{example}\n\n\nThe fact that in general $2$-nonimposition does not hold for committee\nscoring rules is quite disappointing because many results would be far\neasier to prove if we could assume that it is always possible to\nconstruct an election where two arbitrary given committees are the\nonly winning ones. On the other hand, it is possible to construct\nelections where two size-$k$ committees $W_1$ and $W_2$ are the only\nwinning ones, provided that they share $k-1$ candidates (and, indeed,\nthis fact is used in the proof of Lemma~\\ref{lem:unique}).\n\n\nThere are a few more common properties of committee scoring rules. For\nexample, they all satisfy the \\emph{candidate\n  monotonicity} property which requires that if we shift forward a\nmember of a winning committee then, afterward, this candidate still\nbelongs to some winning committee (but possibly quite a different one;\nsee the work of Bredereck et\nal.~\\cite{bre-fal-kac-nie-sko-tal:c:multiwinner-robustness}). Also,\nall committee scoring rules are \\emph{consistent} in the sense that if\ntwo elections $E_1$ and $E_2$ (over the same candidate set) have some\ncommon winning committees, then these are exactly the winning\ncommittees in an election obtained by merging the voter collections of\n$E_1$ and $E_2$.  The former property is related to our discussions in\nSection~\\ref{sec:axiomatic_properties} and the latter one is often\nuseful as a tool when proving various results (and, indeed, it is\ncrucial in characterizing the class of committee scoring rules\naxiomatically~\\cite{sko-fal-sli:t:axiomatic-committee}).\n\n\n\n\n\n\n\\subsection{The T-Shirt Store Example}\n\nIn Section~\\ref{sec:rules-examples} we have provided a number of\nexamples of committee scoring rules and we have discussed some of\ntheir applications, focusing mostly on political elections. However,\ncommittee scoring rules have far more varied applications (see, e.g.,\nthe overview of Faliszewski et\nal.~\\cite{fal-sko-sli-tal:b:multiwinner-trends}), most of which have\nnothing to do with politics. Below we describe a simplified\nbusiness-inspired scenario where committee scoring rules may be\nuseful.  We use this example to guide our way through the different\ntypes of committee scoring rules discussed in this paper.\n\n\n\n\\begin{example}\\label{example:shirts1}\n  Consider a T-shirt store that needs to decide which shirts to put on\n  offer.\n  %\n  %\n  Let $C$ be the set of T-shirts that the store can order from its\n  suppliers ($|C| = m$). Since the store has limited space, it can\n  only put $k$ different T-shirts on display, and it wants to pick\n  them in a way that would maximize its revenue (i.e., the number of\n  T-shirts sold).  We assume that every customer knows all the designs\n  (say, from a website) and ranks all T-shirts from the best one to\n  the worst one.\n  %\n  Let us say\n  that a customer considers a T-shirt to %\n  %\n  be ``very good'' if it is among the top $k$ T-shirts (of course,\n  this is an arbitrary choice, made for the sake of simplifying the\n  example).\n\n  How should the store decide which T-shirts to put on display? This\n  depends on how the customers behave. Consider a customer that ranks\n  the available T-shirts on positions $i_1 < i_2 < \\cdots < i_k$. If\n  this is a very picky customer that only buys a T-shirt if it is the\n  very best among all possible ones (according to his or her\n  opinion) then the number of T-shirts this customer buys is given by\n  $f^{{{\\mathrm{SNTV}}}}_{m,k}(\\row ik) = \\alpha_1(i_1)$.  However, if\n  this customer were to buy one copy of each T-shirt he or she considered as ``very\n  good,'' he or she would buy $f^{{{\\mathrm{Bloc}}}}_{m,k}(\\row ik) =\n  \\textstyle\\sum_{t=1}^{k}\\alpha_k(i_t)$ T-shirts. \n  %\n  It is also possible that a customer would buy only one $T$-shirt,\n  provided he or she considered it as ``very good.'' The number of\n  T-shirts bought by such a customer would be $f^{{{\\alpha_k\\hbox{-}\\mathrm{CC}}}}_{m,k}(\\row\n  ik) = \\alpha_k(i_1)$.  Depending on which type of customers the\n  store expects to have, it should choose its selection of T-shirts\n  either using SNTV, Bloc, or $k$-Approval\n  Chamberlin--Courant. (Surely, other types of customers are possible\n  as well and we will discuss some of them later. It is also likely\n  that the store would face a mixture of different types of customers,\n  but this is beyond our study.)\n\\end{example}\n\n\n\n\\section{Hierarchy of Committee Scoring Rules}\n\\label{sec:classification}\n\nIn this section we describe the classes of committee scoring rules\nthat were studied to date, introduce a new class---the class of\ndecomposable rules---and argue how all these classes relate to each\nother, forming a hierarchy. In Figure~\\ref{fig:hierarchy} we present\nthe relations between the classes discussed in this section, with\nexamples of notable rules.\nThe classes are defined by setting restrictions on the scoring\nfunctions so, in other words, in this section we are interested in the\nsyntactic hierarchy of committee scoring rules.  Later, in\nSection~\\ref{sec:axiomatic_properties}, we will consider semantic\nproperties.\n\n  \n\n\\begin{figure}\n\\center\n\\scalebox{1.0}{\n\\begin{tikzpicture}\n\\tikzstyle{node} = [draw=black,fill=none]\n\n\\fill [node] (0,1.35) ellipse (7 and 4.5);\n\\node [above] at (0,5.05) {\\textbf{committee scoring rules}};\n\\node [above] at (0,4.55) {\\large{$\\max$-threshold rules, $\\ell_p$-Borda}};\n\n\\fill [node] (0,0.675) ellipse (6.5 and 3.75);\n\\node [above] at (0,3.75) {\\large\\textbf{decomposable}};\n\\node [above] at (0,3.25) {\\large multithreshold rules};\n\n\\fill [node] (0,0) ellipse (6 and 3);\n\\node [above] at (0,2.25) {\\large\\textbf{OWA-based}};\n\\node [above] at (0,1.75) {\\large $\\alpha_t$-PAV, $q$-HarmonicBorda};\n\n\\fill [node] (0,-1.425) ellipse (3 and 1.5);\n\\node [above] at (0,-2.1) {\\textbf{representation-focused}};\n\\node [above] at (0,-2.6) {\\small{$\\beta$-CC}};\n\n\\fill [node] (-2.0,0) ellipse (3 and 1.5);\n\\node [above] at (-2.25,0.5) {\\textbf{weakly separable}};\n\\node [above] at (-2.25,0) {{$k$-Borda}};\n\n\\fill [node] (2.0,0) ellipse (3 and 1.5);\n\\node [above] at (2.25,0.55) {\\textbf{top-$\\boldsymbol{k}$-counting}};\n\\node [above] at (2.95,0) {{$\\alpha_k$-PAV, Perfectionist}};\n\n\\node [above] at (0,0.25) {{Bloc}};\n\n\\node [above] at (1.45,-1.2) {{$\\alpha_k$-CC}};\n\n\\node [above] at (-1.45,-1.2) {{SNTV}};\n\n\\node [above] at (0,-0.7) {\\small{Trivial}};\n\n\\end{tikzpicture}\n}\n\\caption{\\label{fig:hierarchy}The hierarchy of committee scoring rules.}\n\\end{figure}\n\n\n\\subsection{Separable and Weakly Separable Rules}\n\nWe say that a family of committee scoring functions $f=(f_{m,k})_{k\\le\n  m}$ is {\\em weakly separable} if there exists a family of\n(single-winner) scoring functions $(\\gamma_{m,k})_{k\\le m}$ with\n$\\gamma_{m,k}\\colon [m] \\to \\mathbb{R}_+$ such that for every $m\\in \\mathbb{N}$ and\nevery committee position $I = (i_1, \\ldots, i_k) \\in [m]_k$ we have:\n\\[\nf_{m,k}(i_1, \\ldots, i_k) = \\textstyle\\sum_{t=1}^k \\gamma_{m,k}(i_t).\n\\]\nA committee scoring rule $\\mathcal{R}_f$ is {\\em weakly separable} if it is\ndefined through a family of weakly separable scoring functions $f$.\nIn other words, if a rule is weakly separable then we can compute the\nscore of each candidate independently, using the single-winner scoring\nfunction $\\gamma_{m,k}$, and pick the $k$ candidates with the highest\nscores.\nIn consequence, it is possible to compute winning committees for all\nweakly separable rules in polynomial time, provided that their\nunderlying single-winner scoring functions are polynomial-time\ncomputable~\\cite{elk-fal-sko-sli:j:multiwinner-properties}.\\footnote{There\n  is a subtlety here as there may be exponentially many winning\n  committees. However, by listing the scores of all the candidates, we\n  provide enough information to, e.g., enumerate all the winning\n  committees in time proportional to the number of these committees,\n  or to perform many other tasks related to winner determination (such\n  as computing the score of a winning committee).}\n\nIf for all $m$ we have $\\gamma_{m,1} = \\cdots = \\gamma_{m,m}$, then we\nsay that the family $f$ and the corresponding committee scoring rule $\\mathcal{R}_f$ are {\\em separable}, without the ``weakly''\nqualification. Thus, separable rules use the same scoring function for each value of the size of a committee to be elected.\nInterestingly, separable rules have some axiomatic properties that\nother weakly separable %\nrules lack~\\cite{elk-fal-sko-sli:j:multiwinner-properties}---we will discuss this further in Section~\\ref{sec:axiomatic_properties}.\n\nThe notion of (weakly) separable rules was introduced by Elkind et\nal.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}; they pointed\nout that SNTV and $k$-Borda are separable, whereas Bloc is only weakly\nseparable.\n\n\\iffalse \nWe note that in the case of Bloc $\\gamma_{m,k}$ does not depend on $m$\nand in the case of $k$-Borda $\\gamma_{m,k}$ does not depend on\n$k$. This is why the following rule, which we call Bloc-Borda is very\ninteresting and could be used as a test rule for statements concerning\nweakly separable voting rules. For this rule\n\\[\n  f_{m,k}(\\row ik) = \\textstyle\\sum_{t=1}^{k}\\beta_m(i_t)\\alpha_k(i_t).\n\\]\nIn this case the score of a committee given by a voter is the sum of\nBorda scores of the members of the committee that appear in the top\n$k$ positions in this voter's ranking.\n\nThe class of (weakly) separable scoring functions was introduced by\nElkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}, who\nalso pointed out that weakly separable rules can have significantly\ndifferent axiomatic properties than the separable ones. Nonetheless,\nthe two classes of rules also have many similarities. For example, it\nis easy to see that if the underlying single-winner scoring functions\n$\\gamma_{m,k}$ are polynomial-time computable, then so are the rules:\nIt suffices to separately compute the score of each candidate and\noutput a committee that consists of the highest-scoring\ncandidates.\\footnote{Technically one has to be careful here. There can\n  be exponentially many committees that tie as winners. However, in\n  polynomial time one can compute the score of every winning\n  committee, a single tied winning committee, or one could provide a\n  function that given a winning committee can in polynomial time\n  output the lexicographically next winning committee.}\n\nElkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties} point\nout that, for example, $k$-Borda and SNTV are separable, whereas Bloc\nis only weakly separable.\n\\fi\n\n\n\\subsection{Representation-Focused Rules}\n\nA family of committee scoring functions $f=(f_{m,k})_{k\\le m}$ is {\\em\n  representation-focused} if there exists a family of (single-winner)\nscoring functions $(\\gamma_{m,k})_{k\\le m}$ such that for every $m\\in\n\\mathbb{N}$ and every committee position $I = (i_1, \\ldots, i_k) \\in [m]_k$ we\nhave:\n\\begin{align*}\n   f_{m,k}(i_1, \\ldots, i_k) = \\gamma_{m,k}(i_1).\n\\end{align*}\nThis means that the score that a committee receives from a voter\ndepends only on the position of the most preferred member of this\ncommittee in the voter's preference ranking---such a member can be\nviewed as a representative of the voter in the committee.  A committee\nscoring rule $\\mathcal{R}_f$ is {\\em representation-focused} if it is\ndefined through a family of representation-focused scoring\nfunctions~$f$.  The notion of representation-focused rules was\nintroduced by Elkind et\nal.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}; $\\beta$-CC is the\narchetypal example of a representation-focused committee scoring rule\nand, in consequence, all the representation-focused rules can be seen as\nvariants of the Chamberlin--Courant rule.\n\n\n\nSNTV is both separable and representation-focused, and it is the only\nnon-degenerate committee scoring rule with this property.\n\n\\begin{proposition}\n  SNTV is the only non-degenerate committee scoring rule that is\n  (weakly) separable and representation-focused.\n\\end{proposition}\n\\begin{proof}\n  It is easy to verify that SNTV is separable and\n  representation-focused.  For the other direction, let $\\mathcal{R}$ be a\n  rule which is separable and representation focused.  It follows that\n  $\\mathcal{R} \\equiv \\mathcal{R}_f \\equiv \\mathcal{R}_g$ for some families of committee\n  scoring functions $f$ and $g$, such that $f_{m,k}(i_1, \\ldots, i_k)\n  = \\phi_{m,k}(i_1) + \\ldots + \\phi_{m,k}(i_k)$ and $g(i_1, \\ldots,\n  i_k) = \\gamma_{m,k}(i_1)$. Every linear transformation of $g$ has\n  the same form (i.e., it only depends on $i_1$), so by\n  Lemma~\\ref{lem:unique} (linearly transforming $g$, if necessary)\n  %\n  we can assume that $f = g$.\n\n  Without loss of generality, we can assume that $m > k$. For each\n  committee positions $(i_1, \\ldots, i_k)$ with $i_1 = 1$, we have\n  that\n  \\begin{align*}\n  \\phi_{m,k}(i_1) + \\ldots + \\phi_{m,k}(i_k) = \\gamma_{m,k}(i_1) \\text{,}\n  \\end{align*}\n  and, so, we can conclude that $f_1(2) = \\cdots = f_1(m)$.  Since\n  $\\mathcal{R}$ is non-degenerate, we have that $f_1(1) > f_1(m)$, and so\n  that $f_1(1) > f_1(2)$.  This is sufficient to conclude that $\\mathcal{R}$\n  is equivalent to SNTV.\n\\end{proof}\n\nGenerally, representation-focused rules are ${\\mathrm{NP}}$-hard to compute\n(SNTV is one obvious exception).  This fact was first shown by Procaccia et\nal.~\\cite{pro-ros-zoh:j:proportional-representation} in the\napproval-based setting, and then by Lu and\nBoutilier~\\cite{bou-lu:c:chamberlin-courant} for ${{{\\beta\\hbox{-}\\mathrm{CC}}}}$.  Since\nthen, various means of computing the results under the\nChamberlin--Courant rule and its variants were studied in quite some\ndetail~\\cite{bet-sli-uhl:j:mon-cc,cor-gal-spa:c:sp-width,sko-fal-sli:j:multiwinner,sko-yu-fal-elk:j:mwsc,pet:t:total-unimodularity,fal-sli-sta-tal:c:cc-mon-clustering,lac-pet:c:spoc,fal-lac-pet-tal:c:csr-heuristics}.\n\n\n\\subsection{Top-$\\boldsymbol{k}$-Counting Rules}\n\nA committee scoring rule $\\mathcal{R}_f$, defined by a family\n$f=(f_{m,k})_{k\\le m}$, is {\\em top-$k$-counting} if there exists\na %\nsequence of nondecreasing functions $(g_{m,k})_{k \\leq m}$, with $g_{m,k} \\colon\n\\{0, \\ldots, k\\} \\rightarrow \\mathbb{R}_{+}$, such that: %\n\\[\n  f_{m,k}( i_1, \\ldots, i_k ) = g_{m,k}\\Big( \\big| \\{ i_t \\mid i_t \\leq k\\} \\big| \\Big). \n\\]\nThat is, the value $f_{m,k}(i_1, \\ldots, i_k)$ depends only on\nthe number of committee members that the given voter ranks among his\nor her top $k$ positions.  We refer to the functions $g_{m,k}$ as the {\\em\n  counting functions}.  Top-$k$-counting rules were introduced by\nFaliszewski et al.~\\cite{fal-sko-sli-tal:c:top-k-counting}.\n\n\\begin{remark}\n  It would be quite natural to require that all counting functions for\n  a given committee size were the same, that is, that for each $k \\in\n  {{\\mathbb{N}}}$ it held that $g_{k,k} = g_{k+1,k} = g_{k+2,k} =\n  \\cdots$. Following Faliszewski et\n  al.~\\cite{fal-sko-sli-tal:c:top-k-counting}, we formally do not make\n  this requirement, but we expect it to hold for all natural\n  top-$k$-counting rules.\n\\end{remark}\n\n\nTop-$k$-counting rules include, for example, the Bloc rule,\n$\\alpha_k$-PAV, and $\\alpha_k$-CC, where Bloc uses the linear\ncounting functions $g_{m,k}^{{{\\mathrm{Bloc}}}}(i) = i$, \n$\\alpha_k$-PAV uses counting functions $g_{m,k}^{{{\\alpha_k\\hbox{-}\\mathrm{PAV}}}}(i) =\n\\sum_{t=1}^i\\frac{1}{t}$, and $\\alpha_{k}$-CC uses counting functions:\n\\begin{align*}\ng_{m,k}^{{{\\mathrm{CC}}}}(i) =\n  \\begin{cases}\n    1       & \\quad \\text{if~} i \\geq 1\\\\\n    0       & \\quad \\text{if~} i = 0.\n  \\end{cases}\n\\end{align*}\nAs an extreme example of a top-$k$-counting rule, Faliszewski et\nal.~\\cite{fal-sko-sli-tal:c:top-k-counting} introduced the\nPerfectionist rule, which uses counting functions:\n\\begin{align*}\ng_{m,k}^{{\\mathrm{Perf}}}(i) =\n  \\begin{cases}\n    1       & \\quad \\text{if~} i = k\\\\\n    0       & \\quad \\text{otherwise}\n  \\end{cases}\n\\end{align*}\nPerfectionist is extreme in the sense that a voter assigns a point to\na committee exactly if he or she ranks all the members of this\ncommittee as $k$ best ones.\n\n\\begin{example}\\label{example:shirts-perfectionist}\n  Let us recall our T-shirt store example\n  (Example~\\ref{example:shirts1}).  Consider a particularly snobbish\n  customer, who is willing to buy a shirt from a store only if he or\n  she views all the available shirts as very good (recall that we\n  defined ``very good'' to mean being ranked among top $k$\n  positions). Then if $i_1, \\ldots, i_k$ are the positions of the\n  available shirts in the customer's ranking, the number of shirts\n  that the store should expect to sell to such a customer is:\n  \\[\n  f_{m,k}(i_1, \\ldots, i_k) = g_{m,k}^{{\\mathrm{Perf}}}\\Big( \\big| \\{ i_t \\mid i_t\n  \\leq k\\} \\big| \\Big) = \\alpha_k(i_k).\n  \\]\n  Thus if the store expects such customers, then it should use the\n  Perfectionist rule to choose its merchandise (and, possibly,\n  should also increase its prices!).\n\\end{example}\n\n\n\n\nBloc is the only nontrivial rule that is both weakly separable and\ntop-$k$-counting, and $\\alpha_k$-CC is the only nontrivial rule that\nis both representation-focused and top-$k$-counting.\n\n\n\n\\begin{proposition} %\n  Bloc is the only nontrivial %\n  rule that is weakly separable and top-$k$-counting.\n\\end{proposition}\n\n\\begin{proof}\n  By combining Lemma~\\ref{lem:unique} and the results of Faliszewski\n  et al.~\\cite{fal-sko-sli-tal:c:top-k-counting}, we obtain that\n  top-$k$-counting rule defined by a family of linear counting\n  functions is the only weakly separable top-$k$-counting rule, and this\n  rule is exactly Bloc.\n\\end{proof}\n\n\\begin{proposition}\\label{thm:cc-hope}\n  $\\alpha_k$-CC is the only nontrivial %\n  rule that is representation-focused and top-$k$-counting.\n\\end{proposition}\n\\begin{proof}\n  It is easy to verify that $\\alpha_k$-CC is top-$k$-counting and\n  representation-focused.  For the other direction, let $\\mathcal{R}$ be a\n  rule which is both top-$k$-counting and representation focused.  It\n  follows that $\\mathcal{R} \\equiv \\mathcal{R}_f \\equiv \\mathcal{R}_g$ for two\n  functions, $f$ and $g$, such that, $f(i_1, \\ldots, i_k) = f_1(i_1)$ and\n  $g(i_1, \\ldots, i_k) = g_1(s)$, where $s = \\big|\\{i_t \\mid i_t \\leq k\\}\\big|$.\n  Since any linear transformation of $f$ has the same form, by the uniqueness %\n  we can assume that $f = g$.\n\n  For each $i \\in [m-k+1]$ let $L(i)$ denote the sequence $(i, m-k+2, m-k+3, \\ldots, m)$. For each $i, j > k$ we have that:\n  \\begin{align*}\n  f_1(i) = f(L(i))= g(L(i)) = g(L(j)) = f(L(j)) = f_1(j) \\text{.}\n  \\end{align*}\n  By the same reasoning, we can prove that for each $i, j \\leq k$ we have $f_1(i) = f_1(j)$.\n  Since the rule is nontrivial, we know that for some $i, j$ it holds that $f_1(i) \\neq f_1(j)$.\n  This is sufficient to claim that $\\mathcal{R}$ is equivalent to $\\alpha_k$-CC.\n\\end{proof}\n\n\nFaliszewski et al.~\\cite{fal-sko-sli-tal:c:top-k-counting} show that\ntop-$k$-counting rules tend to be ${\\mathrm{NP}}$-hard to compute, but point out\nseveral polynomial-time computable exceptions, including Bloc and\nPerfectionist. They also observe that for rules with concave counting\nfunctions there are polynomial-time constant-factor approximation\nalgorithms, whereas for rules with convex counting functions such\nalgorithms may be missing (under standard complexity-theoretic\nassumptions).\n\n\n\\subsection{OWA-Based Rules}\nSkowron et al.~\\cite{sko-fal-lan:c:collective} introduced a class of\nmultiwinner rules based on ordered weighted average (OWA) operators.\nSimilar rules for approval-based ballots were first considered in the\n19th century by Thiele~\\cite{Thie95a} and more recently were studied\nby Aziz et\nal.~\\cite{azi-bri-con-elk-fre-wal:j:justified-representation,azi-gas-gud-mac-mat-wal:c:approval-multiwinner}\nand Lackner and Skowron~\\cite{lac-sko:t:approval-thiele} (see also the\ndiscussion by Kilgour~\\cite{kil-handbook}).  Elkind and\nIsmaili~\\cite{conf\/aldt\/ElkindI15} use OWA operators to define a\ndifferent class of multiwinner rules, which we do not consider in this\npaper.\n\nWe provide intuition for the OWA-based rules by using our T-shirts\nstore example.\n\n\\begin{example}\\label{example:shirts}\n  Let us say that a customer views a T-shirt as ``good enough'' if it\n  is among the top $10\\%$ of the shirts available on the market.  Suppose\n  that a customer identifies the best T-shirt available in the store\n  and buys it with probability 1, provided it is ``good enough''. Then he\n  or she also finds the second best T-shirt and buys it with\n  probability $\\nicefrac{1}{2}$ (again, provided that it is ``good\n  enough''), the third best shirt with probability $\\nicefrac{1}{3}$,\n  and so on, all the way to the $k$'th best T-shirt, which he or she buys\n  with probability $\\nicefrac{1}{k}$ (if it is ``good enough'').  If $i_1,\n  \\ldots, i_k$ are the positions (in the customer's preference order)\n  of the T-shirts that the store puts on display, then the expected\n  number of T-shirts he or she buys is given by the function:\n   \\[\n     f_{m,k}(i_1, \\ldots, i_k) = 1\\cdot \\alpha_{0.10m}(i_1) + \\nicefrac{1}{2}\\cdot \\alpha_{0.10m}(i_2)+\\cdots+\\nicefrac{1}{k}\\cdot \\alpha_{0.10m}(i_k) .\n   \\]\n   Thus, to maximize its revenue, the store should find a winning\n   committee for the election where the T-shirts are the candidates,\n   the voters are the customers, and where we use committee scoring\n   rule $\\mathcal{R}_f$ based on $f = (f_{m,k})_{k \\le m}$. This multiwinner\n   voting rule is $\\alpha_{0.10m}$-PAV, a variant of the Proportional\n   Approval Voting rule.\n\\end{example}\n\n\nNow let us define OWA-based rules formally. An OWA operator $\\Lambda$\nof dimension $k$ is a sequence $\\Lambda = (\\lambda_1, \\ldots,\n\\lambda_k)$ of nonnegative real numbers.\n\n %\n\n\n\\begin{definition}\\label{def:owa-csr}\n  Let $\\Lambda=(\\Lambda^{m,k})_{k \\leq m}$ be a sequence of OWA\n  operators such that $\\Lambda_{m,k} = (\\lambda_1^{m,k}, \\ldots,\n  \\lambda_k^{m,k})$ has dimension $k$. Let $\\gamma =\n  (\\gamma_{m,k})_{k \\le m}$ be a family of single-winner scoring\n  functions. %\n  Then, $\\gamma$ and $\\Lambda$ define a family $f =\n  (f_{m,k})_{k \\le m}$ of committee scoring functions such that for\n  each $(i_1, \\ldots, i_k) \\in [m]_k$ we have:\n  \\[\n    f_{m,k}(i_1, \\ldots, i_k) = \\textstyle\\sum_{t=1}^{k}\\lambda_t^{m,k} \\gamma_{m,k}(i_t).\n  \\]\n  We refer to committee scoring rules $\\mathcal{R}_f$ defined through $f$ in\n  this way as OWA-based.\n\\end{definition}\n\nIt is known that weakly separable, representation-focused, and\ntop-$k$-counting rules are OWA-based. The first class is defined using\nOWA operators $(1, \\ldots, 1)$, the second one uses OWA operators\n$(1,0, \\ldots,0)$, and the last one contains rules that use $k$-Approval single-winner\nscoring functions and any OWA operator (the argument that shows\nthis is due to Faliszewski et\nal.~\\cite[Proposition 3]{fal-sko-sli-tal:c:top-k-counting} and requires a bit more\neffort than for the previous two classes). As a corollary to the\npreceding propositions, we get the following.\n\n\\begin{corollary}\n  Each of the classes of separable, top-$k$-counting, and\n  representation-focused rules is strictly contained in the class of\n  OWA-based rules.\n\\end{corollary}\n\\begin{proof}\n  Containment follows from the paragraph above.\n  Strictness follows as we have Bloc as the unique rule in the intersection of\n  top-$k$-counting and weakly separable; SNTV as the unique rule in the\n  intersection of weakly separable and representation-focused; and\n  $\\alpha_k$-CC as the unique rule in the intersection of\n  top-$k$-counting and representation-focused:\n    it follows that Bloc is not representation-focused; SNTV is not top-$k$-counting; and\n  $\\alpha_k$-CC is not weakly separable.  We get the claim by noticing that\n  Bloc, SNTV, and $\\alpha_k$-CC, are all OWA-based.\n\\end{proof}\n\nNaturally, there are also OWA-based rules that do not belong to any of\nthe above-mentioned classes. For example, this is the case for\n$\\alpha_t$-PAV rules (provided that the parameter $t$ is not equal to\nthe committee size $k$, e.g., if it is fixed as a constant) or for the\nrelated $q$-HarmonicBorda rules (the $q$-HB rules), defined by the\nfollowing scoring functions ($q \\in \\mathbb{R}_{+}$ is a parameter):\n\\begin{align*}\n  f^{{{q\\hbox{-}\\mathrm{HB}}}}_{m,k}(\\row ik) = \\beta_m(i_1) +\n  \\textstyle\\frac{1}{2^q}\\beta_m(i_2) + \\frac{1}{3^q}\\beta_m(i_3) +\n  \\cdots + \\frac{1}{k^q}\\beta_m(i_k) \\text{.}\n\\end{align*}\nThe $q$-HarmonicBorda rules were introduced by Faliszewski et\nal.~\\cite{fal-sko-sli-tal:c:paths}, who were looking for various means\nof achieving a compromise between the $k$-Borda rule and the\nChamberlin--Courant rule ($0$-HB is $k$-Borda, and as $q$ becomes\nlarger and larger, $q$-HB becomes more and more similar to\n$\\beta$-CC).\n\n\\begin{proposition}\\label{pro:owa}\n  Neither $\\alpha_t$-PAV nor $q$-HB is weakly separable, nor\n  representation-focused, nor top-$k$-counting, for any choice of\n  constants $t \\in {{\\mathbb{N}}}$ and $q \\in \\mathbb{R}_+$.\n\\end{proposition}\n\nTo prove Proposition~\\ref{pro:owa} it suffices to show that the\ncommittee scoring functions of these rules cannot be expressed as linear\ntransformations of weakly separable, representation-focused, and\ntop-$k$-counting scoring functions, and invoke Lemma~\\ref{lem:unique}.\nWe omit the details of this simple but somewhat tedious task.\n\n\n\n\\iffalse\n\\begin{proof}\nOne can prove the proposition by using Lemma~\\ref{uniqness_of_implementation} and by observing that the committee scoring function defining $2$-PAV cannot be expressed as a linear transformation of any weakly separable, representation-focused, nor top-$k$-counting committee scoring function. %\n\\fi\n\\iffalse First, we will show that $2$-PAV is not separable.  If\n$2$-PAV was separable, then it would have satisfied non-crossing\nmonotonicity~\\cite{elk-fal-sko-sli:j:multiwinner-properties}. Informally\nspeaking, non-crossing monotonicity requires that if a voter pushes a\nmember of a winning committee up in her preference order so that this\ncandidate does not cross any other member of the winning committee,\nthen the committee should still be winning after such a push (we\nrecall the formal definition of non-crossing monotonicity in\nSection~\\ref{sec:noncrossing_monotonicity},\nDefinition~\\ref{def:noncrossing_monotonicity}).  However, $2$-PAV does\nnot satisfy this property.  To see why, consider an election where\neach permutation of the set of candidates is cast exactly once.  For\nexample, for $k = 2$, and $m = 4$, we have all permutations of the\ncandidates $\\{a, b, c, d\\}$.  From symmetry, all pairs of candidates\nform winning committees; in particular, $\\{a,c\\}$ and $\\{c,d\\}$ are\nboth winning.  Now, take the voter who votes as $a \\succ b \\succ c\n\\succ d$ and shift $c$ forward, to get the vote $a \\succ c \\succ b\n\\succ d$.  As a result, the score of the committee $\\{c,d\\}$ would\nincrease by $1$ while the score of the committee $\\{a,c\\}$ would\nincrease by only $\\nicefrac{1}{2}$; consequently, $\\{a,c\\}$ would not\nbe winning in the shifted election, thus violating the non-crossing\nmonotonicity.\n\n\nSecond, we will show that $2$-PAV is not representation-focused. Consider the following election:\n\\begin{align*}\n  a \\succ c \\succ b \\\\\n  a \\succ c \\succ b \\\\\n  b \\succ c \\succ a\n\\end{align*}\nand let $k = 2$.\nIf $2$-PAV was a representation-focused rule, then it should select $\\{a,b\\}$ as a winning committee.\nHowever, under $2$-PAV, only $\\{a,c\\}$ is winning.\n\nFinally, we will show that $2$-PAV is not top-$k$-counting.\nLet $k = 1$, and consider a simple election which consists of a single vote $a \\succ b \\succ c$.\nIf $2$-PAV were top-$k$-counting then only $\\{a\\}$ would be winning. Yet, according to $2$-PAV both $\\{a\\}$ and $\\{b\\}$ are winning.\n\\end{proof}\n\\fi\n\n\n\nSkowron et al.~\\cite{sko-fal-lan:c:collective} have shown that\nOWA-based rules are typically ${\\mathrm{NP}}$-hard to compute (with the clear\nexception of, e.g., weakly separable rules and the Perfectionist rule). They\nhave also linked the properties of the OWA operators with the ability\nto approximate the rules (generally speaking, if the OWA operators for\na given rule are non-increasing then there are polynomial-time\nconstant-factor approximation algorithms for this rule, and otherwise\nthey are typically missing\\footnote{However, there are exceptions. For\n  example, viewed as an OWA-based rule, Perfectionist uses OWA\n  operators $(0, \\ldots, 0,1)$ but still is polynomial-time\n  computable. This is because, as a top-$k$-counting rule,\n  Perfectionist uses a very restrictive single-winner scoring\n  function, and is not captured by the results of Skowron et\n  al.~\\cite{sko-fal-lan:c:collective}.}).\n\n\n\n\n\n\n\n\\subsection{Decomposable Rules}\nWe introduce the following class that naturally generalizes the class\nof OWA-based rules and resort to our T-shirt store example to help the\nreader rationalize it.\n\n\\begin{definition}\\label{def:decomposable-csr}\n  Let $\\gamma = (\\gamma^{(t)}_{m,k})_{t \\le k \\le\n    m}$ %\n  be a family of single-winner scoring functions. %\n  These functions define\n  a family of committee scoring functions $f = (f_{m,k})_{k \\le m}$\n  such that for each committee position $(i_1, \\ldots, i_k) \\in [m]_k$\n  we have:\n  \\[\n    f_{m,k}(i_1, \\ldots, i_k) = \\textstyle\\sum_{t=1}^{k}\\gamma^{(t)}_{m,k}(i_t).\n  \\]\n  We refer to committee scoring rules $\\mathcal{R}_f$ defined through $f$ in\n  this way as {decomposable}.\n\\end{definition}\n\n\nAt first glance, decomposable rules seem very similar to the weakly\nseparable ones. The difference is that for fixed $m$ and $k$ and two\ndifferent values $t$ and $t'$, for decomposable rules the functions\n$\\gamma^{(t)}_{m,k}$ and $\\gamma^{(t')}_{m,k}$ can be completely\ndifferent. It is apparent that OWA-based rules are decomposable.  \nWe will see that this containment is strict.\n\n\n\n\\begin{example}\\label{example:shirts:decomposable}\n %\n %\n  Let us recall from Example~\\ref{example:shirts} that a customer\n  considers a T-shirt to be ``good enough'' if it is among the best\n  $10\\%$ of all shirts and let us say that a shirt is ``great'' if it\n  is among the top $1\\%$ of all shirts.\n  A customer buys two ``great'' T-shirts, or one ``at least good\n  enough'' T-shirt (if there are no two ``great'' T-shirts on\n  display). Naturally, the customer picks the best T-shirt(s) he can\n  find (respecting the above constraints).  If $i_1, \\ldots, i_k$ are\n  the positions (in the customer's preference order) of the T-shirts\n  that the store puts on display, then the number of T-shirts he or\n  she buys is given by function:\n   \\[\n     f_{m,k}(i_1, \\ldots, i_k) = \\alpha_{0.10m}(i_1) + \\alpha_{0.01m}(i_2).\n   \\]\n   Thus, to maximize its revenue, the store should find a winning\n   committee for the election where the T-shirts are the candidates,\n   the voters are the customers, and where we use decomposable\n   committee scoring rule $\\mathcal{R}_f$ based on $f = (f_{m,k})_{k \\le\n     m}$.\n\\end{example}\n\n\n\nWe refer to decomposable rules defined through committee scoring\nfunctions of the form\n\\[\n  f_{m,k}(i_1, \\ldots, i_k) = \\lambda^k_1 \\alpha_{t_{m,k,1}}(i_1) + \\cdots +\\lambda^k_k \\alpha_{t_{m,k,k}}(i_k),\n\\]\nwhere $\\Lambda_k = (\\lambda^k_1, \\ldots, \\lambda^k_k)$ are OWA\noperators and $t_{m,k,1}, \\ldots, t_{m,k,k}$ are sequences of\nintegers from $[m]$, as \\emph{multithreshold} rules (we put no constraints on\n$t_{m,k,1}, \\ldots, t_{m,k,k}$; both increasing and decreasing\nsequences are natural).\n\n\\begin{proposition}\\label{prop:multithreshold-not-owa}\n  The committee scoring rule defined through the multithreshold functions\n  $f_{m,k}(i_1, \\ldots, i_k) = \\alpha_{p_1}(i_1) + \\alpha_{p_2}(i_2)$,\n  for $p_1, p_2 \\in \\{2, \\ldots, m-k-2\\}$, $p_1 > p_2+1 \\geq 3$, is not\n  OWA-based.\n\\end{proposition}\n\\begin{proof}\n  Let us fix $p_1$, $p_2$, $m$, and $k$ that satisfy the requirements\n  from the statement of the theorem.  For the sake of contradiction,\n  assume that our multithreshold function is OWA-based.  By\n  Lemma~\\ref{lem:unique} we infer that there exist a  committee scoring function\n  $g_{m,k}$ of the form:\n  \\begin{align*}\n    g_{m,k}(i_1, \\ldots, i_k) = \\lambda_1 \\gamma(i_1) + \\lambda_2 \\gamma(i_2) \\text{,}\n  \\end{align*}\n  where $\\lambda_1, \\lambda_2 \\in \\mathbb{R}$ are two numbers and $\\gamma$\n  is a single-winner scoring function, such that for each committee\n  position $I = (i_1, \\ldots, i_k)$ it holds that $f_{m,k}(I) =\n  g_{m,k}(I)$; this follows because, by Lemma~\\ref{lem:unique}, the\n  OWA-based committee scoring functions for our rule have to depend on\n  $i_1$ and $i_2$ only, and by applying appropriate linear\n  transformations, we can assume that these functions equal $f_{m,k}$.\n\n\n\n\n  Let us now consider two committee positions $I' = (p_2, p_1+1,\n  \\ldots)$ and $I'' = (p_2, p_1, \\ldots)$.  We see that:\n  \\begin{align*}\n  f_{m,k}(I') - f_{m,k}(I'') &= \\big(\\alpha_{p_1}(p_2) + \\alpha_{p_2}(p_1+1)\\big) - \\big(\\alpha_{p_1}(p_2) + \\alpha_{p_2}(p_1)\\big) = \\alpha_{p_2}(p_1+1) - \\alpha_{p_2}(p_1) = 0 \\text{,}\n  \\end{align*}\n  and, thus, it must also be the case that:\n  \\begin{align*}\n  g_{m,k}(I') - g_{m,k}(I'') = \\big(\\lambda_1\\gamma(p_2) + \\lambda_{2}\\gamma(p_1+1)\\big) - \\big(\\lambda_{1}\\gamma(p_2) + \\lambda_{2}\\gamma(p_1)\\big) = \\lambda_2\\big(\\gamma(p_1+1) - \\gamma(p_1)\\big) = 0  \n  %\n  \\end{align*}\n  On the other hand, for committee positions $J' = (p_1+1,p_1+2,\n  \\ldots)$ and $J'' = (p_1, p_1+2, \\ldots)$ we have:\n  \\begin{align*}\n  f_{m,k}(J') - f_{m,k}(J'') = \\big(\\alpha_{p_1}(p_1+1) + \\alpha_{p_2}(p_1+2)\\big) - \\big(\\alpha_{p_1}(p_1) + \\alpha_{p_2}(p_1+2)\\big) < 0 \n  \\end{align*}\n  and, consequently:\n  \\begin{align*}\n    g_{m,k}(J') - g_{m,k}(J'') &= \\big(\\lambda_1\\gamma(p_1+1) +\n    \\lambda_{2}\\gamma(p_1+2)\\big) - \\big(\\lambda_{1}\\gamma(p_1) +\n    \\lambda_{2}\\gamma(p_1+2)\\big) \\\\ &= \\lambda_1 \\big(\\gamma(p_1+1) -\n    \\gamma(p_1)\\big) < 0.\n  \\end{align*}\n  Since we have both $\\lambda_2\\big(\\gamma(p_1+1) - \\gamma(p_1)\\big) =\n  0$ and $\\lambda_1 \\big(\\gamma(p_1+1) - \\gamma(p_1)\\big) < 0$, we\n  conclude that $\\lambda_2 = 0$.  However, for committee positions $L'\n  = (p_2-1, p_2+1, \\ldots)$ and $L'' = (p_2-1, p_2, \\ldots)$ we have:\n  \\begin{align*}\n  f_{m,k}(L') - f_{m,k}(L'') = \\big(\\alpha_{p_1}(p_2-1) + \\alpha_{p_2}(p_2+1)\\big) - \\big(\\alpha_{p_1}(p_2-1) + \\alpha_{p_2}(p_2)\\big) < 0\n  \\end{align*}\n  and:\n  \\begin{align*}\n  g_{m,k}(L') - g_{m,k}(L'') &= \\big(\\lambda_1\\gamma(p_2-1) +\n    0 \\cdot \\gamma(p_2+1)\\big) - \\big(\\lambda_{1}(p_2-1) +\n    0 \\cdot (p_2)\\big) = 0 \\text{,}\n  \\end{align*}\n  which is a contradiction  and completes the proof. \n  %\n  %\n  %\n\\end{proof}\n\n\\begin{comment}\n  Let us fix the number of candidates $m$ to be sufficiently large (as\n  will become clear throughout the proof) and the committee size $k$\n  to be sufficiently small (e.g., $k=2$ would suffice).  For the sake\n  of contradiction, let us assume that $\\mathcal{R}_f$ is OWA-based, i.e.,\n  in particular, that there exists $f'$ such that $\\mathcal{R}_f =\n  \\mathcal{R}_{f'}$ (for elections with $m$ candidates and committee size\n  $k$) and such that\n  $f'(i_1, \\ldots, i_k) = \\lambda_1\\gamma(i_1) + \\lambda_2\\gamma(i_2)+\n  \\cdots + \\lambda_k\\gamma(i_k)$, %\n  where $\\gamma$ is a single-winner scoring function and the\n  coefficients $\\lambda_i$ are all nonnegative.\n\n\n  Let $E = (C,V)$ be an election with candidate set $C = \\{c_1,\n  \\ldots, c_m\\}$ and voter collection $V = (v_1, \\ldots, v_{m!})$,\n  with one voter for each possible preference order. By symmetry, each\n  size-$k$ subset $W$ of $C$ is a winning committee of $E$ under\n  $\\mathcal{R}_f$. Let $v$ be an arbitrary fixed voter and let $b$ be the\n  candidate that $v$ ranks on top, $c_1$ be the candidate that $v$\n  ranks on position $(p_1+1)$, and $c_2$ be the candidate that $v$\n  ranks on position $(p_2+1)$. Note that $v$ prefers $b$ to $c_2$ to\n  $c_1$. Let $D_{k-1}$ and $D_{k-2}$ be, respectively, the sets of\n  candidates that $v$ ranks on bottom $k-1$ and $k-2$ positions.  We\n  define three committees: $C_1 = D_{k-1} \\cup \\{c_1\\}$, $C_{1,2} =\n  D_{k-2} \\cup \\{c_1, c_2\\}$, and $C_{b,2} = D_{k-2} \\cup \\{b, c_2\\}$.\n\n  Let $E_1$ be an election obtained from $E$ by shifting $c_1$ one\n  position forward in $v$.  According to $f_{m,k}$, in $E_1$ the score of\n  committee $C_1$ increases by one point (as compared to $E$), and the\n  score of $C_{1,2}$ does not change. Since every other committee\n  gains at most one point, we get that $C_1$ is a winner in $E_1$ and\n  that $C_{1,2}$ is not. However, this means that also under $f'$ the\n  score increase of $C_1$ must have been greater than the score\n  increase of $C_{1,2}$ and, so, we get\n$    \\lambda_1 (\\gamma(p_1) - \\gamma(p_1 + 1)) > \\lambda_2 (\\gamma(p_1) - \\gamma(p_1 + 1))$.\n  It must be that $\\gamma(p_1) - \\gamma(p_1 + 1) > 0$ (otherwise the\n  above inequality would not hold) and we conclude that $\\lambda_1 >\n  \\lambda_2$.\n\n  Next, let $E_2$ be an election obtained from $E$ by shifting $c_2$\n  one position forward in $v$: In $E_2$ the committee $C_{b,2}$ gains one\n  point and by the same reasoning as above we infer that $C_{b,2}$ is\n  a winner in $E_2$. Similarly, $C_{1,2}$ does not gain the additional\n  point and, so, it is not a winner in $E''$. Under $f'$ the score\n  increase of $C_{b,2}$ must be greater than that of $C_{1,2}$, so\n$    \\lambda_2 (\\gamma(p_2) - \\gamma(p_2 + 1)) > \\lambda_1 (\\gamma(p_2) - \\gamma(p_2 + 1))$.\n  This implies that $\\lambda_2 > \\lambda_1$, which gives a\n  contradiction. %\n\\end{comment}\n\n\nWe generally expect decomposable rules to be ${\\mathrm{NP}}$-hard, but even\namong these rules there are polynomial-time computable rules (that are\nnot OWA-based). For example, in their discussion of top-$k$-counting\nrules, Faliszewski et al.~\\cite{fal-sko-sli-tal:c:top-k-counting}\nmention a multithreshold rule that uses scoring functions that mix\nSNTV and Perfectionist:\n\\begin{align*}\n  f_{m,k}^{{{{\\mathrm{SNTV}}}} + {{\\mathrm{Perf}}}}(i_1, \\ldots, i_k) = f_{m,k}^{{{{\\mathrm{SNTV}}}}}(i_1,\n  \\ldots,i_k) + f_{m,k}^{{{\\mathrm{Perf}}}}(i_1, \\ldots,i_k) = \\alpha_1(i_1) +\n  \\alpha_k(i_k) \\text{.}\n\\end{align*}\nBriefly put, each winning committee under this rule is either an SNTV\nwinning committee or is ranked on top $k$ positions by some voter, and\nit suffices to check all such possibilities (thus, e.g., it is\npossible to compute some winning committee in polynomial time). One\ncan show that this rule is not OWA-based using the same approach as in\nProposition~\\ref{prop:multithreshold-not-owa}.\n\n\n\n\\subsection{Beyond Decomposable Rules}\n\nNaturally, there are also committee scoring rules that go beyond the\nclass of decomposable rules. Below we provide two examples, starting\nwith one inspired by our T-shirt store.\n\n\n\\begin{example}\n  In this example, the store does not want to maximize its direct\n  revenue (i.e., the number of T-shirts sold), but the number of happy\n  customers (in hope of increased future revenue). Let us say that a\n  customer is happy if he or she finds at least two ``good enough''\n  T-shirts or at least one ``great'' T-shirt (recall that ``at least\n  good enough'' shirts are among top $10\\%$ of all available ones, and\n  ``great'' shirts are among the top $1\\%$). Then the store should use\n  the committee scoring function\n $$\n   f_{m,k}(i_1, \\ldots, i_k) = \\max( \\alpha_{0.01m}(i_1),   \\alpha_{0.10m}(i_2)).\n $$\n\\end{example}\n\n\n\n\nWe refer to multithreshold rules with summation replaced by the $\\max$\noperator as \\emph{max-threshold} rules. Using an approach similar to that\nfrom Proposition~\\ref{prop:multithreshold-not-owa}, one can show that\nthere are max-threshold rules that are not decomposable (we omit\ndetails).\n\n\\iffalse\n\n\\begin{proposition}\n  The committee scoring rule based on max-threshold function:\n  \\begin{align*}\n    f_{m,k}(i_1, \\ldots, i_k) = \\max\\Big( \\alpha_{p_1}(i_1),\n    \\alpha_{p_2}(i_2)\\Big) \\textrm{,}\n  \\end{align*}\n  for $p_1, p_2 \\in [m-k-2]$, $p_2 > p_1+1$, is not decomposable.\n\\end{proposition}\n\n\\begin{proof}\nThis can be proved in a variety of ways but the shortest one is to use our powerful tool, Lemma~\\ref{uniqness_of_implementation}. \n  \\end{proof}\n  \n\\begin{proof}\n  We could use our powerful tool, Lemma~\\ref{uniqness_of_implementation}, to prove this proposition.\n  However, in order to demonstrate a variety of techniques useful for proving similar statements, we will not refer to Lemma~\\ref{uniqness_of_implementation} directly, but rather present an alternative approach.\n\n  For the sake of contradiction, let us assume that $\\mathcal{R}_f$ is\n  decomposable, i.e., that there exists $f'$ such that $\\mathcal{R}_f =\n  \\mathcal{R}_{f'}$ and such that:\n  \\begin{align*}\n    f'(\\ell_1, \\ldots, \\ell_k) = \\gamma_1(\\ell_1) +\\gamma_2(\\ell_2)+\n    \\cdots + \\gamma_k(\\ell_k) \\textrm{.}\n  \\end{align*}\n\n  Let $E = (C,V)$ be an election with candidate set $C = \\{c_1,\n  \\ldots, c_m\\}$ and $m!$ voters $v_1, \\ldots, v_{m!}$, one for each\n  possible preference order. By symmetry, any size-$k$ subset $W$ of\n  $C$ is a winning committee of $E$ under $\\mathcal{R}_f$. Let $v$ be an\n  arbitrary fixed voter and let $c_1$ and $c_2$ be candidates that\n  stand on positions $(p_1+1)$ and $(p_2+1)$ in $v$, respectively. Let\n  $D_{x}$ be the sets of the last $x$ candidates, according to\n  $v$. Let $C_1 = D_{k-1} \\cup \\{c_1\\}$, $C_2 = D_{k-1} \\cup \\{c_2\\}$,\n  and $C_3 = D_{k-2} \\cup \\{c_1, c_2\\}$. Let $E'$ be the election that\n  is obtained from $E$ by shifting $c_1$ and $c_2$ one position up. In\n  $E'$ the committees $C_1$, $C_2$, and $C_3$ gained one point (in\n  comparison with $E$) according to the scoring function\n  $f_{m,k}$. Since each other committee could have gained at most one\n  point, we infer that $C_1$, $C_2$, and $C_3$ are winners in $E'$ and\n  that, for instance $D_k$ is not. Thus:\n  \\begin{align*}\n    \\gamma_2(p_2) - \\gamma_2(p_2 + 1) = \\gamma_1(p_1) - \\gamma_2(p_1 + 1) = \n    \\gamma_2(p_2) - \\gamma_2(p_2 + 1) + \\gamma_1(p_1) - \\gamma_2(p_1 + 1) \\textrm{.}\n  \\end{align*}\n  Consequently:\n  \\begin{align*}\n   \\gamma_2(p_2) - \\gamma_2(p_2 + 1) = 0 \\quad\\quad \\text{and} \\quad\\quad \\gamma_1(p_1) - \\gamma_2(p_1 + 1) = 0 \\textrm{.}\n  \\end{align*}\n  However, this would mean that the gain of score of $C_1$ in $E'$ is\n  the same as of $D_k$. From this we infer that $D_k$ is also a winner\n  in $E'$, which gives a contradiction.\n\\end{proof}\n\\fi\n\n\nIn their search for rules between $k$-Borda and ${{{\\beta\\hbox{-}\\mathrm{CC}}}}$,\nFaliszewski et al.~\\cite{fal-sko-sli-tal:c:paths} introduced the class\nof $\\ell_p$-Borda rules, based on the following scoring functions ($p\n\\geq 1$ is a parameter):\n\\begin{align*}\n  f^{{{\\ell_p\\hbox{-}\\mathrm{Borda}}}}_{m,k}(\\row ik) = \\sqrt[p]{\\beta_m^p(i_1) + \\cdots + \\beta_m^p(i_k)} \\text{.}\n\\end{align*}\nWhile the motivation for these rules is the same as for the\n$q$-HarmonicBorda rules, they behave quite differently (see the work\nof Faliszewski et al.~\\cite{fal-sko-sli-tal:c:paths} for a detailed\ndiscussion). \n\n\\begin{corollary}\n  There are committee scoring rules that are not decomposable.\n\\end{corollary}\n\n\nThroughout the rest of the paper, we will not venture outside the\nclass of decomposable rules. However, the above two examples show that\nthere are interesting rules there that also deserve to be studied\ncarefully.\n\n\n\n\n\n\n\\section{Axiomatic Properties of Committee Scoring Rules}\n\\label{sec:axiomatic_properties}\n\nAfter exploring the universe of committee scoring rules from a\nsyntactic (structural) perspective, we now consider axiomatic\nproperties of the observed classes.\nSpecifically, we will use two types of monotonicity\nnotions---non-crossing monotonicity (together with its relaxations)\nand committee enlargement monotonicity---to characterize several of\nthe classes and to gain insights regarding some others.\nIndeed, various monotonicity concepts have long been used in social\nchoice (with Maskin monotonicity~\\cite{maskin1999nash} being perhaps the\nmost important example) and we follow this tradition.\n\n\n\n\n\\iffalse\nHaving observed the universe of committee scoring rules from a\nsyntactic (structural) perspective, let us now focus on axiomatic\nproperties of the observed classes of rules.\nVarious monotonicity concepts have been long used in social choice,\namong which the most important one is Maskin\nmonotonicity~\\cite{maskin1999nash}. \nMonotonicity type concepts will be\nextremely useful in this paper as well when we characterize the\nclasses introduced in the previous section.\n\\fi\n\n \n\n\n\n\\subsection{Non-crossing Monotonicity and Its Relaxations}\n\n\n\\label{sec:noncrossing_monotonicity}\n\nElkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}\nintroduced two monotonicity notions for multiwinner rules, namely\ncandidate monotonicity (recall Section~\\ref{sec:basic}) and\nnon-crossing monotonicity. In the former, we require that if we shift\nforward a candidate from a winning committee in some vote, then this\ncandidate still belongs to some winning committee after the shift, but\npossibly to a different one.\nIn the latter monotonicity notion, we require that the whole committee\nremains winning, but we forbid shifts were members of the winning\ncommittee pass each other (i.e., after a shift none of the committee\nmembers gets worse and some get better). More formally, we have the\nfollowing definition.\n\n\\begin{definition}[Elkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}]\\label{def:noncrossing_monotonicity}\n  A multiwinner rule $\\mathcal{R}$ is non-crossing monotone if\n  for each election $E = (C,V)$ and each $k \\in [|C|]$ the following\n  holds: if $c \\in W$ for some $W \\in \\mathcal{R}(E,k)$, then for each $E'$\n  obtained from~$E$ by shifting $c$ forward by one position in some\n  vote without passing another member of $W$, we still have  $W \\in\n  \\mathcal{R}(E',k)$.\n\\end{definition}\n\nElkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties} have\nshown that weakly separable rules are non-crossing monotone, and we\nwill now show that the converse is also true.  However, before we\nproceed to the proof, we introduce the following notation (that will\nalso be useful in further analysis):\n\\begin{enumerate}\n\\item[] Consider an arbitrary number of candidates $m$ and a size of\n  committee $k \\in [m]$.  For each $t \\in [k]$ and $p \\in\n  [m]$, %\n  let $P_{m, k}(t, p)$ be the set of committee positions from $[m]_k$\n  that have their \\hbox{$t$-th} element equal to $p$ and such that\n  they do not include position $p-1$.  We set $P_{m, k}(p) =\n  \\bigcup_{t \\leq k} P_{m, k}(t, p)$.\n\\end{enumerate}\nFor example, if $m=5$ and $k=3$,\nthen\n$P_{5,1}(1, 4) = \\emptyset$,\n$P_{5,3}(2, 4) = \\{(1, 4, 5), (2, 4, 5)\\}$,\n$P_{5,3}(3, 4) = \\{(1, 2, 4)\\}$,\nand\n$P_{5,3}(4) = P_{5,3}(1,4) \\cup P_{5,3}(2,4) \\cup P_{5,3}(3,4) = \\{(1, 4, 5), (2, 4, 5), (1, 2, 4)\\}$.\n\nIntuitively, $P_{m,k}(t, p)$ is a collection of\ncommittee positions  in which the $t$-th committee member\nstands on position $p$ and where shifting him or her %\nwithout passing another committee\nmember is possible.  \nSimilarly, $P_{m,k}(p)$ is a collection of committee positions \nin which there is \\emph{some} committee member on position $p$ and it\nis possible to shift him to position $p-1$ without passing another\ncommittee member.\n\n\\begin{theorem}\\label{thm:weaklysep}\n  Let $\\mathcal{R}_f$ be a committee scoring rule.\n %\n %\n %\n$\\mathcal{R}_f$ \n  is non-crossing monotone if and only if it is weakly\n  separable. %\n\\end{theorem}\n\\begin{proof}\n  Let $\\mathcal{R}_f$ be a committee scoring rule defined through a family\n  $f=(f_{m,k})_{k\\le m}$ of scoring functions $f_{m,k}\\colon [m]_k\\to\n  \\mathbb{R}$.  Due to the results of Elkind et\n  al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}, it suffices to\n  show that if $\\mathcal{R}_f$ is non-crossing monotone then it is weakly\n  separable. So let us assume that $\\mathcal{R}_f$ is non-crossing monotone.\n\n  Let us fix the number of candidates $m$ and the committee size $k\n  \\in [m]$. Let $E = (C,V)$ be an election with candidate set $C = \\{c_1,\n  \\ldots, c_m\\}$ and collection of voters $V = (v_1, \\ldots, v_{m!})$,\n  with one voter for each possible preference order. By symmetry, every\n  size-$k$ subset $W$ of $C$ is a winning committee %\n  under $\\mathcal{R}_f$.\n\n  Consider an arbitrary integer $p \\in \\{2, \\ldots, m\\}$, two\n  arbitrary (but distinct) committee positions $I = (i_1, \\ldots,\n  i_k)$ and $J = (j_1, \\ldots, j_k)$ from $P_{m,k}(p)$, and an\n  arbitrary vote $v$ from the election.  Let $C(I)$ be the set of\n  candidates that $v$ ranks at positions $i_1, \\ldots, i_k$, and let\n  $C(J)$ be defined analogously for the case of~$J$. Let $E'$ be the\n  election obtained by shifting in $v$ the candidate currently in\n  position $p$ one position up. Finally, let $I'$ and $J'$ be\n  committee positions obtained from $I$ and $J$ by replacing the\n  number $p$ with $p - 1$ (it is possible to do so as $I$ and $J$ are\n  both from $P_{m,k}(p)$). %\n\n\n  Since, by assumption, $\\mathcal{R}_f$ is non-crossing monotone, it must be\n  the case that $C(I)$ and $C(J)$ are winning committees under\n  $\\mathcal{R}_f$ also in election $E'$. The difference of the scores of committee $C(I)$ in elections\n  $E'$ and $E$ is $f_{m,k}(I') - f_{m,k}(I)$, and the difference of the scores of committee\n  $C(J)$ in $E'$ and $E$ is $f_{m,k}(J') - f_{m,k}(J)$. It must be the case that:\n  \\begin{align*}\n     f_{m,k}(I') - f_{m,k}(I) = f_{m,k}(J') - f_{m,k}(J)\\ge 0\\text{.}\n  \\end{align*}\n  However, since the choice of $p$ and the choices of $I$ and $J$\n  within $P_{m,k}(p)$ were completely arbitrary, it must be the case\n  that there is a function $h_{m,k}$ such that for each $p \\in \\{2,\n  \\ldots, m\\}$, each sequence $U \\in P_{m,k}(p)$, and each committee\n  position $U'$ obtained from $U$ by replacing position $p$ with\n  $p-1$, we have:\n  \\begin{align*}\n    h_{m,k}(p-1) = f_{m,k}(U') - f_{m,k}(U) \n  \\end{align*}\n  and the values of $ h_{m,k}$ are non-negative.\n\n  Our goal now is to construct a single-winner scoring  function $\\gamma_{m,k}$\n  such that for each committee position $(\\ell_1, \\ldots, \\ell_k) \\in [m]_k$ it holds that:\n  \\begin{align*}\n    f_{m,k}(\\ell_1, \\ldots, \\ell_k) = \\gamma_{m,k}(\\ell_1) +\\gamma_{m,k}(\\ell_2)+ \\cdots + \\gamma_{m,k}(\\ell_k) \\text{.}\n  \\end{align*}\n  We define  $\\gamma_{m,k}$ by requiring that\n  (a) for\n  each $p \\in \\{2, \\ldots, m\\}$, we have $\\gamma_{m,k}(p-1)-\\gamma_{m,k}(p) = h_{m,k}(p-1)$\n  (so $ \\gamma_{m,k}$ is a non-increasing function), and \n  (b) \n  %\n  %\n  %\n  %\n  %\n  $\\gamma_{m,k}(m) $ is such that $\\gamma_{m,k}(m) +\n  \\gamma_{m,k}(m-1) + \\ldots + \\gamma_{m,k}(m-(k-1)) =\n  f_{m,k}(m-(k-1), \\ldots, m-1, m)$ (so that $\\gamma_{m,k}$ indeed\n  correctly describes the $f_{m,k}$-score of the committee ranked at the $k$\n  bottom positions as a sum of the scores of the candidates).\n\n\n  We fix some committee position $(\\ell_1, \\ldots, \\ell_k)$ from\n  $[m]_k$.\n  We know that, due to the choice of $\\gamma_{m,k}(m)$, for $R = (r_1, \\ldots, r_k) = (m-k+1, \\ldots, m)$ it does\n  hold that $f_{m,k}(r_1, \\ldots, r_k) = \\gamma_{m,k}(r_1) + \\cdots +\n  \\gamma_{m,k}(r_k)$. \n  Now we can see that this property also holds for $R' = (r_1-1, r_2,\n  \\ldots, r_k)$. The reason is that\n  \\begin{align*}\n  \\gamma_{m,k}(m-k) - \\gamma_{m,k}(m-k+1) = h_{m,k}(m-k) = f_{m,k}(R') - f_{m,k}(R) \\text{.}\n  \\end{align*}\n  Thus, for $R'$, we have $f_{m,k}(R') = \\gamma_{m,k}(r_1-1) + \\gamma_{m,k}(r_2) + \\cdots + \\gamma_{m,k}(r_k)$.\n  We can proceed in this way, shifting the top member of the\n  committee up by sufficiently many positions, to obtain $R'' = (\\ell_1,\n  r_2, \\ldots, r_k)$ and (by the same argument as above) have:\n  \\begin{align*}\n  f_{m,k}(R'') = \\gamma_{m,k}(\\ell_1) + \\gamma_{m,k}(r_2) + \\cdots + \\gamma_{m,k}(r_k) \\text{.}\n  \\end{align*}\n  Then we can do the same to position $r_2$, and keep decreasing it\n  until we get $\\ell_2$. Then the same for the third position, and so on,\n  until the $k$-th position. Finally, we get:\n\\begin{align*}\n  f_{m,k}(\\ell_1, \\ldots, \\ell_k) = \\gamma_{m,k}(\\ell_1) + \\cdots + \\gamma_{m,k}(\\ell_k) \\text{.}\n\\end{align*}\n  This \n  proves our claim and \n completes the proof.\n\\end{proof}\n\n\n\nNon-crossing monotonicity is particularly natural when we seek\ncommittees of individually excellent candidates (for example, when we\nseek finalists of a competition or where we are interested in some shortlisting\ntasks~\\cite{elk-fal-sko-sli:j:multiwinner-properties,fal-sko-sli-tal:b:multiwinner-trends}). Indeed,\nif we have a committee $W$ where we view each member as good enough to\nbe selected, and one of the members of $W$ improves its performance\nwithout hurting the performance of any of the others, then it is\nperfectly natural to expect that all members of $W$ are still good\nenough to be selected.\nTheorem~\\ref{thm:weaklysep} justifies axiomatically that if we are\nlooking for a committee scoring rule for selecting individually\nexcellent candidates then we should look within the class of weakly\nseparable rules.\nIn fact, Elkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}\npointed out that we should focus on separable rules only, and we will\nprovide axiomatic justification for this view in\nSection~\\ref{sec:com_mon_sep_rules}.\n\n\n\n\n\n\n\\iffalse\nTheorems~\\ref{thm:weaklysep}~and~\\ref{thm:committee_monot_and_sep}\nhave yet another interesting consequence: they imply that committee\nmonotonicity of committee scoring rules implies their non-crossing\nmonotonicity. This is somehow surprising, since the two variants of\nmonotonicity seem almost unrelated: one describes how the result of an\nelection changes if we increase the size of the committee and the\nother one---what happens when we shift a member of a winning committee\nin a preference relation of a voter.  \\fi\n\n\\subsubsection{Prefix Monotonicity and Decomposable Rules}\\label{sec:other_noncrossing_monotonicity}\n\nBased on the idea of non-crossing monotonicity, we can define other\nsimilar notions. In this section we introduce and discuss one of them,\nwhich we call {\\em prefix monotonicity}.  Intuitively, if a rule\nsatisfies the prefix monotonicity condition, then shifting forward a\ngroup of highest-ranked members of a winning committee within a given\nvote never prevents this committee from winning.\n\n\\begin{definition}%\n\\label{def:prefix_monotonicity}\nA multiwinner rule $\\mathcal{R}$ satisfies \\emph{$t$-prefix monotonicity},\n$0 \\leq t \\leq k$, if for each election $E = (C,V)$ and each committee\nsize $k$, $t \\leq k \\leq |C|$ the following holds: For every $W \\in\n\\mathcal{R}(E,k)$, and every $E'$ obtained from~$E$ by shifting in some vote\nthe top-ranked $t$ members of $W$ (according to this vote), then we have\nthat $W \\in \\mathcal{R}(E',k)$. We say that $\\mathcal{R}$ satisfies \\emph{prefix\n  monotonicity} if it satisfies $t$-prefix monotonicity for every $t\n\\in {{\\mathbb{N}}}$.\\footnote{Note that $0$-prefix monotonicity is an empty concept; as such, every rule satisfies it.}\n\\end{definition}\n\n\nPrefix monotonicity is a relaxation of non-crossing monotonicity and,\nin consequence, all weakly separable rules satisfy it.  In the\nremaining part of this section we will show that only decomposable\nrules can be prefix-monotone (and mostly, though not only, those based\non convex functions; we will explain this in the further part of this\nsection).  Before we prove this statement, let us first prove one more\ntechnical lemma, which will allow us to reuse some of the reasoning\nlater on. The lemma uses the same high-level idea as the first part of\nthe proof of Theorem~\\ref{thm:weaklysep}, yet it is more involved and\ndiffers in a number of details.  (Recall that $P_{m, k}(t, p)$ used in\nthe statement of the lemma was defined right before\nTheorem~\\ref{thm:weaklysep}.)\n\n\n\\begin{lemma}\\label{lem:t_prefix_monotonicity}\n  Let $\\mathcal{R}_f$ be a committee scoring rule and let $t$ be an integer\n  such that for each $x \\in [t]$ this rule is $x$-prefix\n  monotone.\n  Then, for every number of candidates $m$ and size of the committee~$k$,\n  there exists a function $h_t$ such that for each $p\\in [m]$,\n  each $U \\in P_{m, k}(t, p)$, with $p\\ge t$, and committee position\n  $U'$ obtained from $U$ by replacing position $p$ with $p-1$, we\n  have:\n\\begin{equation*}\n\\label{functionh_t}\n      h_t(p-1) = f(U') - f(U) \\geq 0 \\textrm{.}\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n  Consider an arbitrary integer $p \\in [m]$ %\n  and two arbitrary (but distinct) committee positions $I = (i_1,\n  \\ldots, i_k)$ and $J = (j_1, \\ldots, j_k)$ from $P_{m,k}(t, p)$,\n  such that $I$ and $J$ have the first $t$ elements equal. Let $I'$\n  and $J'$ be the committee positions obtained from $I$ and $J$,\n  respectively, by replacing the element $p$ with $p-1$ (by the choice\n  of $I$ and $J$, it is possible to do so). Let $I_1$ and $J_1$ be the\n  committee positions obtained from $I$ and $J$, respectively, by\n  increasing every element with value lower than $p$ by one (in particular, when\n  $t=1$ we have $I_1=I$ and $J_1=J$). The way the sequences $I'$ and\n  $I_1$ are constructed from $I$ is depicted in\n  Figure~\\ref{fig:shifting} (in this example $t=4$) and is presented\n  below, also for $J'$ and $J_1$ ($i_t = p$, $i_{t-1} \\leq p-2$, $j_1\n  = i_1, \\ldots, j_{t} = i_{t}$):\n  \\begin{align*}\n    &I = (i_1, i_2 \\ldots, i_{t-1}, p, i_{i+1}, \\ldots, i_k), &  & J = (i_1, i_2 \\ldots, i_{t-1}, p, j_{i+1}, \\ldots, j_k), \\\\\n    &I' = (i_1, i_2 \\ldots, i_{t-1}, p-1, i_{i+1}, \\ldots, i_k), &  & J' = (i_1, i_2 \\ldots, i_{t-1}, p-1, j_{i+1}, \\ldots, j_k), \\\\\n    &I_1 = (i_1+1, i_2+1 \\ldots, i_{t-1}+1, p, i_{i+1}, \\ldots, i_k), &  & J_1 = (i_1+1, i_2+1 \\ldots, i_{t-1}+1, p, j_{i+1}, \\ldots, j_k).\n  \\end{align*}\n\n  \\begin{figure}[tb]\n    \\begin{center}\n\t\\includegraphics[scale=0.60]{proof_shifting}\n      \\end{center}\n      \\caption{An example showing how the sequences of positions $I$,\n        $I'$ and $I_1$ from the proof of\n        Lemma~\\ref{lem:t_prefix_monotonicity} are related.}\n      \\label{fig:shifting}\n    \\end{figure}\n\n    As in the proof of Theorem~\\ref{thm:weaklysep}, we construct an\n    election $E = (C,V)$ with candidate set $C = \\{c_1, \\ldots, c_m\\}$\n    and $m!$ voters $v_1, \\ldots, v_{m!}$, one for each possible\n    preference order. By symmetry, every size-$k$ subset $W$ of $C$ is\n    a winning committee of $E$ under $\\mathcal{R}_f$. Further, consider an\n    arbitrary vote $v$ from the election; let $C(I_1)$ and $C(J_1)$ be\n    the committees that $v$ ranks on positions $I_1$ and $J_1$,\n    respectively. As all other committees, $C(I_1)$ and $C(J_1)$ are\n    winning in $E$. Let us shift in $v$ by one position forward each\n    candidate from $C(I_1)$ that stands on a position with value lower\n    than $p$. After such an operation, committee $C(I_1)$ will have\n    position $I$ and committee $C(I_1)$ will have position $J$. Since,\n    by assumption, $\\mathcal{R}_f$ is $(t-1)$-prefix-monotone, and exactly\n    $t-1$ candidates have changed positions, it must be the case that\n    $C(I_1)$ and $C(J_1)$ are still winning under $\\mathcal{R}_f$. \n    It must be the case that:\n    \\begin{align*}\n      f(I) - f(I_1) = f(J) - f(J_1)\\ge 0\\textrm{.}\n    \\end{align*}\n    By a similar reasoning, using the fact that $\\mathcal{R}_f$ is\n    $t$-prefix-monotone, we also conclude that:\n    \\begin{align*}\n      f(I') - f(I_1) = f(J') - f(J_1)\\ge 0\\textrm{.}\n    \\end{align*}\n    From the two above equalities we get that (the final inequality follows because\n    $J'$ dominates $J$):\n    \\begin{equation}\\label{eq:pm:1}\n      f(I') - f(I) = f(J') - f(J) \\geq 0 \\textrm{.}\n    \\end{equation}\n\n    Recall that in the above equality $I$ and $J$ have the first $t-1$\n    elements equal. We would like to obtain the same relation even if\n    the prefixes of $I$ and $J$ differ. Thus, now we will show how to\n    change one element in the prefix of $I$ and $I'$ to an element\n    different by one, so that the equality still holds. By repeating\n    this operation sufficiently many times, we can conclude that the\n    equality does not depend on the prefix of $I$. For the sake of\n    concreteness, we will show how to change $i_{t-1}$ to $i_{t-1}+1$\n    in the prefix of $I$ (this assumes that $i_{t-1}+1 < p-1$). A\n    change of any other element in the prefix can be performed\n    analogously. We proceed as follows.  Let us define:\n  \\begin{align*}\n    &I_{\\mathrm{new}} = (i_1, i_2 \\ldots, i_{t-1}+1, p, i_{i+1}, \\ldots, i_k), &  & L_{\\mathrm{new}} = (i_1, i_2 \\ldots, i_{t-1}+1, p-1, i_{i+1}, \\ldots, i_k), \\\\\n    &I_{\\mathrm{new}}' = (i_1, i_2 \\ldots, i_{t-1}, p, i_{i+1}, \\ldots, i_k), &  & L_{\\mathrm{new}}' = (i_1, i_2 \\ldots, i_{t-1}, p-1, i_{i+1}, \\ldots, i_k).\n  \\end{align*}\n  In particular, observe that $I_{\\mathrm{new}}' = I$ and that\n  $L_{\\mathrm{new}}' = I'$.  Similarly as before, by using\n  $(t-1)$-prefix-monotonicity and $(t-2)$-prefix-monotonicity, we\n  obtain that:\n    \\begin{equation}\\label{eq:pm:2}\n      f(I_{\\mathrm{new}}') - f(I_{\\mathrm{new}}) = f(L_{\\mathrm{new}}') - f(L_{\\mathrm{new}}) \\geq 0 \\textrm{.}\n    \\end{equation}\n    Adding  inequalities~\\eqref{eq:pm:1} and~\\eqref{eq:pm:2}, we get:\n    \\begin{align*}\n      f(I') - f(I) + f(I_{\\mathrm{new}}') - f(I_{\\mathrm{new}}) =  f(J') - f(J)  + f(L_{\\mathrm{new}}') - f(L_{\\mathrm{new}}) \\textrm{.}\n    \\end{align*}\n    which is equivalent to:\n    \\begin{align*}\n     f(L_{\\mathrm{new}}) - f(I_{\\mathrm{new}}) =  f(J') - f(J)   \\textrm{.}\n    \\end{align*}\n    However, we can see that $L_{\\mathrm{new}}$ and $I_{\\mathrm{new}}$\n    are simply $I'$ and $I$ where one element of the prefix,\n    $i_{t-1}$, is replaced with $i_{t-1}+1$. By our previous\n    discussion, it follows that we can prove that $f(I') - f(I) =\n    f(J') - f(J)$ even if $I$ and $J$ have different prefixes.\n\n    Since the choice of $p$, $I$, and $J$ (within $P_{m,k}(t, p)$) is\n    completely arbitrary, it must be the case that for each $t$ there\n    exists a function $h_t$ such that for each $p \\in \\{t+1, \\ldots,\n    m\\}$, each sequence $U \\in P_{m,k}(t, p)$, and each sequence $U'$\n    obtained from $U$ by replacing position $p$ with $p-1$, we have:\n    \\begin{equation*}\n      h_t(p-1) = f(U') - f(U) \\geq 0. \n    \\end{equation*}\n    The final inequality follows from equation~\\eqref{eq:pm:1}.\n\\end{proof}\n\nWe are ready to show that only decomposable rules can satisfy prefix-monotonicity.\n\n\\begin{theorem}\\label{thm:prefixMonAndDecomposable}\n  Let $\\mathcal{R}_f$ be a committee scoring rule. %\n  If $\\mathcal{R}_f$ is prefix-monotone then it must be decomposable.\n\\end{theorem}\n\n\n\\begin{proof}\n  Let $f = (f_{m,k})_{k \\le m}$ be a family of committee scoring\n  functions such that $\\mathcal{R}_f$ is prefix-monotone. Let us fix the\n  number of candidates $m$ and the committee size $k$.  For each $t\n  \\in [k]$, let $h_t$ be the function constructed in\n  Lemma~\\ref{lem:t_prefix_monotonicity}.\n\n  Our goal is to provide single-winner scoring functions\n  $\\gamma^{(1)}_{m,k}, \\ldots, \\gamma^{(k)}_{m,k}$ such that for each\n  committee position $(\\ell_1, \\ldots, \\ell_k)$ we have:\n  \\begin{equation}\\label{eq:fpm:1}\n  f_{m,k}(\\ell_1, \\ldots, \\ell_k) = \\gamma^{(1)}_{m,k}(\\ell_1) +\\gamma^{(2)}_{m,k}(\\ell_2)+\n    \\cdots + \\gamma^{(k)}_{m,k}(\\ell_k) \\textrm{.}\n  \\end{equation}\n  To this end, for each $t \\in [k]$,  we define\n   %\n   %\n  $\\gamma^{(t)}_{m,k}: \\{t, \\ldots, m - k + t\\}\n  \\rightarrow \\mathbb{R}$ so that:\\footnote{Formally, $\\gamma_t$ must\n    be defined on $[m]$ but it actually never has a chance to\n    calculate values $\\gamma_t(s)$, where $s<t$ or $s>m - k + t$, so\n    these values of $\\gamma_t$ can be chosen arbitrarily.}\n  \\begin{enumerate}\n  \\item The values $\\gamma_k(m), \\gamma_{k-1}(m - 1), \\ldots,\n    \\gamma_1(m-(k-1))$ are such that $ f(m-(k-1), \\ldots, m-1, m) =\n    \\gamma_k(m) + \\gamma_{k-1}(m - 1) + \\ldots + \\gamma_1(m-(k-1))$\n    (so equation~\\eqref{eq:fpm:1} holds for the committee position\n    where the candidates are ranked at the $k$ bottom positions).\n  \\item For each $p \\in \\{t+1, \\ldots, m - k + t\\}$, we have\n    $\\gamma^{(t)}_{m,k}(p-1)-\\gamma^{(t)}_{m,k}(p) = h_t(p-1)$.\n    (By Lemma~\\ref{lem:t_prefix_monotonicity}, we have $h_t(p-1) \\geq 0$, so \n    $\\gamma^{(t)}_{m,k}$ is nonincreasing.)\n\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  \\end{enumerate}\n  There may be many different ways to define functions\n  $\\gamma^{(1)}_{m,k}, \\ldots, \\gamma^{(k)}_{m,k}$ satisfying the\n  above conditions and we choose one of them arbitrarily.\n\n  To show that equation~\\eqref{eq:fpm:1} holds, we use the same\n  approach as in the second part of the proof of\n  Theorem~\\ref{thm:weaklysep}.  Specifically, we note that if equation\n  \\eqref{eq:fpm:1} holds for some committee position $R = (r_1, \\ldots,\n  r_k)$ and $R' = (r_1, \\ldots, r_t-1, \\ldots, r_k)$ also is a valid\n  committee position for some $t \\in [k]$, then (by definition of $h_t$) we have:\n  \\begin{align*}\n    f_{m,k}(R') & = f_{m,k}(R) + h_t(r_t-1) \\\\ \n     & = \\gamma^{(1)}_{m,k}(r_1) + \\cdots + \\gamma^{(t-1)}_{m,k}(r_{t-1})+ \\bigg( \\gamma^{(t)}_{m,k}(r_{t})+ h_t(r_t-1) \\bigg) + \\gamma^{(t+1)}_{m,k}(r_{t+1}) +  \\cdots + \\gamma^{(k)}_{m,k}(r_k)  \\\\\n     & = \\gamma^{(1)}_{m,k}(r_1) + \\cdots + \\gamma^{(t-1)}_{m,k}(r_{t-1})+ \\gamma^{(t)}_{m,k}(r_{t}-1) + \\gamma^{(t+1)}_{m,k}(r_{t+1}) +  \\cdots + \\gamma^{(k)}_{m,k}(r_k) \\textrm{.} \n  \\end{align*}\n  Since equation~\\eqref{eq:fpm:1} holds for committee position\n  $(m-(k-1), \\ldots, m)$, applying the above argument inductively\n  proves that equation~\\eqref{eq:fpm:1} holds for all committee\n  positions.\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  \\end{proof}\n\n  Theorem~\\ref{thm:prefixMonAndDecomposable} states that\n  decomposability is a necessary condition for a committee scoring\n  rule to be prefix-monotone. However, as the following example shows,\n  it is not sufficient.\n\n\\begin{example}\n  The $k$-Approval Chamberlin--Courant rule ($\\alpha_k$-CC), defined by committee\n  scoring functions $f^{{{\\alpha_k\\hbox{-}\\mathrm{CC}}}}_{m,k}(i_1, \\ldots, i_k) =\n  \\alpha_k(i_1)$, is a decomposable\n  rule %\n  that is not prefix-monotone. Indeed, consider $k=2$ and an election\n  with four candidates $\\{a, b, c, d\\}$ that includes one vote for\n  each possible ranking of these four candidates. This election\n  contains $4! = 24$ votes and, in particular, vote $v\\colon a \\succ b\n  \\succ c \\succ d$.  By the symmetry of the rule we see that for such\n  election each committee is winning, including $W = \\{b,c\\}$ and $W'\n  = \\{c,d\\}$. If $\\alpha_k$-CC were prefix-monotone, then shifting $b$\n  and $c$ by one position forward in $v$ (to obtain $b \\succ c \\succ a\n  \\succ d$) should keep $W$ winning. Doing so, however, does not change\n  the score of $W$ and increases the score of $W'$, so $W$ no longer\n  wins. This shows that $\\alpha_k$-CC is not prefix-monotone.\n\\end{example} \n\nOn the other hand,\nif we assume that the single-winner scoring functions underlying a\ndecomposable rule are, in a certain sense, convex, then we obtain a\nsufficient condition for this rule to be prefix-monotone.\n\n \n\\begin{proposition}\\label{thm:prefixMonAndDecomposable2}\n  Let $\\mathcal{R}_f$ be a decomposable committee scoring rule defined\n  through a family of scoring functions\n  $\n    f_{m,k}(i_1, \\ldots, i_k) = \\gamma^{(1)}_{m,k}(i_1) +\\gamma^{(2)}_{m,k}(i_2)+\n    \\cdots + \\gamma^{(k)}_{m,k}(i_k) \\textrm{,}\n  $\n  where $\\gamma = (\\gamma^{(t)}_{m,k})_{t \\leq k \\leq m}$ is a family\n  of single-winner scoring functions.  A sufficient condition for\n  $\\mathcal{R}_f$ to be prefix-monotone is that for each $m$ and each $k \\in\n  [m]$ we have that:\n  \\begin{enumerate}\\item[(i)] for each $i \\in [k]$ and each $p, p' \\in [m-1]$, $p <\n  p'$, it holds that:\n  \\begin{equation}\n  \\label{first_condition}\n    \\gamma^{(i)}(p) - \\gamma^{(i)}(p+1) \\geq \\gamma^{(i)}(p') - \\gamma^{(i)}(p'+1) \\text{, and} \n  \\end{equation} \n  \\item[(ii)] for each $i, j \\in [k]$, $j>i$, and each $p \\in [m]$, $j \\leq p < m - (k-i)$, \n  it holds that \n  \\begin{equation}\n  \\label{second_condition}\n    \\gamma^{(i)}(p) - \\gamma^{(i)}(p+1) \\geq \\gamma^{(j)}(p) - \\gamma^{(j)}(p+1) \\text{.}\n  \\end{equation}\n  \\end{enumerate}\n\\end{proposition}\n\nIntuitively, condition (i) says that the functions in the  family $\\gamma$\nare convex, and condition (ii) says that, for each $m$ and $k$, if $i\n< j$ then $\\gamma^{(i)}_{m,k}$ decreases not faster than\n$\\gamma^{(j)}_{m,k}$. %\n                      %\n\n\n\\begin{figure}[tb]\n   \\hspace{-3cm}\\includegraphics[scale=0.60]{proof_shifting2}\n   \\caption{Illustration of the notation used in Proposition~\\ref{thm:prefixMonAndDecomposable2}.}\n   \\label{fig:shifting2}\n\\end{figure}\n\n\n\\begin{proof}%\n  Let $\\mathcal{R}_f$ be defined as in the statement of the proposition and\n  fix the number of candidates $m$ and the committee size $k$.\n  Consider an election $E$ where a committee $W$ is a winner. Let $j$\n  be a number from $[k]$ and let $E'$ be an election obtained from $E$\n  by shifting forward by one position each of the first $j$ members of\n  $W$ in some vote $v$. We will show that $W$ is a winning committee\n  in~$E'$. Let $(\\ell_1, \\ldots, \\ell_k)$ be the committee position of\n  $W$ in $v$ (in election $E$). In comparison with $E$, in~$E'$ the\n  score of $W$ is increased by:\n  \\begin{align*}\n    \\sum_{t=1}^j \\big(\\gamma_t(\\ell_t - 1) - \\gamma_t(\\ell_t)\\big)\n    \\textrm{.}\n  \\end{align*}\n  Let us now assess by how much the score of some other committee,\n  $W'$, can increase. Let us fix $t\\in[j]$, and let $c_t$ be the\n  candidate standing at position $\\ell_t$ in $v$ (in particular, $c_t\\in W$). If $c_t \\notin W'$,\n  then shifting $c_t$ one position up has no positive effect on the\n  score of $W'$. Consider the case when $c_t \\in W'$. Let $x_t$ denote\n  the position of $c_t$ within $W'$ according to $v$ (for instance, if\n  $c_t$ is the most preferred among members of $W'$ in $v$, then $x_t =\n  1$). This notation is illustrated in Figure~\\ref{fig:shifting2}.\n  Now, we consider two cases:\n  \n  \\begin{description}\n  \\item[Case 1 ($\\boldsymbol{x_t \\geq t}$).] The condition~\\eqref{second_condition} implies that:\n    \\begin{align*}\n \\gamma_{x_t}(\\ell_t - 1) - \\gamma_{x_t}(\\ell_t) \\leq \\gamma_t(\\ell_t - 1) - \\gamma_t(\\ell_t) \\text{.}\n    \\end{align*}\n    Thus the increase of the score of $W'$ due to shifting $c_t$ one\n    position up is not greater than the increase of the score of $W$\n    due to shifting $c_t$ one position up. We assign $c_t$ in $W$ to\n    $c_t$ in $W'$; this assignment is shown with a bold dashed arrow\n    in Figure~\\ref{fig:shifting2}) and, intuitively, it means that the\n    increase of the score of $W$ due to shifting $c_t$ ``compensates\n    for'' the increase of the score of $W'$ due to shifting the\n    assigned candidate.\n\n    \\item[Case 2 ($\\boldsymbol{x_t < t}$).] Now, we observe that due to \\eqref{first_condition} we have:\n    \\begin{align*}\n     \\gamma_{x_t}(\\ell_t - 1) - \\gamma_{x_t}(\\ell_t) \\leq\n      \\gamma_{x_t}(\\ell_{x_t} - 1) -\n      \\gamma_{x_t}(\\ell_{x_t}).\n    \\end{align*}\n    Thus the increase of the score of $W'$ due to shifting $c_t$ one\n    position up is not greater than the increase of the score of $W$\n    due to shifting  the candidate at position $\\ell_{x_t} <\n    \\ell_t$, call such a candidate $c$, one position up. We assign\n    $c_t$ in $W'$ to $c$ in $W$; this assignment is depicted with a\n    solid arrow in Figure~\\ref{fig:shifting2}.\n  \\end{description}\n\n  From the above reasoning we see that for each $t\\in[j]$ the increase\n  of the score of $W'$ due to shifting $c_t$ one position up is no\n  greater than the increase of the score of $W$ due to shifting some\n  other candidate $c_{r}$ ($r \\leq t$) one position up; in such case\n  we say that $c_{r}$ is assigned to $c_t$ and that $c_{r}$\n  compensates for $c_t$. Further, we note that each candidate $c_{t}\n  \\in W'$ is assigned to a different ``compensating'' candidate (see\n  Figure~\\ref{fig:shifting2} and consider how the assignment is\n  defined, starting from the highest values of~$t$ and decreasing $t$\n  one by one).  We conclude that the score of $W'$ increases in $E'$\n  by a value that is not greater than the increase of the score of\n  $W$. Since $W'$ was chosen arbitrarily, we get that $W$ is a winner\n  in $E'$, which completes the proof.\n\\end{proof}\n\n\nAs far as applications of multiwinner voting goes, prefix monotonicity\ndoes not seem to have as clear-cut interpretation as non-crossing\nmonotonicity. Nonetheless, in the next section we will see how its\nrelaxed variant is useful in characterizing representation-focused\nrules (and how this characterization can be interpreted in the context\nof diversity-oriented committee elections).\n\n\n\n\n\n\\subsubsection{Top-Member Monotonicity and Representation-Focused  Rules}\n\\label{sec:top-member_monotonicity}\n\nOur goal in this section is to provide an axiomatic characterization\nof representation-focused rules. The first tool that we employ for\nthis task is $1$-prefix monotonicity (recall\nDefinition~\\ref{def:prefix_monotonicity}), which we rename as\n\\emph{top-member monotonicity}. Intuitively, top-member monotonicity\nrequires that if in some vote $v$ we shift forward the highest-ranked\nmember of a given winning committee, then this committee remains\nas a winning one.\nSince top-member monotonicity is a relaxed variant of\nnon-crossing monotonicity (and of prefix monotonicity), it is satisfied\nby all weakly separable rules and alone is insufficient to\ncharacterize representation-focused rules.\nThus we will also use the notion of narrow-top consistency, defined\nbelow (which, in fact, is a relaxed form of the solid coalitions\nproperty of Elkind et\nal.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}, itself motivated\nby a much stronger notion of Dummet~\\cite{dum:b:voting}).\n\n\n\\begin{definition} %\n  A multiwinner rule $\\mathcal{R}$ satisfies \\emph{narrow-top consistency}\n  if for each election $E = (C,V)$ and each $k \\in [|C|]$ the\n  following holds: If there exists a set of at most $k$ candidates\n  $S$, such that each voter in $V$ ranks some candidate from $S$\n  first and each member of $S$ is ranked first by some voter,\n  then for each $W \\in \\mathcal{R}(E,k)$ it holds that $S \\subseteq\n  W$.\n\\end{definition}\n\nTogether, top-member monotonicity and narrow-top consistency exactly\ncharacterize the class of representation-focused rules (within the\nclass of committee scoring rules). We prove this result formally\nbelow, and then we explain the roles of both our axioms intuitively.\n\n\n\n\n\n\\begin{theorem}\\label{thm:representation_focused}\n  Let $\\mathcal{R}_f$ be a committee scoring rule.\n %\n %\n %\n %\n  $\\mathcal{R}_f$ is representation-focused if and only if it \n  satisfies top-member monotonicity and narrow-top consistency. \n\\end{theorem}\n\n\\begin{proof}\n  It is apparent that each representation-focused rule satisfies both\n  top-member monotonicity and narrow-top consistency.  Suppose that\n  $\\mathcal{R}_f$ is a committee scoring rule, defined through a family\n  $f=(f_{m,k})_{k\\le m}$ of scoring functions $f_{m,k}\\colon [m]_k\\to\n  \\mathbb{R}_{+}$, that satisfies these two properties. We will show that\n  $\\mathcal{R}_f$ is representation-focused.\n\n  Let us fix the number of candidates $m$ and the committee size\n  $k$. Since $\\mathcal{R}_f$ satisfies $1$-prefix monotonicity (top-member\n  monotonicity), by Lemma~\\ref{lem:t_prefix_monotonicity} we have that\n  there exists a function $h$ such that for each $p\\in [m]$, each $U\n  \\in P_{m, k}(1, p)$ and the committee position $U'$, obtained from\n  $U$ by replacing position $p$ with $p-1$, we have $h(p-1) = f(U') -\n  f(U)\\ge 0$.\n\n  Let $I = (i_1, \\ldots, i_k)$ and $J = (j_1, \\ldots, j_k)$ be such\n  that $i_1 = j_1$. We will show that $f_{m,k}(I) = f_{m,k}(J)$, which is\n  sufficient to prove that $\\mathcal{R}_f$ is representation focused. For\n  the sake of contradiction, let us assume that this is not the case,\n  and without loss of generality, let us assume that $f_{m,k}(I) >\n  f_{m,k}(J)$. There exists a positive integer $\\eta$ such that $\\eta f_{m,k}(I) >\n  \\eta f_{m,k}(J) + k f_{m,k}(1, \\ldots, k)$.\n\n  Let us fix a vote $v$ and let $W$ and $W'$ denote the committees\n  that stand in $v$ on positions $I$ and $J$, respectively. Note that\n  they have a common member $d$ who stands on position $i_1 = j_1$ and\n  is highest ranked by $v$ in both committees. Consider an election\n  $E$ with $\\eta$ copies of vote $v$ and with $k$ votes such that for\n  each candidate $c \\in W'$ there is one vote who ranks $c$ first and\n  the remaining candidates in some fixed, arbitrary way. In this\n  election the score of $W$ is at least equal to $\\eta f(I)$ and the\n  score of $W'$ is at most equal to $\\eta f(J) + k f(1, \\ldots k)$.\n  Thus the score of $W$ is higher than the score of $W'$. If $i_1 =\n  j_1=1$ we get a contradiction immediately since by the narrow-top\n  consistency $W'$ must be winning.\n\n  If $i_1 = j_1\\ne 1$, we construct election $E'$ by shifting, in each\n  copy of $v$, the candidate $d\\in W\\cap W'$ to the top position. In\n  comparison to $E$, the scores of committees $W$ and $W'$ in $E'$\n  increase by the same value $\\eta\\big(h(1) - h(i_1)\\big)$. As a\n  result, $W$ has a higher score than $W'$ also in $E'$. This,\n  however, contradicts narrow-top consistency, since all top positions\n  in this profile are occupied by candidates from $W'$. This proves\n  that $f(I) = f(J)$, and completes the reasoning.\n\\end{proof}\n\n\nLet us now explain intuitively the interplay between top-member monotonicity and\nnarrow-top consistency in the characterization of representation-focused\nrules. If we applied similar reasoning as we used in the\nproofs of Theorems~\\ref{thm:weaklysep}\nand~\\ref{thm:prefixMonAndDecomposable} to top-member monotonicity,\nthen we could show that if a committee scoring rule $\\mathcal{R}_f$ is\ntop-member monotone then its scoring functions are of the form:\n\\begin{equation}\\label{eq:tmm}\n  f_{m,k}(i_1, \\ldots, i_k) = \\gamma_{m,k}(i_1) + g_{m,k-1}(i_2, \\ldots, i_k) \\textrm{,}\n\\end{equation}\nwhere $\\gamma = (\\gamma_{m,k})_{k \\leq m}$ is a family of\nsingle-winner scoring functions and $g = (g_{m,k-1})_{k-1 \\le m}$ is a\nfamily of committee scoring functions. Requiring that $\\mathcal{R}_f$ is\nalso narrow-top consistent ensures that the functions $g_{m,k}$ are,\nin fact, constant, and in consequence gives that $\\mathcal{R}_f$ is\nrepresentation-focused. Since all decomposable committee scoring rules\nare already of the form presented in equation~\\eqref{eq:tmm} (and, in\nfact, they are of a far more restricted form), we have the following\ncorollary (the proof follows directly from the preceding reasoning,\nbut straightforward calculations also show it directly; we omit these details).\n\\begin{corollary}\n  If a decomposable committee scoring rule is narrow-top consistent\n  then it is representation focused.\n\\end{corollary}\n\n\n\nRepresentation-focused rules generally, and the Chamberlin Courant\nrule specifically, are often considered in the context of selecting\ndiverse\ncommittees~\\cite{elk-fal-sko-sli:j:multiwinner-properties,fal-sko-sli-tal:b:multiwinner-trends}. While\nthere is no clear definition of what a ``diverse committee'' is,\nresearchers often use this term intuitively, to mean that as many\nvoters as possible can find a committee member that they rank highly\n(if a voter $v$ ranks some committee member $c$ highly, then we could\nsay that $c$ ``covers'' the views of $v$, so some authors speak of\n``diversity\/coverage''; see the works of Ratliff and\nSaari~\\cite{rat-sar:j:diverse-committees}, Bredereck et\nal.~\\cite{bre-fal-iga-lac-sko:c:diverse-committees}, Celis et\nal.~\\cite{cel-hua-vis:diverse-committees}, and Izsak et\nal.~\\cite{interclass} for a different view regarding diverse\ncommittees).\nTheorem~\\ref{thm:representation_focused} justifies the use of\nrepresentation-focused rules to seek committees that are diverse in\nthis sense. Indeed, if there is a committee such that every voter\nranks one of its members on top, then certainly this committee\n``covers'' the ``diverse'' views of all the voters; narrow-top\nconsistency ensures that this committee is selected. On the other\nhand, if there is a committee $W$ and we agree that it ``covers'' the\nviews of sufficiently many voters, then if some voter ranks his or her\nhighest-ranked committee member even higher (i.e., this voter realizes\nthat the candidate represents his or her views even better), then\ncertainly we should still view $W$ as ``covering'' the views of\nsufficiently many voters; this is ensured by top-member monotonicity.\n\n\n\n\n\n\\subsection{Committee Enlargement Monotonicity and Separable Rules}\n\\label{sec:com_mon_sep_rules}\n\nIn this section we consider the committee enlargement monotonicity\naxiom. While it is markedly different from the notions that we used in\nthe previous sections, it still has a clear monotonicity flavor:\nInformally speaking, it requires that if $W$ is a size-$k$ winning\ncommittee for some election, then there also is a size-$(k+1)$ winning\ncommittee for this election that includes all the members of $W$ (the\nactual definition is more complicated due to possible ties; its exact\nform is due to Elkind et\nal.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}, but it was\nalready studied by Barber\\`a\nand~Coelho~\\cite{bar-coe:j:non-controversial-k-names} for resolute\nmultiwinner rules, and in the literature on apportionment rules it is\nwell known as \\emph{house monotonicity}~\\cite{Puke14a,\n  bal-you:b:polsci:representation}).\n\n\n\n\\begin{definition}[Elkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}]\n  A multiwinner election rule $\\mathcal{R}$ satisfies \\emph{committee monotonicity} if for each $m$ and $k$, $1 \\leq k < m$, and for each election $E$ the following two conditions hold: \n  \\begin{itemize}\n  \\item[(1)] for each $W \\in \\mathcal{R}(E,k)$ there exists $W' \\in \\mathcal{R}(E,k+1)$ such that $W \\subseteq W'$; \n  \\item[(2)] for each $W \\in \\mathcal{R}(E,k+1)$ there exists $W' \\in \\mathcal{R}(E,k)$ such that $W' \\subseteq W$.\n  \\end{itemize}\n\\end{definition} \n\nThis section is almost completely dedicated to showing that in the\nclass of committee scoring rules, committee enlargement monotonicity\ncharacterizes exactly the class of separable rules.\n\n\n\\begin{theorem}\\label{thm:committee_monot_and_sep}\n  Let $\\mathcal{R}_f$ be a committee scoring rule.\n  $\\mathcal{R}_f$ is committee-enlargement monotone if and only if $\\mathcal{R}_f$ is\n  separable.\n\n\\end{theorem}\n\\noindent\nBefore we provide the proof of\nTheorem~\\ref{thm:committee_monot_and_sep}, we first introduce useful\nnotation and tools. Given two elections $E_1 = (C,V_1)$ and $E_2 =\n(C,V_2)$, by $E_1 + E_2$ we mean election $(C, V_1+V_2)$, whose voter\ncollection is obtained by concatenating the voter collections of $E_1$\nand $E_2$. For an election $E = (C,V)$ and a positive integer\n$\\lambda$, by $\\lambda E$ we mean election $(C, \\lambda V)$, whose\nvoter collection consists of $\\lambda$ concatenated copies of $V$.\nWe will heavily rely on the\nproperties of the following elections\n(let $C$ be some set of $m$ candidates; this set will always be clear\nfrom the context when we use the notation introduced below):\n\\begin{enumerate}\n\\item For each candidate $c \\in C$, by $\\zeta(c)$ we denote the\n  election with $(m-1)!$ voters who all rank $c$ as their most\n  preferred candidate, followed by each possible permutation of the\n  remaining $m-1$ candidates.\n\n\\item For each subset $S \\subseteq C$, we define election $\\zeta(S)$\n  to be $\\sum_{c \\in S} \\zeta(c)$ (i.e., it is a concatenation of the\n  elections $\\zeta(c)$ for each $c \\in S$).\n\n\\end{enumerate}\nThe next two lemmas describe which committees win in elections\n$\\zeta(c)$ and $\\zeta(S)$.\n\n\n\n\n\\begin{lemma}\\label{lemma:profile_zeta1}\n  Fix $m$ and $k$, and consider a non-degenerate committee scoring rule\n  $\\mathcal{R}$ defined through a scoring function $f_{m,k}$.  The set of\n  winners for $\\zeta(c)$ consists of all committees that contain $c$.\n\\end{lemma}\n\\begin{proof}\n  Since $\\mathcal{R}$ is non-degenerate, there exists $i$ such that\n  $f_{m,k}(i+1, \\ldots, i+k) > f_{m,k}(i+2, \\ldots, i+k+1)$. By the\n  fact that election $\\zeta(c)$ is symmetric with respect to all the\n  candidates except $c$, we see that all committees that contain $c$\n  have the same $f_{m,k}$-score. Similarly, all committees that do not\n  contain $c$ also have the same score.  Consider a committee $W$ such\n  that $c \\notin W$. Let $c'$ be an arbitrary member of $W$ and let\n  $W' = (W \\setminus \\{c'\\}) \\cup \\{c\\}$. Naturally, in each vote the\n  position of committee $W'$ dominates that of $W$. Further, there\n  exists a vote where $W$ has committee position $(i+2, \\ldots,\n  i+k+1)$, and $W'$ has position $(1, i+2, \\ldots, i+k)$. From this\n  vote $W$ gets score $f_{m,k}(i+2, \\ldots, i+k+1)$ and $W'$ gets\n  score $f_{m,k}(1, i+2, \\ldots, i+k) > f_{m,k}(i+1, \\ldots,\n  i+k)$. Thus the score of $W'$ in $\\zeta(c)$ is higher than that of\n  $W$. This completes the proof.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lemma:profile_zeta2}\n  Fix $m$, $k$, and $S \\subseteq C$, and consider a non-degenerate\n  committee scoring rule $\\mathcal{R}$ defined through a scoring function\n  $f_{m,k}$. If $|S| \\geq k$ then the set of winning committees of\n  $\\zeta(S)$ consists of all the committees $W$ such that $W \\subseteq\n  S$.  Otherwise, it consists of all the committees $W$ such that $S\n  \\subseteq W$.\n\\end{lemma}\n\\begin{proof}\n  Consider election $\\zeta(c)$ and let $x$ and $y$ denote the scores\n  of committees, respectively, containing $c$ and not containing\n  $c$. From Lemma~\\ref{lemma:profile_zeta1} it follows that $x >\n  y$. Consider the case when $|S| \\geq k$ (the proof for the other\n  case follows by analogous reasoning). The score of a committee $W$\n  such that $W \\subseteq S$ is equal to $kx + (|S| - k)y$. For each\n  committee $W'$ with $W' \\not\\subseteq S$, its score is at most equal\n  to $(k-1)x + (|S| - k+1)y < kx + (|S| - k)y$.\n\\end{proof}\n\nIn the following observation we analyze the scores of candidates and\ncommittees in the elections we will be using in the proof of\nTheorem~\\ref{thm:committee_monot_and_sep}.\n\n\n\n\\begin{observation}\\label{obs:zeta_construction2}\n  Consider two committees, $W_1$ and $W_2$, with $W_1 \\setminus W_2 =\n  \\{c_1\\}$ and $W_2 \\setminus W_1 = \\{c_2\\}$.  By symmetry of our\n  construction, for each single-winner scoring function $f_{m,1}$, the\n  $f_{m,1}$-scores of the candidates $c_1$ and $c_2$ \n  are the same in election $\\zeta(W_1 \\cup W_2)$,\n  are the same in election $\\zeta(W_1 \\cap W_2)$,\n  and are the same in election $\\zeta\\big(\\{c_1,\n  c_2\\}\\big)$.\n  Further, in each of these three elections, the $f_{m,1}$-scores of\n  any two candidates $c, c' \\in W_1 \\cap W_2$ are equal. If $f_{m,1}$\n  is nontrivial, then in $\\zeta(W_1 \\cup W_2)$, $\\zeta(W_1 \\cap W_2)$\n  and $\\zeta\\big(\\{c_1, c_2\\}\\big)$ the $f_{m,1}$-scores of candidates\n  $c_1$ and $c_2$ are, respectively, the same, lower, and higher than\n  the $f_{m,1}$-score of any other candidate $c\\in W_1\\cup W_2$.\n  Also, for each committee scoring function $f_{m,k}$, the\n  $f_{m,k}$-scores of committees $W_1$ and $W_2$ are the same in\n  $\\zeta(W_1 \\cup W_2)$, are the same in $\\zeta(W_1 \\cap W_2)$, and\n  are the same in $\\zeta\\big(\\{c_1, c_2\\}\\big)$.  In $\\zeta\\big(W_1\n  \\cap W_2 \\setminus \\{c\\}\\big)$, where $c\\neq c_1, c_2$, the $f_{m,k}$-scores\n  of $W_1$ and $W_2$ are equal, and the $f_{m,1}$-score of $c$ is lower than the\n  $f_{m,1}$-score of any other candidate from $W_1 \\cap W_2$.\n\\end{observation}\n\nIn the next lemma we handle the possibility that the rule $\\mathcal{R}_f$ in\nTheorem~\\ref{thm:committee_monot_and_sep} may be trivial.\n(Note that\nit is not the case that if a committee-enlargement monotone\nmultiwinner rule always outputs all size-$1$ committees then it\nalso always outputs all size-$k$ committees for larger values of~$k$.\nThe result below excludes this behavior for the subclass of committee scoring rules.)\n\n\\begin{lemma}\n  \\label{triviality}\n  Suppose that $\\mathcal{R}_f$ is a committee scoring rule defined by a\n  family $f=(f_{m,k})_{k\\le m}$ of scoring functions, such that\n  $\\mathcal{R}_f$ is committee-enlargement monotone and $f_{m,1}$ is\n  constant. Then $f_{m,k}$ is constant for every $k\\le m$.\n\\end{lemma}\n\n\\begin{proof}\n  $\\mathcal{R}_f$ is trivial for $k=1$ and we will show that, in fact, it is\n  trivial for all $k$. The proof follows by induction. Let us assume\n  that $\\mathcal{R}_f$ is trivial for some $k = p - 1$, i.e., that\n  $f_{m,p-1}$ is constant. For the sake of contradiction let us assume\n  that $f_{m,p}$ is not trivial, hence $f_{m,p}(1, \\ldots, p) > f_{m,\n    p}(m-p+1, \\ldots m)$. Let $i$ be the smallest positive integer\n  such that $f_{m,p}(i+1, \\ldots, i + p) > f_{m, p}(i+2, \\ldots\n  i+p+1)$. Consider an election where a certain candidate $c$ is\n  always in position $i + p + 1$, the positions $i+p+2, \\ldots, m$ are\n  also always occupied by the same candidates, and on positions $1,\n  \\ldots, i + p$ there are always the same candidates, call the set of\n  these candidates $S$, but in all possible permutations. We can see\n  that the $f_{m,p}$-scores of committees that consists only of\n  candidates from $S$ are higher than the $f_{m,p}$-scores of\n  committees that contain $c$ (the reasoning is very similar to the\n  one given in the proof of Lemma~\\ref{lemma:profile_zeta1}). This,\n  however, contradicts committee monotonicity, since by our inductive\n  assumption, for $k = p-1$ all committees were winning, and so for\n  $k=p$ there should be at least one winning committee containing $c$.\n\\end{proof}\n\nWe are nearly ready to present the proof of\nTheorem~\\ref{thm:committee_monot_and_sep}. The final piece of notation\nthat we will need is as follows. Given two committee positions $I =\n(i_1, \\ldots, i_k)$ and $J = (j_1, \\ldots, j_k)$, we will sometimes\ntreat them as sets rather than sequences. For example, by $|I \\cap J|$\nwe will mean the number of single-candidate positions that occur\nwithin both $I$ and $J$, and we will say that $i \\in I$ if there is\nsome $t$ such that $i = i_t$.\n\n\n\n\\newcommand{f_{m,1}\\hbox{-}\\score}{f_{m,1}\\hbox{-}{{\\mathrm{score}}}}\n\\newcommand{f_{m,p}\\hbox{-}\\score}{f_{m,p}\\hbox{-}{{\\mathrm{score}}}}\n\\newcommand{f_{m,p-1}\\hbox{-}\\score}{f_{m,p-1}\\hbox{-}{{\\mathrm{score}}}}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:committee_monot_and_sep}] \n  Each separable committee scoring rule is committee-enlargement\n  monotone and we focus on proving the converse.\n\n  Let $\\mathcal{R}_f$ be the committee-enlargement monotone committee\n  scoring rule defined through a family $f=(f_{m,k})_{k\\le m}$ of\n  scoring functions.  Let us fix the number of candidates in our\n  elections to be $m$. $\\mathcal{R}_f$ assigns a score to each committee of\n  each size and, in particular, for $k = 1$, given an election\n  $E=(C,V)$ it assigns $f_{m,1}$-score to each candidate (singleton):\n  \\begin{align*}\n    f_{m,1}\\hbox{-}\\score_E(c) = \\sum_{v_i \\in V}f_{m,1}({{{\\mathrm{pos}}}}_{v_i}(c)) \\text{,}\n  \\end{align*}\n  We will show by induction on $k$ that for each election $E$, a\n  size-$k$ committee $W$ is winning under $\\mathcal{R}_f$ if and only if it\n  consists of candidates with the $k$ highest $f_{m,1}\\hbox{-}\\score\\mathrm{s}$.\n  \n\n\\iffalse\n  %\n  %\n  Thus for each election $E$ it induces a weak order $\\succeq_E$ over\n  the set of candidates $C$, so that for each $c, c' \\in C$ we have $c\n  \\succeq_E c'$ whenever $f_{m,1}\\hbox{-}\\score_E(c) \\geq\n  f_{m,1}\\hbox{-}\\score_E(c')$. If $f_{m,1}\\hbox{-}\\score_E(c) = f_{m,1}\\hbox{-}\\score_E(c')$ then we\n  say that the candidates $c, c' \\in C$ are tied, denoted $c\\sim_E\n  c'$.\n\n  Being tied is an equivalence relation so let\n  $\\mathrm{cand}({\\succeq_E},i)$ be the $i$th equivalence class\n  relative to ${\\sim_E}$ and suppose that the classes are enumerated\n  so that, for any $i<j$ and any $x\\in \\mathrm{cand}({\\succeq_E},i)$\n  and $y\\in \\mathrm{cand}({\\succeq_E},j)$, we have $x\\succ_E y$.\n  %\n  %\n  %\n  %\n  %\n  %\n  Further, let $\\mathrm{top}({\\succeq_E}, i) = \\big\\{c \\in C \\colon c\n  \\succ_E c' \\text{~for each~} c' \\in \\mathrm{cand}({\\succeq_E},\n  i)\\big\\}$ be the set of candidates which precede members of\n  $\\mathrm{cand}({\\succeq_E}, i)$ in ${\\succeq_E}$.  We will show by\n  induction that for each $k$, each winning size-$k$ committee for $E$\n  consists of all elements from $\\mathrm{top}({\\succeq_E}, k)$ and\n  some elements from $\\mathrm{cand}({\\succeq_E}, k)$. This will prove\n  the theorem since then the rule $\\mathcal{R}_f$ will be separable since it\n  can be also represented as $\\mathcal{R}_g$ for the family\n  $g=(g_{m,k})_{k\\le m}$, where\n  \\[\n  g_{m,k}(i_1, \\ldots, i_k) = \\textstyle\\sum_{t=1}^k\n  f_{m,1}(i_t). %\n  \\]\n\n  \\fi\n\n  The base for the induction, for $k=1$, follows immediately from\n  the definition of $\\mathcal{R}_f$.  Now, to prove the inductive step, let us\n  assume that for each $k < p$ and for each election $E$ it holds that\n  $W \\in \\mathcal{R}_f(E, k)$ if and only if it consists of $k$ candidates\n  with the highest $f_{m,1}\\hbox{-}\\score\\mathrm{s}$.\n  We will show that this is also the case for $k = p$.  By\n  Lemma~\\ref{triviality} we may assume that $\\mathcal{R}_f$ for $k=1$ is\n  nontrivial.\\medskip\n\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n  %\n\n  Our first task is to show that  whenever:\n  \\begin{equation}\n    \\label{ind_hyp} \n    f_{m,p}(1, \\ldots, p) = f_{m,p}(i+1, \\ldots, i+p)\n  \\end{equation}\n  for some $i\\in [m]$, then $f_{m,1}(1) = \\cdots = f_{m,1}(i+p)$. For\n  the sake of contradiction let us assume that $f_{m,p}(1, \\ldots, p) = f_{m,p}(i+1,\n  \\ldots, i+p)$ and $f_{m,1}(1) > f_{m,1}(i+p)$. We first show that it\n  must hold that $f_{m,1}(1) = \\cdots = f_{m,1}(i+p-1)$. To see why\n  this is the case, consider an election with a single vote $c_1 \\succ\n  c_2 \\succ \\cdots \\succ c_m$. In such an election, committee $\\{c_1,\n  \\ldots, c_p\\}$ always wins and, since $f_{m,p}(1, \\ldots, p) =\n  f_{m,p}(i+1, \\ldots, i+p)$, we have that committee $W_i = \\{c_{i+1}, \\ldots, c_{i+p}\\}$ also wins. By\n  committee-enlargement monotonicity we know that some\n  size-$(p-1)$ subcommittee of $W_i$ wins for committee size $p-1$\n  and, in particular, by weak dominance we get that\n  certainly $W'_i = \\{c_{i+1}, \\ldots, c_{i+p-1}\\}$ wins. Thus,\n  by the inductive hypothesis it must be the case that:\n  \\begin{align*}\n    f_{m,1}(1) + \\ldots + f_{m,1}(p-1) = f_{m,1}(i+1) + \\ldots + f_{m,1}(i+p-1)\n  \\end{align*} \n  which implies that $f_{m,1}(1) = f_{m,1}(i+p-1)$ and, thus, that\n  $f_{m,1}(1) = f_{m,1}(2) = \\cdots = f_{m,1}(i + p - 1)$. Since we assumed that $f_{m,1}(1) > f_{m,1}(i+p)$,\n  it must be the case that $f_{m,1}(i+p-1) > f_{m,1}(i+p)$.\n\n  Now we show that the assumption that $f_{m,1}(i+p-1) > f_{m,1}(i+p)$ also leads\n  to a contradiction.\n  %\n  %\n  Consider an election $E$ with two votes:\n  \\begin{align*}\n    v_1\\colon& c_1 \\succ c_2 \\succ \\ldots \\succ c_{i+p-1} \\succ c_{i+p} \\succ \\ldots \\succ c_m \\text{,} \\\\\n    v_2\\colon& c_1 \\succ c_2 \\succ \\ldots \\succ c_{i+p} \\succ c_{i+p-1} \\succ \\ldots \\succ c_m \\text{,}\n  \\end{align*}\n  which differ only in the order of $c_{i+p-1}$ and $c_{i+p}$.  For\n  each $j < i+p-1$, we have $f_{m,1}\\hbox{-}\\score_E(c_j)=2f_{m,1}(j)$ and this\n  value is higher than the $f_{m,1}$-scores of $c_{i+p-1}$ and\n  $c_{i+p}$. By the inductive hypothesis, this means that for $k =\n  p-1$ there is no winning committee that contains either $c_{i+p-1}$\n  or $c_{i+p}$. Thus, by committee-enlargement monotonicity, we infer\n  that no winning committee for $k = p$ contains both $c_{i+p-1}$ and\n  $c_{i+p}$. On the other hand, for $k = p$ due to \\eqref{ind_hyp} the\n  $f_{m,p}$-score of committee $\\{c_{i+1}, \\ldots, c_{i+p-1}, c_{i+p}\\}$ is\n  the highest among committees of size $p$, which gives a\n  contradiction.\\medskip\n\n  Next, let $i$ be the smallest value such that $f_{m,p}(i + 1,\n  \\ldots, i + p) > f_{m,p}(i+2, \\ldots, i+p+1)$. We will show that\n  $f_{m,1}(i+p) > f_{m,1}(i+p+1)$. Again, for the sake of\n  contradiction, let us assume that this is not the case and $f_{m,1}(i+p) = f_{m,1}(i+p+1)$. By our\n  previous reasoning we have that $f_{m,1}(1) = \\cdots =\n  f_{m,1}(i+p)$. Consider an election where a fixed candidate $c$\n  stands on position $i+p+1$ and some set of $i+p$ candidates stands\n  on the first $i+p$ positions in all possible permutations. In such\n  an election there is no winning committee of size $p$ that contains\n  $c$. However, by the inductive hypothesis, a winning committee of\n  size $p-1$ containing $c$ does exist. This contradicts\n  committee-enlargement monotonicity.\\medskip\n\n  By the above reasoning, we can find two committee positions $I^*$\n  and $J^*$, for committees of size $p$, such that $|I^* \\cap J^*| =\n  p-1$, $f_{m,p}(I^*) > f_{m,p}(J^*)$, and $\\sum_{i \\in I^*}f_{m,1}(i)\n  > \\sum_{i \\in J^*}f_{m,1}(i)$.  Let us arrange all committee\n  positions from $[m]_k$ in a sequence $\\mathcal{S}$ so that for each two\n  consecutive elements $I$ and $J$ in $\\mathcal{S}$ it holds that $|I \\cap\n  J| = p-1$. This is possible (see the construction based on Johnson\n  graphs in Lemma 8 of the work of\n  Skowron~et~al.~\\cite{sko-fal-sli:t:axiomatic-committee}).  We claim\n  that for each two consecutive elements of sequence $\\mathcal{S}$, call\n  them $I$ and $J$, it holds that:\n  \\begin{align}\\label{eq:equalDerivatives}\n    \\frac{f_{m,p}(I) - f_{m,p}(J)}{f_{m,p}(I^*) - f_{m,p}(J^*)} =\n    \\frac{\\sum_{i \\in I}f_{m,1}(i) - \\sum_{i \\in J}f_{m,1}(i)}{\\sum_{i\n        \\in I^*}f_{m,1}(i) - \\sum_{i \\in J^*}f_{m,1}(i)} \\text{.}\n  \\end{align}\n  (Note that the above expression is well defined. There is no\n  division by zero because we selected $I^*$ and $J^*$ so that\n  $f_{m,p}(I^*) \\neq f_{m,p}(J^*)$ and $\\sum_{i \\in I^*}f_{m,1}(i)\n  \\neq \\sum_{i \\in J^*}f_{m,1}(i)$.)  For the sake of contradiction,\n  let us assume that equality~\\eqref{eq:equalDerivatives} does not\n  hold for some $I$ and $J$, and let us assume that\n  %\n  there exist $x, y \\in {{\\mathbb{N}}}$ such that:\n  \\begin{align}\\label{eq:nonequalDerivatives}\n    \\frac{f_{m,p}(I) - f_{m,p}(J)}{f_{m,p}(I^*) - f_{m,p}(J^*)} >\n    \\frac{x}{y} > \\frac{\\sum_{i \\in I}f_{m,1}(i) - \\sum_{i \\in\n        J}f_{m,1}(i)}{\\sum_{i \\in I^*}f_{m,1}(i) - \\sum_{i \\in\n        J^*}f_{m,1}(i)}\n  \\end{align}\n  Let $W_1$ and $W_2$ be two fixed committees with $|W_1 \\cap W_2| =\n  p-1$. Let $W_1 \\setminus W_2 = \\{c_1\\}$ and $W_2 \\setminus W_1 =\n  \\{c_2\\}$. We construct election $Q$ in which there are $x$ votes\n  where $W_1$ stands on position $I^*$ and $W_2$ on position $J^*$,\n  and $y$ votes where $W_1$ stands on position $J$ and $W_2$ on\n  position $I$. In $Q$ the score of $W_2$ is equal to $xf_{m,p}(J^*) +\n  yf_{m,p}(I)$, and the score of $W_1$ is equal to $xf_{m,p}(I^*) +\n  yf_{m,p}(J)$. By inequality~\\eqref{eq:nonequalDerivatives}, we see\n  that the $f_{m,p}$-score of $W_2$ in $Q$ is greater than the\n  $f_{m,p}$-score of $W_1$, yet the sum of the $f_{m,1}$-scores of\n  members of $W_2$ is lower than that of the members of $W_1$, which\n  means that the $f_{m,1}$-score of $c_1$ in $Q$ is greater than the\n  $f_{m,1}$-score of~$c_2$ (these are the only candidates in which the\n  two committees differ).  If\n  inequality~\\eqref{eq:nonequalDerivatives} were reversed (i.e., if we\n  replaced both occurrences of ``$>$'' with ``$<$'') then the same\n  construction would still work but we would have to reverse the roles\n  of $W_1$ and $W_2$ and of $c_1$ and $c_2$.\n\n\n  We construct election $Q_s$ by taking each possible permutation\n  $\\sigma$ of the candidates from $W_1 \\cap W_2$ and by concatenating\n  all elections of the form $\\sigma(Q)$ (where $\\sigma(Q)$ is an\n  election that results from applying $\\sigma$ to the candidates in\n  all the preference orders within $Q$).  Thus, intuitively, $Q_s$ can\n  be viewed as a symmetric version of $Q$, where symmetry is with\n  respect to the candidates in $W_1 \\cap W_2$.  In particular in $Q_s$\n  it holds that:\n  \\begin{enumerate}[(a)]\n  \\item the $f_{m,p}$-score of $W_2$ is higher than the $f_{m,p}$-score of $W_1$,\n  \\item the $f_{m,1}$-score of $c_1$ is higher than that of $c_2$, and\n  \\item the $f_{m,1}$-scores of all candidates from $W_1 \\cap W_2$ are equal.\n  \\end{enumerate}\n\n\\begin{figure}[t!]\n  \\begin{center}\n    \\includegraphics[scale=0.55]{proof_transformation1}\n  \\end{center}\n  \\caption{The construction of election $Q_2$ from $Q_1$ in case\n    (i). The ``$x$-axis'' corresponds to candidates and the\n    ``$y$-axis'' corresponds to their $f_{m,1}$-scores.  Here,\n    $\\delta_1$ and $\\delta_2$ denote, respectively, the differences\n    between the scores of $c$ and $c_1$ and the difference between the\n    scores of $c$ and $c_2$. The shape of the election\n    $\\zeta\\big(\\{c_1, c_2\\}\\big)$  is\n    justified in Observation~\\ref{obs:zeta_construction2}.}\n  \\label{fig:proof_transformation1}\n  \\begin{center}\n    \\includegraphics[scale=0.55]{proof_transformation2}\n  \\end{center}\n  \\caption{The construction of election $Q_2$ from $Q_1$ in case (ii);\n    the interpretation of the figure is the same as for\n    Figure~\\ref{fig:proof_transformation1}. The shape of the election\n    $\\zeta(W_1 \\cap W_2)$ is justified in\n    Observation~\\ref{fig:proof_transformation1}.}\n  \\label{fig:proof_transformation2}\n  \\end{figure}\n\n\n  There exists $\\lambda \\in {{\\mathbb{N}}}$ such that in election $Q_1 =\n  \\lambda \\zeta(W_1 \\cup W_2) + Q_s$ each candidate from $W_1 \\cup\n  W_2$ has higher $f_{m,1}$-score than each candidate outside of $W_1\n  \\cup W_2$. By Observation~\\ref{obs:zeta_construction2}, it is clear\n  that the $f_{m,1}$-score of candidate $c_1$ in $Q_1$ is higher than\n  that of candidate $c_2$. Intuitively, this transformation allows us\n  to focus only on the candidates from $W_1 \\cup W_2$.\n\n  Now, let $c$ be a fixed arbitrary candidate from $W_1 \\cap W_2$.  We\n  construct election $Q_2$ using $Q_1$ in the following way.\n  \\begin{enumerate}[(i)]\n  \\item If in $Q_1$ the $f_{m,1}$-score of $c$ is higher than the\n    $f_{m,1}$-score of $c_1$, then we define $Q_2$ as a linear\n    combination $Q_2 = \\lambda_1 Q_1 + \\lambda_2 \\zeta\\big(\\{c_1,\n    c_2\\}\\big)$ (this is depicted in\n    Figure~\\ref{fig:proof_transformation1}).\n  \\item Otherwise, i.e., if in $Q_1$ the $f_{m,1}$-score of $c_1$ is at least as\n    high as the $f_{m,1}$-score of $c$, then we define $Q_2$ as a linear\n    combination $Q_2 = \\lambda_1 Q_1 + \\lambda_2 \\zeta(W_1 \\cap W_2)$\n    (this is depicted in Figure~\\ref{fig:proof_transformation2}).\n  \\end{enumerate}\n  In each of these two cases we choose the coefficients $\\lambda_1$\n  and $\\lambda_2$ so that in $Q_2$ it holds that the $f_{m,1}$-score of $c_1$\n  is higher than that of $c$, which is higher than the $f_{m,1}$-score of $c_2$.\n  Further, we choose $\\lambda_1$ and $\\lambda_2$ so that the\n  difference between the $f_{m,1}$-scores of $c$ and $c_1$ is smaller than the\n  difference between the $f_{m,1}$-scores of $c$ and $c_2$. Formally:\n  \\begin{align}\\label{eq:proper_differences}\n    f_{m,1}\\hbox{-}\\score_{Q_2}(c_1) - f_{m,1}\\hbox{-}\\score_{Q_2}(c) < f_{m,1}\\hbox{-}\\score_{Q_2}(c) - f_{m,1}\\hbox{-}\\score_{Q_2}(c_2) \\text{.}\n  \\end{align}\n  Why is it possible to choose such $\\lambda_1$ and $\\lambda_2$? We\n  will give a formal argument for Case (i) and it will be clear that\n  this reasoning can be repeated for Case (ii). Let:\n  \\begin{align*}\n    \\Delta_1 = f_{m,1}\\hbox{-}\\score_{Q_1}(c) - f_{m,1}\\hbox{-}\\score_{Q_1}(c_1) \\quad \\text{and}\n    \\quad \\Delta_2 = f_{m,1}\\hbox{-}\\score_{Q_1}(c_1) - f_{m,1}\\hbox{-}\\score_{Q_1}(c_2).\n  \\end{align*}\n  Further, let $\\Delta_3$ denote the difference between the\n  $f_{m,1}$-scores of the candidates from $\\{c_1, c_2\\}$ and the\n  $f_{m,1}$-scores of the candidates outside of $\\{c_1, c_2\\}$ in\n  $\\zeta\\big(\\{c_1, c_2\\}\\big)$. Naturally, there exist natural\n  numbers $p, q \\in {{\\mathbb{N}}}$ such that:\n  \\begin{align*}\n    \\Delta_1 < \\frac{p}{q} \\Delta_3 < \\Delta_1 + \\frac{1}{2} \\Delta_2\n    \\text{.}\n  \\end{align*}\n  We set $\\lambda_1 = q$ and $\\lambda_2 = p$, and from the above\n  inequality we get that:\n  \\begin{align}\\label{eq:lambda1lambda2set}\n    \\lambda_1 \\Delta_1 < \\lambda_2 \\Delta_3 < \\lambda_1\\left(\\Delta_1 +\n    \\frac{1}{2} \\Delta_2\\right) \\text{.}\n  \\end{align}\n  Observe that:\n  \\begin{align*}\n    f_{m,1}\\hbox{-}\\score_{Q_2}(c_1) - f_{m,1}\\hbox{-}\\score_{Q_2}(c) &= - \\lambda_1 \\Delta_1 + \\lambda_2 \\Delta_3 > 0 \\text{,} \\\\\n    f_{m,1}\\hbox{-}\\score_{Q_2}(c) - f_{m,1}\\hbox{-}\\score_{Q_2}(c_2) &= \\lambda_1 (\\Delta_1 + \\Delta_2) - \\lambda_2 \\Delta_3 > - \\lambda_1 \\Delta_1 + \\lambda_2 \\Delta_3 \\\\\n    &= f_{m,1}\\hbox{-}\\score_{Q_2}(c_1) - f_{m,1}\\hbox{-}\\score_{Q_2}(c) \\text{.}\n  \\end{align*}\n  (The second inequality above is equivalent to $2\\lambda_1\\Delta_1 +\n  \\lambda_1\\Delta_2 - 2\\lambda_2 \\Delta_3 > 0$, and thus follows from\n  inequality~\\eqref{eq:lambda1lambda2set}.)  \n  \\begin{figure}[t!]\n    \\begin{center}\n      \\includegraphics[scale=0.55]{proof_transformation3}\n    \\end{center}\n    \\caption{Illustration of election $Q_3$. The interpretation of the\n      figure is the same as for\n      Figure~\\ref{fig:proof_transformation1}.}\n    \\label{fig:proof_transformation3}\n  \\end{figure}\n  Next, we construct $Q_3$ as $Q_3 = \\lambda_4 Q_2 + \\zeta\\big(W_1\n  \\cap W_2 \\setminus \\{c\\}\\big)$, where $\\lambda_4$ is a very large\n  number so that in $Q_3$ we still have that $f_{m,1}\\hbox{-}\\score_{Q_3}(c_1)\n  > f_{m,1}\\hbox{-}\\score_{Q_3}(c) > f_{m,1}\\hbox{-}\\score_{Q_3}(c_2)$ and that in $Q_3$\n  inequality~\\eqref{eq:proper_differences} still holds, yet the\n  $f_{m,1}$-score of $c$ is slightly lower than the $f_{m,1}$-scores\n  of the other candidates from $W_1 \\cap W_2$. Election $Q_3$ is\n  depicted in Figure~\\ref{fig:proof_transformation3}.\n\n  Given the $f_{m,1}$-scores of the candidates in $Q_3$, by our\n  inductive assumption the unique size-$(p-1)$ winning committee\n  consists of $c_1$ and all the candidates from $W_1 \\cap W_2\n  \\setminus \\{c\\}$. By committee-enlargement monotonicity, we conclude\n  that all size-$p$ winning committees for $Q_3$ are of the form\n  $\\{c_1\\} \\cup (W_1 \\cap W_2 \\setminus \\{c\\}) \\cup \\{c'\\}$, where\n  $c'$ is some other candidate. Let $W'$ be one such winning\n  committee. We know that it cannot be the case that $c' = c$\n  (i.e., $W_1$ cannot be winning in $Q_3$). This is so, because in\n  $Q_3$ the $f_{m,p}$-score of $W_2$ is higher than the\n  $f_{m,p}$-score of $W_1$ (since it was higher already in $Q_s$, and\n  we added only elections which are symmetric with respect to $W_1$\n  and $W_2$---this symmetry follows from\n  Observation~\\ref{obs:zeta_construction2}).  Thus, by the properties\n  of the $f_{m,1}$-scores of the candidates (see\n  Figure~\\ref{fig:proof_transformation3}), $c'$ must be some candidate\n  such that:\n  \\begin{align*}\n    f_{m,1}\\hbox{-}\\score_{Q_3}(c') \\leq f_{m,1}\\hbox{-}\\score_{Q_3}(c_2) \\text{,}\n  \\end{align*}\n  and, in particular $c'$ may simply be $c_2$ (but it also may be some\n  other candidate).  From the above inequality and from\n  inequality~\\eqref{eq:proper_differences} (which holds for $Q_3$ as\n  well) we get that:\n  \\begin{align*}\n    f_{m,1}\\hbox{-}\\score_{Q_3}(c_1) - f_{m,1}\\hbox{-}\\score_{Q_3}(c) < f_{m,1}\\hbox{-}\\score_{Q_3}(c) -    f_{m,1}\\hbox{-}\\score_{Q_3}(c') \\text{.}\n  \\end{align*}\n  We note that in $Q_3$ committee $W'$ has higher score than any\n  committee containing $c$ (otherwise, since $W'$ is winning in $Q_3$\n  and by the above analysis, it would mean that $W_1$ is winning in\n  $Q_3$, which is not the case).  \n\n  Next, we construct election $Q_3'$ by swapping candidates $c_1$ and\n  $c'$ in each vote in $Q_3$. Committee $W'$ is also winning in $Q_3'$\n  and thus it has higher score in $Q_3'$ than any committee containing\n  $c$. Similarly, by symmetry, we infer that:\n  \\begin{align*}\n    f_{m,1}\\hbox{-}\\score_{Q_3'}(c') - f_{m,1}\\hbox{-}\\score_{Q_3'}(c) < f_{m,1}\\hbox{-}\\score_{Q_3'}(c) - f_{m,1}\\hbox{-}\\score_{Q_3'}(c_1) \\text{.}\n  \\end{align*}\n  Finally, we construct election $Q_4$ by taking one copy of $Q_3$ and\n  one copy of $Q_3'$. Observe that:\n  \\begin{align*}\n    f_{m,1}\\hbox{-}\\score_{Q_4}(c) &= f_{m,1}\\hbox{-}\\score_{Q_3}(c) + f_{m,1}\\hbox{-}\\score_{Q_3'}(c) \\\\\n    &> f_{m,1}\\hbox{-}\\score_{Q_3}(c_1) - f_{m,1}\\hbox{-}\\score_{Q_3}(c) + f_{m,1}\\hbox{-}\\score_{Q_3}(c') \\\\\n    &\\qquad + f_{m,1}\\hbox{-}\\score_{Q_3'}(c') - f_{m,1}\\hbox{-}\\score_{Q_3'}(c) + f_{m,1}\\hbox{-}\\score_{Q_3'}(c_1) \\\\\n    &= f_{m,1}\\hbox{-}\\score_{Q_4}(c_1) + f_{m,1}\\hbox{-}\\score_{Q_4}(c') - f_{m,1}\\hbox{-}\\score_{Q_4}(c).\n  \\end{align*}\n  We can rewrite the above inequality as:\n  \\begin{align*}\n    f_{m,1}\\hbox{-}\\score_{Q_4}(c) > \\frac{1}{2}\\Big( f_{m,1}\\hbox{-}\\score_{Q_4}(c_1) +\n    f_{m,1}\\hbox{-}\\score_{Q_4}(c') \\Big) \\text{.}\n  \\end{align*}\n  Since $f_{m,1}\\hbox{-}\\score_{Q_4}(c_1) = f_{m,1}\\hbox{-}\\score_{Q_4}(c')$ (election\n  $Q_4$ is symmetric with respect to $c_1$ and $c'$) we get that\n  $f_{m,1}\\hbox{-}\\score_{Q_4}(c) > f_{m,1}\\hbox{-}\\score_{Q_4}(c')$ and\n  $f_{m,1}\\hbox{-}\\score_{Q_4}(c) > f_{m,1}\\hbox{-}\\score_{Q_4}(c_1)$. Thus, the\n  $f_{m,1}$-score of $c$ in $Q_4$ is among the $f_{m,1}$-scores of the\n  $(p-1)$ top-scored candidates. From our inductive assumption and\n  from committee-enlargement monotonicity we infer that each winning\n  committee in $Q_4$ must contain $c$. This contradicts the fact that\n  $W'$ is winning in $Q_4$, and proves\n  equation~\\eqref{eq:equalDerivatives}. \n  Thus setting: $$\\alpha =\n  \\frac{f_{m,p}(I^*) - f_{m,p}(J^*)}{\\sum_{i \\in I^*}f_{m,1}(i) -\n    \\sum_{i \\in J^*}f_{m,1}(i)},$$ we get that for any two consecutive\n  elements, $I$ and $J$, on path $\\mathcal{S}$ it holds that:\n  \\begin{align}\n    f_{m,p}(I) - f_{m,p}(J) = \\alpha\\Big(\\sum_{i \\in I}f_{m,1}(i) -\n    \\sum_{i \\in J}f_{m,1}(i)\\Big) \\text{.}\n  \\end{align}\n  By a simple induction over the path $\\mathcal{S}$ we can show that the\n  above equality holds for any $I$ and $J$ ($I$ and $J$ do not have to\n  be consecutive elements in $\\mathcal{S}$). Consequently, we get that\n  $f_{m,p}$ is a linear transformation of the function $g_{m,p} (I) =\n  \\sum_{i \\in I}f_{m,1}(i)$, thus they yield the same committee\n  scoring rule. This proves our inductive step, and completes the\n  proof.\n\n\\end{proof}\n\n\n\nElkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties} and\nBarber\\`a and~Coelho~\\cite{bar-coe:j:non-controversial-k-names} point\nout that committee-enlargement monotonicity is an extremely natural\nrequirement for multiwinner rules whose role is to select committees\nof individually excellent candidates. For example, if some $k$\ncandidates are good enough to be shortlisted for receiving some award,\nthen increasing $k$ should not lead to any of them losing their\nnominations.\nThus, intuitively, Theorem~\\ref{thm:committee_monot_and_sep} says that\nif one is interested in a committee scoring rule for choosing\nindividually excellent candidates, then one should look within the\nclass of separable rules. This refines and reinforces the\nrecommendation provided by Theorem~\\ref{thm:weaklysep}, which\nsuggested looking among weakly separable rules.\n\n\n\n\nYet, one could challenge this recommendation. For example, SNTV is\nseparable, but it is also representation-focused and there is some\nevidence that its behavior is closer to that of the\nChamberlin--Courant rule (which is seen as selecting committees\nrepresenting a diverse spectrum of opinions; recall the discussion\nafter Theorem~\\ref{thm:representation_focused}), than to that of, say,\n$k$-Borda (which is seen as selecting individually excellent\ncandidates). Such evidence is provided, for example, by Elkind et\nal.~\\cite{elk-fal-las-sko-sli-tal:c:2d-multiwinner}, who evaluated a\nnumber of committee scoring rules experimentally, by computing their\nresults on elections obtained from several two-dimensional Euclidean\nmodels and presenting them graphically.\nNonetheless, in real-life settings even SNTV is sometimes used for\nchoosing individually excellent candidates. As a piece of anecdotal\nevidence, let us mention that while preparing this paper, we have ran\ninto a news article that listed $10$ best ski-jumpers of all time. The\ncriterion for inclusion on that list was the number of times a given\nsportsman had won an individual competition of the ski-jumping World\nCup. In other words, all the individual World Cup competitions that\never took place were seen as ``voters,'' ranking all the sportsmen\nfrom the winner to the loser, and then SNTV was used to select the\n``committee'' of $10$ best ski-jumpers of all\ntime.\\footnote{Naturally, the sets of ski-jumpers participating in the\n  contests were often different. Formally, we would say that the\n  participating sportsmen were ranked according to their result in the\n  competition and all the non-participating ones were ranked below, in\n  some arbitrary order. Since we are using SNTV, the order in which\n  the non-participants are ranked is irrelevant.}\n\n\n\n\n\n\n\nTheorems~\\ref{thm:weaklysep}~and~\\ref{thm:committee_monot_and_sep}\nhave yet another interesting consequence. They imply that\ncommittee-enlargement monotonicity of a committee scoring rule implies\nits non-crossing monotonicity. This is somehow surprising, since the\ntwo variants of monotonicity seem almost unrelated as one describes\nhow the result of an election changes if we increase the size of the\ncommittee and the other one---what happens when we shift a member of a\nwinning committee in a preference relation of a voter.\n\\begin{corollary}\n  If a committee scoring rule is committee-enlargement monotone, then it\n  is also non-crossing monotone.\n\\end{corollary}\n\n\n\\skippedproof{\n\n{\\large \\bf The original theorem statement comes below.}\n\n\\begin{theorem}\\label{thm:prefixMonAndDecomposable2}\n  Let $\\mathcal{R}_f$ be a decomposable committee scoring rule, i.e., let\n  $\\{\\gamma_i\\}_{i\\in[k]}$ be a family of scoring functions such that\n  for each sequence of $k$ positions $(\\ell_1, \\ldots, \\ell_k)$ it\n  holds that:\n  \\begin{align*}\n    f(\\ell_1, \\ldots, \\ell_k) = \\gamma_1(\\ell_1) +\\gamma_2(\\ell_2)+\n    \\cdots + \\gamma_k(\\ell_k) \\textrm{.}\n  \\end{align*}\n  \\begin{enumerate}\n  \\item A necessary condition for $\\mathcal{R}_f$ to be prefix-monotone is\n    that for each $i, j \\in [k]$, $j>i$, and each $p \\in [m]$, $j+1\n    \\leq p < m - (k-i)$ it must hold that:\n    \\begin{align*}\n      \\gamma_i(p) - \\gamma_i(p+1) \\geq \\gamma_j(p) - \\gamma_j(p+1)\n      \\textrm{.}\n\\end{align*}\n\\item A sufficient condition for $\\mathcal{R}_f$ to be prefix-monotone is that for each $i, j \\in [k]$, $j>i$, and each $p \\in [m]$, $j \\leq p < m - (k-i)$, it must hold that:\n\\begin{align*}\n\\gamma_i(p) - \\gamma_i(p+1) \\geq \\gamma_j(p) - \\gamma_j(p+1) \\textrm{.}\n\\end{align*}\nand that for each $i \\in [k]$ and each $p, p' \\in [m-1]$, $p < p'$, it must hold that:\n\\begin{align*}\n\\gamma_i(p) - \\gamma_i(p+1) \\geq \\gamma_i(p') - \\gamma_i(p'+1) \\textrm{.}\n\\end{align*}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n  First, let us prove the first statement from the thesis of this\n  theorem. For the sake of contradiction let us assume that there\n  exist integers $i, j \\in [k]$, $j>i$, and $p \\in [m]$, $j+1 \\leq p <\n  m - (k-i)$, such that it holds that:\n  \\begin{align*}\n    \\gamma_i(p) - \\gamma_i(p+1) < \\gamma_j(p) - \\gamma_j(p+1)\n    \\textrm{.}\n  \\end{align*}\n\n  Let $E = (C,V)$ be an election with candidate set $C = \\{c_1,\n  \\ldots, c_m\\}$ and $m!$ voters $v_1, \\ldots, v_{m!}$, one for each\n  possible preference order. By symmetry, any size-$k$ subset $W$ of\n  $C$ is a winning committee of $E$ under $\\mathcal{R}_f$. Further, consider\n  an arbitrary vote $v$ from the election; let $C_1$ be the set of\n  candidates that $v$ ranks at positions $(2, \\ldots, j, p+1,\n  m-(k-j)+1, \\ldots, m)$ and let $C_2$ be the set of candidates that\n  $v$ ranks at positions $(2, \\ldots, i, p+1, m-(k-i)+1, \\ldots,\n  m)$. Let $E'$ be the election obtained by shifting one position up\n  each candidate standing on a position from $\\{2, \\ldots, i, p+1\\}$\n  in $v$. In comparison with $E$, in $E'$ the score of $C_1$ increased\n  by:\n  \\begin{align*}\n    \\sum_{t=1}^{i-1} \\big(\\gamma_t(t) - \\gamma_t(t+1)\\big) +\n    \\big(\\gamma_j(p) - \\gamma_j(p+1)\\big) \\textrm{.}\n  \\end{align*}\n  The score of $C_2$ increased by:\n  \\begin{align*}\n    \\sum_{t=1}^{i-1} \\big(\\gamma_t(t) - \\gamma_t(t+1)\\big) +\n    \\big(\\gamma_i(p) - \\gamma_i(p+1)\\big) \\textrm{.}\n  \\end{align*}\n  Since the score of $C_1$ increased more, $C_2$ cannot be the winner\n  in $E'$. This, however, contradicts prefix-monotonicity.\n\n\n  Now, let us prove that the second condition from the thesis of this\n  theorem is sufficient for $\\mathcal{R}_f$ to be prefix-monotone. Consider\n  elections where a committee $W$ is a winner. Let $j \\in [k]$ and let\n  $E'$ be elections obtained from $E$ by shifting forward by one\n  position each of the first $j$ members of $W$ in some vote $v$. We\n  will show that $W$ is a winning committee in $E'$. Let $(\\ell_1,\n  \\ldots, \\ell_k)$ be the increasing sequence of positions that the\n  members of $W$ take in $v$. In comparison with $E$, in $E'$ the\n  score of $W$ increased by:\n  \\begin{align*}\n    \\sum_{t=1}^j \\big(\\gamma_t(\\ell_t - 1) - \\gamma_t(\\ell_t)\\big)\n    \\textrm{.}\n  \\end{align*}\n  Let us now assess by how much the score of some other committee,\n  $W'$, increased. Let us fix $t$, $t\\in[j]$. Let $c_t$ be the\n  candidate standing on position $\\ell_t$ in $v$. If $c_t \\notin W'$,\n  then shifting $c_t$ one position up has no positive effect on the\n  score of $W'$. Consider the case when $c_t \\in W'$. Let $x_t$ denote\n  the position of $c_t$ within $W'$ according to $v$ (for instance, if\n  $c_t$ is most preferred among members of $W'$ in $v$, then $x_t =\n  1$). Now, we consider two cases:\n  \\begin{description}\n  \\item[$\\ell_t = x_t$.] This means that the first $t$ members of $W'$\n    stand on $t$ first positions in $v$. Thus, shifting $c_t$ one\n    position up has no effect on the score of $W'$ (effectively, such\n    a shift is an operation that swaps $c_t$ with some other member of\n    $W'$).\n\n  \\item[$\\ell_t \\geq x_t + 1$.] We observe that $\\ell_t < m - (k-t)$\n    (there are $(k-t)$ members of $W$ standing on positions greater\n    than $\\ell_t$). If $x_t \\geq t$, then from the second condition of\n    the theorem, we get that $\\big(\\gamma_{x_t}(\\ell_t - 1) -\n    \\gamma_{x_t}(\\ell_t)\\big)$ is not greater than\n    $\\big(\\gamma_t(\\ell_t - 1) - \\gamma_t(\\ell_t)\\big)$. Thus, the\n    increase of the score of $W'$ due to shifting $c_t$ one position\n    up is not greater than the increase of the score of $W$ due to\n    shifting $c_t$ one position up. If $x_t < t$, then we observe\n    that:\n    \\begin{align*}\n      \\big(\\gamma_{x_t}(\\ell_t - 1) - \\gamma_{x_t}(\\ell_t)\\big) \\leq\n      \\big(\\gamma_{x_t}(\\ell_{x_t} - 1) -\n      \\gamma_{x_t}(\\ell_{x_t})\\big)\n    \\end{align*}\n    Thus, the increase of the score of $W'$ due to shifting $c_t$ one\n    position up is not greater than the increase of the score of $W$\n    due to shifting the candidate on position $\\ell_{x_t}$ one\n    position up.\n  \\end{description}\n  From the above reasoning we see that for each $t\\in[j]$ the increase\n  of the score of $W'$ due to shifting $c_t$ one position up is no\n  greater than the increase of the score of $W$ due to shifting some\n  other candidate $c_{t'}$ ($t' \\leq t$) one position up. Further,\n  from the construction in the two cases above, we observe that if we\n  assign to each $c_t \\in W'$ a candidate $c_{t'}$ as done before,\n  then each $c_{t'}$ will be assigned to a single candidate\n  only. Thus, we get that the score of $W'$ increases in $E'$ by a\n  value that is not greater than the increase of the score of\n  $W$. Since $W'$ was chosen arbitrarily, we get that $W$ is a winner\n  in $E'$, which completes the proof.\n\\end{proof}\n}\n\n\n\n\n\n\n\n\\iffalse\n\n\\section{Computational Remarks}\n\\label{sec:comp_rem}\n\nMany committee scoring rules are ${\\mathrm{NP}}$-hard to compute.  Procaccia et\nal.~\\cite{pro-ros-zoh:j:proportional-representation}, Lu and\nBoutilier~\\cite{bou-lu:c:chamberlin-courant}, and Betzler et\nal.~\\cite{bet-sli-uhl:j:mon-cc} show hardness of winner determination\nfor various representation-focused rules,\nSkowron et al.~\\cite{sko-fal-lan:c:collective} and Aziz et\nal.~\\cite{azi-gas-gud-mac-mat-wal:c:approval-multiwinner} show strong\nhardness results for large families of OWA-based rules, and\nFaliszewski et al.~\\cite{fal-sko-sli-tal:c:top-k-counting} do the same\nfor many top-$k$-counting rules.  (On the other hand, %\nall weakly separable rules are polynomial-time computable.)\n\n\n\nFortunately, many of the papers cited above\nalso provide means to circumvent those hardness results. We add the\nfollowing result to this literature.\n\n\n\\begin{proposition}\n  Let $\\mathcal{R}_f$ be a decomposable committee scoring rule defined\n  through a family of scoring functions $ f_{m,k}(i_1, \\ldots, i_k) =\n  \\gamma^{(1)}_{m,k}(i_1) +\\gamma^{(2)}_{m,k}(i_2)+ \\cdots +\n  \\gamma^{(k)}_{m,k}(i_k) \\textrm{,} $ where $\\gamma =\n  (\\gamma^{(t)}_{m,k})_{t \\leq k \\leq m}$ is a family of\n  polynomial-time computable single-winner scoring functions.  If for\n  each $m$, each $k \\in [m]$, and each $t \\in [k - 1]$ it holds that\n  $\\left(\\gamma_{m,k}^{(t)} - \\gamma_{m,k}^{(t+1)}\\right)$ is\n  nonincreasing, then there is a polynomial-time algorithm that, given\n  an election $E$ with $m$ candidates and a committee size $k$,\n  outputs a committee $W$ such that:\n  \\begin{align*}\n  f_{m,k}\\hbox{-}{{\\mathrm{score}}}_E(W) \\geq \\left(1-\\frac{1}{e}\\right)\\max_{S \\subseteq C, |S| = k}f_{m,k}\\hbox{-}{{\\mathrm{score}}}_E(S) \\text{.}\n  \\end{align*}\n\\end{proposition}\n\n\\begin{proof}\n  Note that the definition of function $f_{m,k}$ can be equivalently\n  rewritten as:\n  \\begin{align*}\n    f_{m,k}(i_1, \\ldots, i_k) = \\sum_{t = 1}^{k}\n    \\gamma^{(k)}_{m,k}(i_t) + \\sum_{t = 1}^{k-1}\n    \\Big(\\gamma^{(k-1)}_{m,k}(i_t) - \\gamma^{(k)}_{m,k}(i_t)\\Big) +\n    \\ldots + \\sum_{t = 1}^{1} \\Big(\\gamma^{(1)}_{m,k}(i_t) -\n    \\gamma^{(2)}_{m,k}(i_t)\\Big) \\text{.}\n  \\end{align*}\n  Note that the function:\n  \\begin{align*}\n    g_{m,k}(i_1, \\ldots, i_k) = \\sum_{t = 1}^{k}\n    \\gamma^{(k)}_{m,k}(i_t)\n  \\end{align*}\n  is weakly-separable, and so it is submodular. Further, for each $p\n  \\in [k-1]$ the function:\n  \\begin{align*}\n    h_{m,k}(i_1, \\ldots, i_k) = \\sum_{t = 1}^{p}\n    \\Big(\\gamma^{(p)}_{m,k}(i_t) - \\gamma^{(p+1)}_{m,k}(i_t)\\Big) =\n    \\sum_{t = 1}^{p} \\Big(\\gamma^{(p)}_{m,k} -\n    \\gamma^{(p+1)}_{m,k}\\Big)(i_t)\n  \\end{align*}\n  is OWA-based (with the OWA vector having ones in the first $p$\n  positions, and zeros in the remaining ones) and so it is also\n  submodular~\\cite{sko-fal-lan:c:collective}. Thus, our function\n  $f_{m,k}$, as a linear combination of submodular functions, is also\n  submodular. Since all the functions in the family $\\gamma$ are\n  positive, the scoring function $f_{m,k}$ is nondecreasing.  Thus, by\n  a famous result of Nemhauser et al.~\\cite{nem-wol-fis:j:submodular},\n  it follows that a greedy algorithm, which iteratively add candidates\n  to the committee, based on the increase of score they incur, has the\n  claimed approximation guarantee.\n\\end{proof}\n\n\n\n\\fi\n\n\n\\section{Related Work}\n\\label{sec:related}\n\nOver the last few years, multiwinner voting has attracted significant\ninterest within the computational social choice literature, but it has\nalso been studied for much longer within social choice theory and\nwithin economics. Below we briefly review this literature (for a more\ndetailed review, we point the readers to the overview of Faliszewski\net al.~\\cite{fal-sko-sli-tal:b:multiwinner-trends}; we have also\nmentioned many related papers in the context of respective results).\n\n\nAxiomatic studies of voting rules were initiated by\nArrow~\\cite{arr:b:polsci:social-choice}, and in a somehow more narrow\nframework, by May~\\cite{mayAxiomatic1952}.  Single-winner scoring\nrules are perhaps the best understood among single-winner election\nsystems.  Axiomatic characterizations of this class were provided,\ne.g., by G{\\\"a}rdenfors~\\cite{gardenfors73:scoring-rules},\nSmith~\\cite{smi:j:scoring-rules}, and\nYoung~\\cite{you:j:scoring-functions}, and in a more general setting,\nby Myerson~\\cite{Myerson1995} and Pivato~\\cite{pivato2013variable}.\nMore specific axiomatic characterizations of single-winner scoring\nrules include those of the Borda rule~\\cite{youngBorda, hansson76,\n  fishburnBorda, smi:j:scoring-rules}, of the Plurality\nrule~\\cite{RichelsonPlurality, Ching1996298}, and of the Antiplurality\nrule~\\cite{Barbera198249} (see also the overviews of Chebotarev and\nShamis~\\cite{che-sha:j:scoring-rules} and of\nMerlin~\\cite{merlinAxiomatic}).  Classic works on axiomatic properties\nof multiwinner rules include those of Dummett~\\cite{dum:b:voting},\nGehrlein~\\cite{geh:j:condorcet-committee}, Felsenthal and\nMaoz~\\cite{fel-mao:j:norms}, Debord~\\cite{deb:j:k-borda},\nRatliff~\\cite{ratliff2003some}, and Barber\\`a and\nCoelho~\\cite{bar-coe:j:non-controversial-k-names}.\n\nOur work mostly builds on that of Elkind et\nal.~\\cite{elk-fal-sko-sli:j:multiwinner-properties} where the authors\nintroduced the class of committee scoring rules and many of the\nnotions on which we rely, such as candidate monotonicity, non-crossing\nmonotonicity and committee enlargement-monotonicity (regarding the\nlatter one, see also the work of Barber\\`a and\nCoelho~\\cite{bar-coe:j:non-controversial-k-names}).  In particular,\nElkind et al.~\\cite{elk-fal-sko-sli:j:multiwinner-properties}\nidentified the classes of (weakly) separable and\nrepresentation-focused rules and provided some of their basic\nfeatures.  OWA-based rules were introduced by\nSkowron et al.~\\cite{sko-fal-lan:c:collective}, who analyzed their\ncomputational properties (and who, in fact, studied a somewhat more\ngeneral model).\nFaliszewski et\nal.~\\cite{fal-sko-sli-tal:c:top-k-counting,fal-sko-sli-tal:c:paths}\nintroduced the class of top-$k$-counting rules and the $\\ell_p$-Borda\nand $q$-HarmonicBorda rules.\n\nRecently Skowron et~al.~\\cite{sko-fal-sli:t:axiomatic-committee}\ncharacterized the class of committee scoring rules using the axioms of\nconsistency, symmetry, continuity, and weak efficiency.  Our paper can\nbe seen as complementary to theirs: They study committee scoring rules\nas opposed to all the other multiwinner rules, whereas we focus on the\ninternal structure of the class.\n\n\n\nAziz et\nal.~\\cite{azi-bri-con-elk-fre-wal:j:justified-representation,azi-gas-gud-mac-mat-wal:c:approval-multiwinner}\nstudied a class of approval-based rules that is very similar to the\nclass of committee scoring rules (the class was first introduced by\nThiele~\\cite{Thie95a} in the 19th century, but was forgotten for some\ntime; some of Thiele's rules were recalled by\nKilgour~\\cite{kil-handbook} and then by Aziz et al.).  Lackner and\nSkowron~\\cite{lac-sko:t:approval-thiele} studied axiomatic\nproperties of these rules and highlighted their axiomatic similarity\nto committee scoring rules.\nRecently, monotonicity notions similar to those studied in this paper\nwere also considered in the context of approval-based multiwinner\nrules~\\cite{san-fis:t:approval-monotonicity,lac-sko:t:approval-thiele}.  For more general\ndiscussions of the properties of approval-based rules we point the\nreader to the work of Kilgour and\nMarshall~\\cite{kil-mar:j:minimax-approval}.\n\n\nThe study of computational properties of committee scoring rules in\ngeneral, and of specific rules, such as Chamberlin--Courant and\nProportional Approval Voting, has attracted significant\nattention. This line of work has started with the paper of Procaccia\net al.~\\cite{pro-ros-zoh:j:proportional-representation}, who have\nshown that an approval-based variant of Chamberlin--Courant is\n${\\mathrm{NP}}$-hard to compute. The same result for the classic, Borda-based\nvariant was shown by Lu and\nBoutilier~\\cite{bou-lu:c:chamberlin-courant}. Betzler et\nal.~\\cite{bet-sli-uhl:j:mon-cc} considered parameterized complexity of\nthe rule, whereas the study of approximation algorithms was initiated\nby Lu and Boutilier~\\cite{bou-lu:c:chamberlin-courant}, who have given\na polynomial-time $(1-\\frac{1}{e})$-approximation algorithm (this\nalgorithm, based on the greedy procedure of Nemhauser et\nal.~\\cite{nem-wol-fis:j:submodular} for submodular functions, has\nsince then been adapted to other committee scoring rules as well). Skowron et\nal.~\\cite{sko-fal-sli:j:multiwinner} improved this result by providing\na polynomial-time approximation scheme for the Borda-based variant;\nSkowron and Faliszewski~\\cite{sko-fal:j:maxcover} gave an FPT\napproximation scheme for the approval-based variant (and argued why\nthe $(1-\\frac{1}{e})$-approximation algorithm is the best we can hope\nfor among polynomial-time algorithms).\nThe complexity of Chamberlin--Courant was also studied in much depth\nfor various restricted domains, including the single-peaked\ndomain~\\cite{bet-sli-uhl:j:mon-cc,cor-gal-spa:c:sp-width,pet:t:total-unimodularity},\nthe single-crossing domain~\\cite{sko-yu-fal-elk:j:mwsc}, and a number\nof\nothers~\\cite{yu-cha-elk:c:multiwinner-trees,lac-pet:c:spoc,elk-pet:c:nice-trees}.\nFaliszewski et\nal.~\\cite{fal-sli-sta-tal:c:cc-mon-clustering,fal-lac-pet-tal:c:csr-heuristics}\nconsidered a number of heuristic algorithms.\n\n\nProportional Approval Voting (PAV) received a bit less attention than\nthe Chamberlin--Courant rule, but due to the work of Aziz et\nal.~\\cite{azi-bri-con-elk-fre-wal:j:justified-representation} on\njustified representation, it is now being studied with increasing\ninterest (briefly put, Aziz et al\\@. have shown that PAV is remarkably\ngood at providing committees that represent the voters proportionally,\nas also confirmed by Brill et al.~\\cite{bri-las-sko:c:apportionment};\nsee the work of Brill et al.~\\cite{aaai\/BrillFJL17-phragmen} for\nanother rule with similar properties). The rule was shown to be\n${\\mathrm{NP}}$-hard to\ncompute~\\cite{azi-gas-gud-mac-mat-wal:c:approval-multiwinner,sko-fal-lan:c:collective},\nbut the standard greedy $(1-\\frac{1}{e})$-approximation algorithm\nworks for it. Very recently, Byrka et\nal.~\\cite{byr-sko-sor:t:pav-approx} have shown a different, apparently\nmuch more powerful algorithm.  The rule was also considered in the\ncontext of restricted domains~\\cite{pet:t:total-unimodularity}. FPT approximation schemes\nfor PAV and other OWA-based rules were provided by Skowron~\\cite{sko:j:submodular}.\n\n\nMore general computational results regarding committee scoring rules\nwere provided by Skowron et al.~\\cite{sko-fal-lan:c:collective}, who\nstudied the complexity and approximability of OWA-based rules, by\nFaliszewski et al.~\\cite{fal-sko-sli-tal:c:top-k-counting}, for\ntop-$k$-counting rules, by Peters~\\cite{pet:t:total-unimodularity},\nfor OWA-based rules in the single-peaked domain, and by Faliszewski et\nal.~\\cite{fal-lac-pet-tal:c:csr-heuristics}, who introduced several\ngeneral-purpose heuristic algorithms.\n\n\n\n\n\nNaturally, there exist many interesting multiwinner rules beyond the\nclass of committee scoring rules.  These include, for example, Single\nTransferable Vote (see, e.g., the work of Tideman and\nRichardson~\\cite{tid-ric:j:stv}), a number of rules based on the\nCondorcet\nprinciple~\\cite{AEFLS17:multiwinner-condorcet,bar-coe:j:non-controversial-k-names,dar:j:condorcet-hard,journals\/scw\/ElkindLS15,fis:j:condorcet-committee,fis:j:majority-committees,geh:j:condorcet-committee,ratliff2003some,sek-sik-xia:c:condorcet-bundling},\nMonroe's rule~\\cite{mon:j:monroe}, and different variants of the rule\ninvented by\nPhragm\\'en~\\cite{Phra94a,Phra95a,Phra96a,Janson16arxiv,aaai\/BrillFJL17-phragmen}.\nFor an overview of electoral systems used to select committees of\nrepresentatives in practice, we refer the reader to the book of\nLijphart and Grofman~\\cite{grofmanChoosingElectoral}.\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:concl}\n\nWe have studied the class of committee scoring rules and explored the\ninteresting hierarchy formed by its subclasses studied to date\n(including the class of decomposable rules introduced in this paper).\nWe have highlighted several fundamental properties of committee\nscoring rules, ranging from the nonimposition property (i.e., that for\nevery committee and every nontrivial committee scoring rule, there is\nan election where this committee wins uniquely under this rule), to\nquite a varied landscape of monotonicity notions.  This allowed us to\npartially match syntactic properties of such rules to their normative\nproperties.\n\nThere is a number of follow-up directions for this research. For\nexample, whole-committee monotonicity (where all the members of the\ncommittee are shifted forward) is an interesting property. The\naxiomatic characterization of OWA-based rules remains an open\nproblem. Further, it is interesting to see whether there exist other\nproperties, e.g., those which relate to proportionality of\nrepresentation, that can be used to characterize different (subclasses\nof) committee scoring rules, or other multiwinner election systems\n(the works of Aziz et\nal.~\\cite{azi-bri-con-elk-fre-wal:j:justified-representation} and\nLackner and Skowron~\\cite{lac-sko:t:approval-thiele} made some headway\nin this direction). A formal axiomatic study which would allow to\ncompare committee scoring rules to other multiwinner election systems\nis an important, yet challenging question.\n \n\n\n\n\\subsubsection*{Acknowledgments.} \nPiotr Faliszewski was supported by the National Science Centre,\nPoland, under project 2016\/21\/B\/ST6\/01509.  Arkadii Slinko was\nsupported by the Royal Society of NZ Marsden Fund UOA-254. Piotr\nSkowron was supported by ERC-StG 639945 and by a Humboldt Research\nFellowship for Postdoctoral Researchers.  Nimrod Talmon was supported\nby a postdoctoral fellowship from I-CORE ALGO.\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\subsubsection*{\\bibname}}\n\\bibliographystyle{apalike}\n\n\\section{Introduction}\n\\allowdisplaybreaks\nBuilding private histograms is a task that underpins a variety of machine learning and data analytics tasks. \nHistograms enable building usable discrete representations, distributions and marginals.\nMaterializing histograms is thus a core subroutine in instantiating graphical models for synthetic data generation~\\citep{NIST21}, and hence they support numerous statistical analyses and inference tasks. \nThe problem has been heavily studied in the setting of differential privacy, with a number of results shown under variant models, such as the central model~\\citep{Dwork06,XuZXYY12, DworkR14}, local model~\\citep{bassilysmith,wangetal,hadamardresponse} and shuffle model~\\citep{ESA20,BalcerCheu20,DUMP20}.\n\nIn this paper, we revisit this foundational question, and show how differential privacy can be obtained via a simple sample-and-threshold mechanism, which can be readily implemented in a distributed setting.  \nImportantly, all the randomization needed for privacy is derived from the sampling operator: there is no additional explicit addition of noise. \nThis is particularly beneficial in scenarios when sampling is inherent, i.e., federated settings when only a uniformly chosen fraction of users are contacted. \nIn this case, privacy essentially comes ``for free''. \nEquipped with an efficient mechanism for histogram computation, we can apply it to a range of core analytics tasks (quantiles and heavy hitters), which in turn enable a broad spectrum of other computations.  \n\n\\paragraph{Our contributions.}\nIn this paper, we present a histogram mechanism that extends prior work as follows: \n\n\\begin{itemize}\n\\item We show that a simple sample-and-threshold approach provides an $(\\epsilon, \\delta)$-differential privacy guarantee for histograms. \n    \\item We show that the resulting mechanism can also answer heavy hitter, quantile and range queries.\n    \\item We show that the associated counts provide accurate frequency estimates for items from the input.\n    \\item Our proofs are compact and self-contained. \n\\end{itemize}\n\nIn more detail, we show that a Poisson sampling based approach is sufficient to provide differential privacy. \nThe key is to choose a sampling rate that is not too large compared to population size, and to prune items with low frequency in the sample, so that the presence of an item in the pruned sample does not indicate exactly how many instances were in the original population. \nWhile prior work has considered the ability of sampling to amplify the privacy bounds of a differentially private mechanism, in this work we show that sampling itself provides a DP histogram mechanism, similar to the pioneering work of \\cite{TrieHH} on heavy hitters.\nConsequently, the sample-and-threshold histogram mechanism can be implemented effectively while requiring very little effort from participating users. \nThe chief points of comparison are results in the shuffle model of differential privacy. \nWe claim that the sampling step is arguably simpler than many shuffle approaches (which require users to perturb their inputs, or add additional ``chaff'' messages to mask their values), while being of equivalent complexity to implement the server-side aggregation of messages. \nDeployed federated systems~\\citep{BonawitzEGHIIKK19} already implicitly sample from a large collection of eligible users, so the mechanism does not introduce any significant additional overhead or error. \n\n\\section{Preliminaries}\n\nWe consider the case where there are $n$ individuals who each hold \na value $x_i$, so that the collection of all user inputs defines a dataset $D$. \nOur goal is to construct a histogram of the frequency distribution according to a\nfixed set of buckets $B$. \nFor convenience, we assume that each input $x_i$ is already mapped into its corresponding bucket, and that the buckets are indexed by integers, so that each $x_i \\in [B]$. \nWe will describe a randomized mechanism, $\\mathcal{M}$, that can process datasets $D$ to give a distribution over output histograms, $H$. \n\nThe objective is to ensure that a sampled output histogram, $H$,  is close to the true histogram $H^*$, while ensuring that the output meets $(\\epsilon, \\delta)$-differential privacy (DP) \\citep{DworkR14}. \nFormally, we require\n\\begin{equation}\n\\Pr[ \\mathcal{M}(D) \\in \\mathbb{H}] \\leq \\exp(\\epsilon)\\Pr[\\mathcal{M}(D') \\in \\mathbb{H}] + \\delta\n\\label{eq:dp}\n\\end{equation}\n\\noindent\nfor any subset of possible output histograms $\\mathbb{H}$ and\nfor neighboring inputs $D$, $D'$ that differ in the input value of one individual. \nAs usual for $(\\epsilon, \\delta)$-DP, we expect $\\delta$ to be small, typically much less than $1\/n$. \n\n\\paragraph{Computational model.}\nOur mechanism is designed to operate in a federated (distributed) setting, where each client sends a message based on their input to a server, which then combines this information before reporting it to an analyst. \nSpecifically, the server aggregates the messages to produce the multiset of values reported (i.e., builds the frequency histogram of messages), and deletes some values which fall below a threshold $\\tau$. \nThis model sits between the shuffle and centralized DP model: \nthe  procedure is conceptually similar to the `shuffling' procedure, but with the minor additional step of removing small counts; meanwhile, it is easy to implement in the central DP model with a trusted aggregator. \nTo fully achieve the benefits of this model, \nwe assume for convenience that there is an entity which aggregates the data, similar to a shuffler in the shuffle model. \nFor a shuffler, applying the threshold would be a trivial final step before the shuffler releases the histogram. \nIndeed, we anticipate that this would be natural to do in any system that implements aggregation via secure hardware (e.g., SGX extensions). \nThen the data analyst only sees output under differential privacy, and is shielded from seeing any intermediate results without a formal privacy property.  \nAlternatively, the aggregation and thresholding step could be performed using techniques from multiparty computation on shares of the input gathered by two or more servers, wherein secure comparison to a public constant is a relatively lightweight operation~\\cite{Nishide:Ohta:07}. \nThe model can also be compared to the early notion of ``$k$-anonymity'', where the output is constrained so that every output item corresponds to at least $k$ individuals in the input~\\citep{Samarati:Sweeney:98}.  \nHere, we obtain $k$-anonymity for $k = \\tau$, the threshold value. \nAlthough $k$-anonymity has been criticized as a weak privacy notion, it carries an intuitive appeal for many lay users, and here we show that in this case we also achieve differential privacy. \n\n\\section{Sampling-Based Histogram Mechanism}\n\nIn the ($B$-bucket) histogram problem, each client $i$ holds a single item $x_i$ corresponding to a bucket $b_i \\in [B]$, and our aim is to produce a private histogram of item frequencies, \nsuch that a frequency associated with $x$ in the private histogram approximates the frequency of $x$ over the input distribution. \n\nThe algorithm is based on Bernoulli sampling. \nEach client out of $n$ is sampled with probability $p_s = m\/n$, so the expected size of the sample is $m$ (we later discuss different ways to implement this sampling). \nOur subsequent analysis will set an upper bound on the sample size $m$ in order to give a required privacy guarantee. \nThe algorithm makes use of a threshold $\\tau$, so that items whose sampled counts are at least $\\tau$ are reported in the histogram, while items \nwhose count falls below $\\tau$ are omitted from the histogram. \nNote then that the mechanism introduces no spurious items into the output: any item which is not present in the input cannot appear in the output histogram. \nIn addition, the error bounds of the algorithm are independent of the dimensionality of the underlying histogram, $B$. \n\n\\begin{theoremEnd}{lemma}\nThe probability that the number of samples of an item is more than $\\tau$ times its expectation is at most\n$\\delta$, for $\\tau = 3 + \\ln 1\/\\delta$. \n\\label{lem:pthetanew}\n\\end{theoremEnd}\n\n\\begin{proofEnd}\nGiven an item that occurs $k$ times in the input, \neach occurrence has probability $p_s = m\/n$ of being picked. \nThe expected number of sampled occurrences is then $k p_s = km\/n$. \n\nLet $X$ denote the random variable that counts the number of successes (times the prefix is picked) out of the $k$ trials, \nso $\\mathsf{E}[X] = km\/n$. \nThen, \n$X$ is a sum of $k$ Bernoulli random variables with parameter $p_s$. \nWe do a case split on $p_s$:\n\n\\paragraph{Case: $p_s \\leq 1\/k$.}\nIf $p_s \\leq 1\/k$, \nwe apply an (additive) Chernoff-Hoeffding bound to the mean of the $k$ trials: \n\\begin{align*} \n\\Pr[ X \\ge \\tau ] & = \\Pr\\Bigg[ \\frac{1}{k}X - \\frac{1}{k}\\mathsf{E}[X] \\ge\\left(\\tau p_s - p_s\\right) \\Bigg] \n\\\\ &\n\\le \n\\exp\\left(- D\\left( \\frac{\\tau }{k } \\middle\\| \\frac{1}{k}\\right)k\\right) . \n\\end{align*}\nHere, $D(p\\|q)$ denotes the K-L divergence (relative entropy) between the (Bernoulli) distributions with parameters $p$ and $q$. \nWe have\n\\begin{align*}\n-D(p\\|q) k  & = -{\\tau} \\ln\\left(\\frac{\\tau}{k} \\cdot \\frac{k}{1}\\right) - (k-\\tau)\\ln\\left(\\frac{k - \\tau}{k} \\cdot \\frac{k}{k-1}\\right)\n\\\\\n& =  {-\\tau} \\ln \\tau - (k-\\tau) \\ln\\left(1 - \\frac{\\tau-1}{k-1}\\right)\\\\\n& = -\\tau \\ln \\tau + (k-\\tau) \\ln\\left( \\frac{k-1}{k -\\tau}\\right) \\\\\n& = -\\tau \\ln \\tau + (k-\\tau) \\ln \\left(1 + \\frac{\\tau-1}{k-\\tau}\\right) \\\\\n& \\leq - \\tau \\ln\\tau + \\tau - 1\n\\end{align*}\n\\begin{equation}\n\\text{Hence, } \\qquad\n   \\Pr[ X \\ge \\tau ] \\leq \\exp(-\\tau \\ln \\tau + \\tau - 1)\n\\end{equation}\nFor this case, \nto achieve a target error bound $\\delta$, we rearrange to obtain\n$\\frac{\\tau}{e} \\ln \\frac{\\tau}{e}  = \\frac{1}{e} \\ln(1\/e\\delta)$, and apply Lambert's W function. \nThis gives \n$\\frac{\\tau}{e} = W(\\frac{1}{e} \\ln(1\/e\\delta)$, i.e., \n$\\tau = e W(\\frac{1}{e} \\ln \\frac{1}{e\\delta})$. \nNote that this case corresponds to the scenario where we do not publish the counts, but only indicate which items occurred more than $\\tau$ times in the sample. \n\n\\paragraph{Case: $p_s > 1\/k$.} \nIf $p_s > 1\/k$, we apply a (multiplicative) Chernoff bound: \n\\begin{align*} \n\\Pr[ X \\geq \\tau \\mathsf{E}[X]] & \\leq \\exp(-(\\tau-1)^2 \\mathsf{E}[X]\/(1 + \\tau)) \\\\ \n& = \\exp(-(\\tau-1)^2 k p_s\/(1+\\tau)) \\\\ & \\leq \\exp(-(\\tau-1)^2\/(1+\\tau))\n\\end{align*}\nIn this case, to achieve a target error bound $\\delta$, we can pick \n$\\tau = 3 + \\ln(1\/\\delta)$, \nand obtain \n\\[\\exp(-(2 + \\ln 1\/\\delta)^2\/(4 + \\ln 1\/\\delta)) < \\exp(-\\ln(1\/\\delta)) = \\delta.\\] \nThe second case is stricter for all $\\tau >1$, so we will use this setting of $\\tau$ in what follows. \n\\end{proofEnd}\n\nThe proof of this claim, and of most technical lemmas, is deferred to the Appendix. \nWe next give a bound on the ratio of probabilities of seeing the same output on neighboring inputs. \n\n\\begin{theoremEnd}{lemma}\nGiven two neighboring inputs $D$, $D'$, such that $D$ differs in one item from $D'$, \nthe ratio of probabilities of seeing a cell with a given value $v$ is bounded by $\\frac{k+1}{k + 1 - v}$, \nwhere $k+1$ is the number of copies of the given item in input $D$\nand $k > v$. \n\\label{lem:ratio}\n\\end{theoremEnd}\n\n\\begin{proofEnd}\nThe case to focus on is when input $D$ has one extra copy of a particular item compared to $D'$, at some intermediate stage of the algorithm. \nFor notation, we will write \n$S_k(n, s, v)$ to denote the number of ways to succeed in collecting exactly $v$ instances of the target item while picking $s$ items out of $n$, when there are $k$ total instances of the item. \nWe can observe that there is a simple combinatorial expression for this quantity: we count the number of combinations where we pick a particular subset of size $v$ from the $k$ instances, and a particular subset of size $s - v$ from the remaining $n-k$ examples. \n\\begin{equation}\n    S_k(n, s, v) = {k \\choose v} {n-k \\choose s-v}\n\\end{equation}\nOur goal is to bound the ratio of probabilities of seeing a count of $v$ copies of the item in the output of $D$, who has $k+1$ copies of the item, and of $D'$ who holds $k$ copies.   \nThe probability that the sample size is exactly $s$ is given by\n$P_s = p_s^{s}(1-p_s)^{n-s}$. \nFor a given sample size $s$, \nthe probability for $D$ is $S_{k+1}(n,s,v) P_s$, and for \n$D'$ it is $S_{k}(n, s, v) P_s$. \nThen this ratio of probabilities is given by\n\\begin{align*}\n\\frac{S_{k+1}(n, m , v)P_s}{S_{k}(n, m, v)P_s} & = \\frac{ {k+1 \\choose v} { n - k - 1 \\choose m - v} }{ {k \\choose v } { n - k \\choose m - v }} \n  = \\frac{ (k+1) ( n -k -m -v) }{ (n - k) (k  + 1 - v) }\n\\\\  = & \\left(1 - \\frac{m+v}{n -k}\\right)\\left(\\frac{k+1}{k+1-v}\\right)\n\\leq \\frac{k + 1}{k + 1 - v}\n\\end{align*} \n\nThen we can bound this ratio across all sample sizes as simply \n$\\sum_{s = 0}^{n} \\frac{k+1}{k+1 - v} P_s = \\frac{k+1}{k+1-v}$.\n\\eat{\nFor $D'$, we can break their successful instances into (a) those where they sample item $n$ and need to collect $\\tau$ instances with their remaining $m-1$ samples out of $n-1$ items, or (b) where they do not sample item $n$, and thus need to collect $\\tau$ instances from $m$ samples out of $n$. Thus, \n\\begin{equation}\nS_k(n, m, \\tau) = S_k(n-1, m, \\tau) + S_k(n-1, m-1, \\tau)\n\\label{eq:dprime}\n\\end{equation}\n\nFor $D$, we first derive a useful relationship that corresponds to the case that $D$ does not pick index $n$. \nThis is equivalent to considering their success on a smaller instance without the $n$'th item. \n$D$ will succeed if they pick exactly $\\tau$ instances of the target prefix from the $k$ instances\n(and $m-\\tau$ from any of the remaining $n-1-k$ items), or pick more than $\\tau$.\nWe can express this in our notation as\n\\[ S_{k}(n-1, m, \\tau) = {k \\choose \\tau} {n-k-1 \\choose m-\\tau} + S_k(n-1, m, \\tau+1)\n\\]\n\nRearranging, we obtain that\n\\[ {n-k-1 \\choose m - \\tau} \\leq  {k \\choose \\tau}^{-1} S_{k}(n-1, m, \\tau)\n\\]\n\nThe set of routes to success for $D$ look similar to those for $D'$, with one extra case: \n(a) $D$ does not pick item $n$, and succeeds with $m$ samples over the $n-1$ remaining items;\n(b) $D$ does pick $n$, and picks $\\tau$ or more items from the remaining $n-1$;\n(c) or else $D$ picks index $n$ and exactly $\\tau-1$ copies of the target prefix from among the \n$k$ copies, along with $m - \\tau$ from the remaining $(n-1) - k$ items. \nThis gives\n\\begin{align*}\n    S_{k+1}(n, m, \\tau) & =  S_k(n-1, m, \\tau) + S_k(n-1, m-1, \\tau) + {k \\choose \\tau-1} {n-k-1 \\choose m-\\tau} \\\\\n    & \\leq  S_k(n-1, m, \\tau) + S_k(n-1, m-1, \\tau) + {{k \\choose \\tau-1}\\over {k \\choose \\tau}} S_k(n-1, m, \\tau) \\\\\n    & =  S_k(n-1, m-1, \\tau) + \\left(1 + \\frac{\\tau}{k - \\tau +1}\\right)S_k(n-1, m, \\tau)\\\\\n    & \\leq  \\left(1 + \\frac{\\tau}{k-\\tau+1}\\right)S_k(n, m, \\tau) \n\\end{align*}\n\n\\noindent\nwhere the last line follows from \\eqref{eq:dprime}.}\n\\end{proofEnd}\n\n\\begin{theorem}\n\\label{thm:dphist}\nThe resulting histogram obeys $(\\epsilon, \\delta)$-differential privacy, \nfor $\\delta = O(\\exp(-\\tau))$\nand $\\epsilon = O(\\frac{m}{n} \\ln (1\/\\delta)) \\le 1$. \n\\end{theorem}\n\n\\begin{proof}\nConsider the treatment of an item $x$ between two neighboring inputs $D$ and $D'$. \nIf $f_x = f'_x$, i.e., the number of copies of $x$ is the same in both inputs, then $x$ is treated identically in both cases. \nOtherwise, wlog we are looking at an $x$ such that $f_x = f'_x + 1 = k + 1$.\nWe condition on the event that the number of samples of the item $x$ is not more than $\\tau$ times its expectation. \nCall this event $E$. \nBy Lemma~\\ref{lem:pthetanew}, event $E$ holds except with probability $\\delta = \\exp(-(\\tau-1)^2\/(\\tau +1) ) = O(\\exp(-\\tau))$. \nWe condition on $E$ holding, and just account for this probability in our final reckoning. \n\nSuppose that the count of $x$ for $D$ is less than $n\/m$. \nThen, by our assumption of $E$, $D$ will not sample $\\tau$ copies of $x$, \nand so both $D$ and $D'$ would output the same histogram. \nHence, the probability of all outputs are equal on $D$ and on $D'$.\n\nOtherwise, the count of $x$ ($k+1$ for $D$) is at least $n\/m$, and by our assumption $D$ samples at most \n$v \\leq \\tau m(k+1)\/n$ copies of $x$. \nThen, by Lemma~\\ref{lem:ratio}, we can state that for the mechanism $M$, the probability of seeing a given output histogram $H$ satisfies: \n\\begin{align}\n\\frac{\\Pr[ M(D) = H | E ] }{\\Pr[M(D') = H | E]} & \\leq \\frac{k+1}{k+1 -v} \\leq \\frac{k+1}{k+1 - \\tau(k+1) m\/n} \\nonumber \\\\ &\n= \\frac{n}{n - \\tau m}\n\\label{eq:thetamn}\n\\end{align}\nWe will assume that $m = c_\\epsilon \\frac{n}{\\tau}$ for a constant $c_\\epsilon \\leq 1 - 1\/e$ that depends on $\\epsilon$. \nThe effect is to ensure that the sample size $m$ is a small fraction of $n$. \nSubstituting this assumption in \\eqref{eq:thetamn}, we conclude \n\\begin{equation}\n    \\label{eq:finalepsbound}\n\\frac{\\Pr[M(D) = H | E]}{\\Pr[M(D') = H | E]} \n= \\frac{n}{n - c_\\epsilon n} = \\frac{1}{1-c_\\epsilon} := \\exp(\\epsilon)\n\\end{equation}\n\nThat is, except with probability $\\delta$, we have $\\epsilon$-differential privacy~\\eqref{eq:dp}. \nRearranging, we set $c_\\epsilon = 1 - \\exp(-\\epsilon)$. \nFor small $\\epsilon$, we can approximate $c_\\epsilon = \\epsilon$.\nWe can also write \n\\[\\epsilon = \\ln\\frac{1}{1 - c_\\epsilon} = \\ln 1 + \\tfrac{c_\\epsilon}{1 - c_\\epsilon} = \\ln 1 + \\tfrac{m\\tau\/n}{1-c_\\epsilon} \\leq \\ln(1 + \\tfrac{e m\\tau}{n})\n\\]\nwhere the last step uses $c_\\epsilon \\leq 1 - 1\/e$ for $\\epsilon \\leq 1$. \nThis proves $(O(\\frac{m\\tau}{n}), O(\\exp({-\\tau }))$-differential privacy, as claimed. \n\\end{proof}\n\n\\subsection{Fixed sized sampling}\n\nFor practical efficiency, we would often like to work with a fixed size sample. \nHowever, the above histogram protocol performs Poisson sampling instead.\nThe reason is that if the fixed size of the sample, $m$, is known, then we are effectively also releasing the number of samples that were suppressed by the $\\tau$ threshold (by adding up the released counts, and subtracting from $m$). \nThis potentially leaks information. \nConsider the case where $D'$ contains $n$ copies of the same item, while $D$ contains $n-1$ copies of the same item, and one unique item. \nWith probability $m\/n$, the mechanism on input $D$ samples the unique item along with $m-1$ other items, and so produces a sample of size $m-1$. \nBut on input $D'$, there is zero probability of producing a sample smaller than $m$. \nThis forces $\\delta \\geq m\/n$, which is typically too large for $(\\epsilon, \\delta)$-DP (we usually seek $\\delta \\ll 1\/n$). \n\nPerforming Poisson sampling with $p_s$ addresses this problem: the expected sample size is the same, but we no longer leak the true size of the sample before thresholding.  \nIndeed, we can see that the (observable) size of the sample is differentially private: given two inputs $D$ and $D'$ such that $D$ has one additional unique item, the distribution of sample sizes are close, up to a factor of $1 + p_s\/(1-p_s) = 1 + m\/(n-m)$, which is below $\\exp(\\epsilon_i)$ by~\\eqref{eq:finalepsbound}. \n\nImplementing Poisson sampling may appear costly: naively, the server would contact $n$ clients instead of $m$, where we expect $n \\gg m$. \nHowever, we can perform the sampling by contacting much fewer clients, since the size of the sample is tightly concentrated around its expectation.  \n\n\\begin{theoremEnd}{lemma}\nSampling $m + O(\\sqrt{m})$ clients is sufficient to apply the sample-and-threshold mechanism, \nwith high probability. \n\\end{theoremEnd\n\n\\begin{proofEnd}\nObserve that, with high probability, the size of the (Poisson) sample will be close to expected value of $m$. \nIn particular, by a Chernoff bound, the probability that the sample size is more than $c\\sqrt{m}$ larger than $m$ is\n\\[ \\Pr[ s > (1 + c\/\\sqrt{m})m] \\leq \\exp(-c^2\/3)  . \\]\nHence, for $c$ a suitable constant (say, 10), this probability is negligibly small.  \nTo realize this sampling, we contact a fixed size number of clients $s = m + c\\sqrt{m}$, and then have each client perform a Bernoulli test on whether to participate: with probability $m\/s$, it participates, otherwise it abstains. \nAn abstaining client can, for example, vote for a unique element (e.g., an item based on a hash of its identifier), and so be automatically discounted from the protocol, without revealing this information to the aggregator. \n\\end{proofEnd}\n\n\\subsection{Accuracy Bounds}\n\nThe histogram produced by the mechanism is ultimately based on sampling and pruning, so for any item whose frequency is sufficiently above the pruning threshold, \nthen its frequency within the histogram is an (almost) unbiased estimate of its true frequency. \nThere is a small gap, since even for an item with high frequency, there is a small chance that it is not sampled often enough, and so its estimate will fall below the threshold $\\tau$ (in which case we do not report the item). \n\n\\paragraph{Probability of omitting a heavy item.}\nWe first consider the probability that a frequent item is not reported by the algorithm. \n\\begin{theoremEnd}{lemma}\nThe sample and threshold histogram protocol omits an item whose true absolute count is $W$ with probability\nat most $\\exp(-(\\frac{Wm}{n} - \\tau)^2\\frac{n}{2Wm})$. \n\\end{theoremEnd}\n\n\\begin{proofEnd}\nFor an item with (absolute) frequency $W$ out of the $n$ input items, it is reported \nif the number of sampled occurrences exceeds $\\tau$. \nSimilar to the analysis above, we can apply a Chernoff-Hoeffding bound to the random variable $X$ that counts the number of occurrences of the item. \nNow, the probability of each sample picking the item is $W\/n$, and the expected number in the sample is\n$Wm\/n > \\tau$. \nFor convenience, we will write $w = Wm\/n$ for this expectation. \nWe have that\\footnote{Here we are sampling without replacement.  However, the bounds for sampling with replacement are still valid here.}%\n\\begin{align*}\n    \\Pr[ X \\leq \\tau] & = \\Pr\\left[ X \\leq \\frac{\\tau}{w}w\\right] \\\\ & = \n    \\Pr\\left[X  \\leq \\left(1 - \\frac{w - \\tau}{w}\\right)\\mathsf{E}[X] \\right]  \\\\ & = \\exp\\left(-\\frac{(w-\\tau)^2}{2w}\\right)\n\\end{align*}\\end{proofEnd}\n\\paragraph{Numeric Example.}\nWhen $w := Wm\/n$ is sufficiently bigger than $\\tau$, this gives a very strong probability.  \nFor example, consider the case $n=10^6$, $\\epsilon = 1$, and we set $\\tau = 20$ to obtain a $\\delta$ of $10^{-8}$. \nThe expected sample size $m = 31,606$, and for an item that occurs 0.2\\% of the time in the input, we expect to sample it\n$w = 63$ times. \nThis gives a bound of $\\exp(-14) < 10^{-7}$ that such an item is not detected.\n\n\\paragraph{Frequency estimation bounds.}\nMore generally, we can use the frequency of any item in the histogram as an estimate for its true occurrence rate. \n\n\\begin{theoremEnd}{lemma}\nWe can estimate the frequency of any item whose relative frequency is $\\phi$ within $\\gamma$ relative error with probability \n$O(\\exp(-\\gamma^2 \\phi m))$.\n\\end{theoremEnd}\n\n\\begin{proofEnd}\nApplying the same Chernoff-Hoeffding bound as above, we have for $\\gamma < 1$, \n\\[ \n\\Pr[ |X - \\mu| > \\gamma\\mu] = 2\\exp( - \\gamma^2 \\mu\/3) = \\beta \\]\nRearranging, we obtain \n$\\mu = \\frac{3}{\\gamma^2} \\ln (1\/2\\beta)$. \nSuppose we aim to find all items whose frequency is at least $\\phi$, and estimate their frequency with relative error at most $\\gamma$. \nThen we have $\\mu = \\phi m \\geq \\phi \\frac{\\epsilon n}{\\tau} = \\frac{3}{\\gamma^2} \\ln (1\/2\\beta)$. \n\\end{proofEnd}\n\n\\paragraph{Numeric Example.}\nWe can substitute values into this expression to explore the space. \nFor example, if we set\n$\\epsilon = 1$, $\\ln (1\/2\\beta) = 10$, $\\tau = 10$ and $\\gamma = 1\/\\sqrt{10}$, then \nwe obtain $\\phi = 3 \\times 10^3\/n$ --- in other words, provided $n > 3 \\times 10^5$, we can accurately find \nestimates of frequencies that occur 1\\% of the time (except with vanishingly small probability). \n\n\\paragraph{Remark.}\nIt is instructive to compare these bounds to those that hold for the shuffle model. \nAccording to \\cite{BalcerCheu20}, \naddition of appropriately parameterized Bernoulli random noise to reports from $n$ clients yields\n$(\\epsilon, \\delta)$-DP, with error that scales as $O(\\frac{1}{\\epsilon^2 n} \\log (1\/\\delta))$ for $\\epsilon \\leq 1$, provided $n$ is large enough. \nExpressing our bound on the estimate of any frequency, we obtain error $O(1\/\\sqrt{m}) = O(\\sqrt{\\frac{\\ln 1\/\\delta}{\\epsilon n}})$ from sampling, plus\nerror from rounding small values down to zero, which is bounded by $O(\\tau\/m) = O(\\ln(1\/\\delta)\/m) = O(\\ln^2(1\/\\delta)\/(\\epsilon n))$. \nNaively, it might seem that the shuffle bounds are preferable, due to the stronger dependence on $n$ ($O(1\/n)$ vs.\\ $O(1\/\\sqrt{n})$). \nHowever, this misses the point that in practical federated computing settings, the server can contact only a fixed size cohort of $m$ clients out of a much larger (and sometimes unknown) population $n$. \nIn this case, results in both the shuffle and sample-and-threshold paradigms incur \\textit{the same} sampling error of \n$O(1\/\\sqrt{m})$. \nThen shuffling introduces additional noise of $O(\\frac{1}{\\epsilon^2 m} \\log (1\/\\delta))$, whereas sample-and-threshold\nincurs zero additional noise on items that exceed the $\\tau$ threshold, and at most\n$O(\\ln(1\/\\delta)\/m)$ on small items. \nHence, we argue that when shuffling implicitly samples from the input, the sample-and-threshold approach has superior error guarantees. \nWe confirm this observation empirically in Section~\\ref{sec:expts}, where we compare accuracy of both approaches. \n\n\n\\section{Heavy hitters and Quantiles via Histograms}\n\n\\subsection{Heavy Hitters}\nWe next show how to use the basic histogram protocol to find the (hierarchical) heavy hitters from the input. \nThis result follows the outline and notation of the TrieHH algorithm \\citep{TrieHH}, to allow easy comparison. \n\nThe heavy hitters algorithm proceeds over $L$ levels, to build up a trie of depth $L$. \nAt each level, we materialize a histogram of those prefixes of items from the input that extend the current trie. \nThis allows us to add items to the current trie based on the threshold $\\tau$, and include the observed count of prefix for each node in the trie, provided it is more than $\\tau$. \nWe can view the TrieHH protocol as materializing a histogram at each level, with progressively finer cells. \nIn the protocol as originally described, cells whose ancestor in a previous level did not exceed the $\\tau$ threshold are not eligible for consideration. \nHowever, the privacy proof still applies if we do not enforce such restrictions. \nWe denote our version of the protocol using the new histogram protocol as TrieHH++, to indicate that the trie is augmented with count information. \n\n\\begin{theoremEnd}{lemma}\n\\label{lem:hh}\nThe TrieHH++ protocol satisfies $(\\epsilon, \\delta)$-DP for \n$\\delta = L \\exp(-\\tfrac{(\\tau-1)^2}{1+\\tau})$\nand \n$\\epsilon = O(L m \\ln(1\/\\delta) \/ n)$. \n\\end{theoremEnd\n\nThe essence of the proof is that the output of the algorithm is the $L$-fold composition of a differentially private mechanism, with some post-processing.  \nBy the differential privacy of the basic histogram protocol (Theorem~\\ref{thm:dphist}), the result follows. \n\n\\begin{proofEnd}\nIn more detail, \nwe can view the protocol as publishing a histogram at each level, where the granularity of the cells is refined in each round. \nThe protocol enforces that if a prefix is not included at a particular level, then none of its extensions are published in any subsequent level. \nHowever, we can view this as ``post-processing'', and analyze the simpler algorithm that does not enforce this constraint. \nApplying Theorem~\\ref{thm:dphist}, we have that each round satisfies $(\\epsilon_i, \\delta_i)$-DP for some $\\epsilon_i$ and $\\delta_i$.  \n\n\n\n\n\n\n\n\nThen we argue that the output of the full protocol is the $L$-fold composition of the mechanisms $M_i$. \nAssuming $\\epsilon_i = \\epsilon'$ and $\\delta_i = \\delta'$ for all $i$, then \nusing basic composition, we obtain a bound of $(L\\epsilon', L\\delta')$-differential privacy, leading to the result stated in the theorem claim. \nFor $\\epsilon_i = \\epsilon' < 1$, we can also obtain a tighter bound, of \n$(L\\epsilon'^2 + \\epsilon' \\sqrt{L \\log 1\/(\\delta' L)}, 2L \\delta')$ using advanced composition~\\citep{DworkR14}. \n\\end{proofEnd}\n\n\\paragraph{Remark.}\nWe remark that if the objective is only to find the heavy hitters, then the factor of $L$ can be dropped from these bounds. \nThat is, instead of proceeding in rounds, we simply apply the basic histogram protocol to the full inputs, and report the items which survive the thresholding process (along with their associated counts if desired). \nFollowing the above analysis, the resulting output is $(\\epsilon, \\delta)$-differentially private, \nwhen setting $\\tau = 3 + \\ln 1\/\\delta$ and\n$m \\leq \\frac{\\epsilon n}{\\tau}$ to get $\\epsilon = \\tau m\/n$.\nThe motivation for having $L$ rounds given by~\\cite{TrieHH} is to reduce the exposure of the server to private information: they only observe prefixes from clients that extend shorter prefixes that are already known to be popular.  However, this does not impact on the formal differential privacy properties of the output. \n\n\n\n\n\\eat{\n\\begin{figure*}[t]\n\\subcaptionbox{Accuracy: $\\epsilon = 0.1, B = 100$ \\label{fig:bin:eps0.1b100}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=0.1, domain_size=100, population=1e+06, sample_size=1806, distribution=Binomial, top_k=50.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 1.0, B = 100$ \\label{fig:bin:eps1.0b100}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=1.0, domain_size=100, population=1e+06, sample_size=29509, distribution=Binomial, top_k=50.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 0.1, B = 10000$ \\label{fig:bin:eps0.1b10000}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=0.1, domain_size=10000, population=1e+06, sample_size=1806, distribution=Binomial, top_k=5000.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 1.0, B = 10000$ \\label{fig:bin:eps1.0b10000}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=1.0, domain_size=10000, population=1e+06, sample_size=29509, distribution=Binomial, top_k=5000.png}}\n\\caption{Top-$k$ accuracy results for Binomially distributed data}\n\\label{fig:bin}\n\\end{figure*}\n}\n\n\\subsection{Quantiles}\n\nFinding the quantiles is a common analytics task to describe the distribution of values held by the clients.\nWe describe two approaches to finding quantiles, both making use of our histogram mechanism. \n\n\\paragraph{Single quantiles via interactive search.}\nGiven client inputs which fall in the range $[0, 1]$, we seek a value $f$ such that \nthe fraction of clients whose value is below $f$ is (approximately) $\\phi$. \n\n\\begin{theoremEnd}{lemma}\nGiven a $\\phi > \\tau\/m$, \nwe can use $h$ applications of the histogram mechanism to \nfind a value $f$ such that \n$f \\pm 2^{-h}$ is a $\\phi \\pm O(m^{-1\/2})$ quantile, \nwith $(O(hm \\ln(1\/\\delta)\/n, h\\delta)$-DP. \n\\end{theoremEnd}\n\n\\begin{proofEnd}\nThe quantile query can be carried out by a binary search: \nwe begin by creating a histogram with buckets $[0, \\frac12], [\\frac12, 1]$, and recursively try different split points $[0, t], [t, 1]$ until \nwe obtain a result with approximately a $\\phi$ fraction of points in the first bucket, at which point we can report $t$ as the $\\phi$-quantile. \nProvided $\\phi$ is sufficiently larger than $\\tau\/m$ (and smaller than $1 - \\tau\/m$), then we are unlikely to hit any cases where a bucket count is removed. \nAs a result, the error will primarily the error from sampling, which is $O(1\/\\sqrt{m})$~\\citep{Lane:03}, plus the error from rounding, which is $2^{-h}$ if we perform $h$ steps of binary search.  \nThat is, we find a result $t$ such that there is a point \nin the range $[t-2^{-h}, t+2^{-h}] := t \\pm 2^{-h}$ that dominates $\\phi \\pm O(1\/\\sqrt{m})$. \nThe privacy guarantee is $\\epsilon = O(h m \\ln (1\/\\delta)\/n)$, from the composition of $h$ queries.  \n\\end{proofEnd}\n\nThis approach is very effective for single queries, but is less desirable when we have a large number of quantile queries to answer in parallel, in which case the hierarchical histogram approach is preferred. \n\n\\paragraph{Quantiles and range queries via hierarchical histograms.}\nA common technique to answer quantile and range queries in one-dimension is to make use of hierachical histograms: histograms with geometrically decreasing bucket sizes, so that any range can be expressed as the union of a small number of buckets.  \nWe can observe that the trie built as part of the TrieHH++ protocol is exactly such a hierarchical histogram, and hence can be used to answer quantile queries, with the same privacy (and similar accuracy) guarantees as for heavy hitters. \n\nAssume again that each client has an input value in the range $[0,1]$ (say), and we can interpret these as prefixes, corresponding to subranges. \nIf we set the branching factor of the trie, $\\alpha$, to 4, then the input value\n$\\frac{1}{3}$ falls in the range $[0.25, 0.5]$ for a prefix of length 1; \nand in the range $[\\frac{5}{16}, \\frac{6}{16}]$ for a prefix of length 2. \nWith this mapping of input values to prefixes, the algorithm proceeds as before, and outputs the (DP) trie with weights on nodes. \n\nTo answer a range query $[0, r]$, we decompose the range greedily into chunks that can be answered by the trie. \nFor example, if $\\alpha=4$, and we want the range $[0,0.7]$, \nwe find the chunks\n$[0, \\frac14], [\\frac14, \\frac24]$ at level 1;\n$[\\frac{8}{16}, \\frac{9}{16}], [\\frac{9}{16}, \\frac{10}{16}], [\\frac{10}{16}, \\frac{11}{16}]$ at level 2; \nand so on. \nIf the trie has $L$ levels, then any prefix query can be answered with $L(\\alpha - 1)$ probes to the histograms ($\\alpha - 1$ for each level). \nMoreover, quantile queries are answered by finding range queries whose weight is (approximately) the desired quantile $\\phi$. \n\nDue to the pruning, we will not have information on any ranges whose sampled weight is less than $\\tau$, corresponding to \na $\\tau\/m$ fraction of mass.  \nThis will give a worst-case error bound of $(\\alpha -1)\\tau\/m$ per level, and so $L(\\alpha-1)\\tau\/m$ over all levels. \nBased on our setting of $m$ proportional to $\\epsilon n \/ (L\\tau)$, we obtain a total error of \n$ (\\alpha - 1)(L\\tau)^2\/\\epsilon n$.\nIn summary, as a consequence of the privacy guarantee from Lemma~\\ref{lem:hh}, we can state: \n\n\n\n\n\\begin{lemma}\nWe can build a set of $L$  $(O(Lm \\ln(1\/\\delta)\/n, L\\delta)$-private histograms to answer any quantile query $\\phi$ \nto find a value $f$ such that \n$f \\pm 2^{-L}$\nis a $\\phi \\pm O((L\\tau)^2\/\\epsilon n)$ quantile. \n\\end{lemma}\n\n\n\\paragraph{Numeric Example.}\nPicking similar test values as above shows that this can give reasonable accuracy for $n$ large enough. \nFor $\\tau = 10$, $L = 10$, $\\alpha = 2$, $\\epsilon = 1$, the error bound yields\n$10^4\/n$.  \nSo for $n > 10^6$, we obtain rank queries (and quantiles) in this space with error around $0.01$. \n\n\\eat{\n\\begin{figure*}[t]\n\\subcaptionbox{Accuracy: $\\epsilon = 0.1, B = 100$ \\label{fig:geom:eps0.1b100}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=0.1, domain_size=100, population=1e+06, sample_size=1806, distribution=Geometric, top_k=50.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 1.0, B = 100$ \\label{fig:geom:eps1.0b100}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=1.0, domain_size=100, population=1e+06, sample_size=29509, distribution=Geometric, top_k=50.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 0.1, B = 10000$ \\label{fig:geom:eps0.1b10000}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=0.1, domain_size=10000, population=1e+06, sample_size=1806, distribution=Geometric, top_k=5000.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 1.0, B = 10000$ \\label{fig:geom:eps1.0b10000}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=1.0, domain_size=10000, population=1e+06, sample_size=29509, distribution=Geometric, top_k=5000.png}}\n\\caption{Top-$k$ accuracy results for Geometrically distributed data}\n\\label{fig:geom}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\subcaptionbox{Accuracy: $\\epsilon = 0.1, B = 100$ \\label{fig:unif:eps0.1b100}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=0.1, domain_size=100, population=1e+06, sample_size=1806, distribution=Uniform, top_k=50.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 1.0, B = 100$ \\label{fig:unif:eps1.0b100}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=1.0, domain_size=100, population=1e+06, sample_size=29509, distribution=Uniform, top_k=50.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 0.1, B = 10000$ \\label{fig:unif:eps0.1b10000}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=0.1, domain_size=10000, population=1e+06, sample_size=1806, distribution=Uniform, top_k=5000.png}}%\n\\subcaptionbox{Accuracy: $\\epsilon = 1.0, B = 10000$ \\label{fig:unif:eps1.0b10000}}{\n\\includegraphics[width=0.25\\textwidth]{figs\/Absolute error at eps=1.0, domain_size=10000, population=1e+06, sample_size=29509, distribution=Uniform, top_k=5000.png}}\n\\caption{Top-$k$ accuracy results for Uniformly distributed data}\n\\label{fig:unif}\n\\end{figure*}\n}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[trim=580 25 580 25,clip,width=0.9\\textwidth]{figs\/hist-legend.pdf}\n\\subcaptionbox{ Binomial, $B=2^6$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Binomial-D64.pdf}}%\n\\subcaptionbox{ Geometric, $B=2^6$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Geometric-D64.pdf}}%\n\\subcaptionbox{ Shakespeare, $B=2^6$ \\label{fig:acc:shake:64}}{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Shakespeare-D64.pdf}}\n\\subcaptionbox{ Binomial, $B=2^{10}$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Binomial-D1024.pdf}}%\n\\subcaptionbox{ Geometric, $B=2^{10}$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Geometric-D1024.pdf}}%\n\\subcaptionbox{ Shakespeare, $B=2^{10}$ \\label{fig:acc:shake:1024}}{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Shakespeare-D1024.pdf}}\n\\subcaptionbox{ Binomial, $B=2^{14}$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Binomial-D16384.pdf}}%\n\\subcaptionbox{ Geometric, $B=2^{14}$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Geometric-D16384.pdf}}%\n\\subcaptionbox{ Shakespeare, $B=2^{14}$ \\label{fig:acc:shake:16384}}{\\includegraphics[width=0.33\\textwidth]{figs\/hist-acc-Shakespeare-D16384.pdf}}\n\\caption{Accuracy results on Binomial, Geometric and Shakespeare datasets}\n\\label{fig:accuracy}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[trim=580 25 580 25,clip,width=0.9\\textwidth]{figs\/hist-legend.pdf}\n\\subcaptionbox{ Binomial, $B=2^8$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-rec-Binomial-D256.pdf}}%\n\\subcaptionbox{ Geometric, $B=2^8$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-rec-Geometric-D256.pdf}}%\n\\subcaptionbox{ Shakespeare, $B=2^8$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-rec-Shakespeare-D256.pdf}}\n\\subcaptionbox{ Binomial, $B=2^{12}$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-rec-Binomial-D4096.pdf}}%\n\\subcaptionbox{ Geometric, $B=2^{12}$ }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-rec-Geometric-D4096.pdf}}%\n\\subcaptionbox{ Shakespeare, $B=2^{12}$ \\label{fig:rec:shake:4096} }{\\includegraphics[width=0.33\\textwidth]{figs\/hist-rec-Shakespeare-D4096.pdf}}\n\n\\caption{Top-$k$ recall results on Binomial, Geometric and Shakespeare datasets}\n\\label{fig:recall}\n\\end{figure*}\n\n\\section{Experiments}\n\\label{sec:expts}\nTo validate our theoretical understanding, we performed a set of experiments on synthetic data using the histogram mechanism that we have developed. \nThe mechanism performs sampling on a population of size $n$ for a target sample size $m$, and applies an appropriate threshold to the resulting sample, to achieve an $(\\epsilon, \\delta)$-DP guarantee. \nWe compared against alternative mechanisms that also provide the same level of privacy when applied to the sampled set of clients: \ncentral differential privacy, via Laplace noise addition, local differential privacy based on Hadamard encoding of elements from the domain~\\citep{hadamardresponse}, and a shuffling-approach which adds Bernoulli noise~\\citep{BalcerCheu20}. \n\nWe worked with the text from the complete works of Shakespeare\\footnote{\\url{http:\/\/shakespeare.mit.edu\/}},\nwhere we extract each word, consistently map the words to one of the $B$ buckets, and count the total number of words in each bucket. \nWe also use synthetic data generated by distributions providing different frequency distributions: Geometric and Binomial distributions over the $B$ cells of the histogram. \nFor the Binomial data, each client draws from the Binomial distribution with $n=B$ and $p=0.5$ to choose a histogram bucket. \nFor the Geometric data, each client draws from a Geometric distribution with $p=1\/\\sqrt{B}$ to pick a histogram bucket. \nThese parameters are chosen to model the non-uniform frequency distributions seen in practice, where the most popular items occur approximately 1-5\\% of the time. \n\nWe experimented with a range of privacy parameters $\\epsilon$, $\\delta$, histogram sizes $B$, and population sizes $n$. \nWe pick a default $\\delta = 10^{-8}$, which yields a threshold $\\tau = 21$. \nWe simulate a population of $n = 10^6$ clients, \nand measure the accuracy of recovering the frequencies for each mechanism. \nWe compare the absolute difference of the estimated frequencies to those from the full population, \nand also measure the recall for the top-$k$ heaviest buckets for $k=B\/10$, i.e., the largest 10\\% of frequencies. \nIn the plots, we focus on showing results for the range of $\\epsilon = 0.1$ (high privacy) to $\\epsilon = 1.0$ (medium privacy) regimes, consistent with the range where all the mechanisms have privacy guarantees.\nWe vary the size of the histograms ($B$) from tens up to tens of thousands. \nError bars show the standard error over 10 repetitions of each mechanism. \nPlots for other parameter settings are withheld for brevity, but support the same conclusions.  \n\n\\paragraph{Accuracy results.}\nOur results on accuracy are shown in Figure~\\ref{fig:accuracy}. \nEach row shows results for a different histogram size, from small ($B=2^6$), to large ($B=2^{14}$); \neach column shows results on a different dataset (Binomial, Geometric or Shakespeare data). \nThe y-axis shows the absolute error, expressed as a fraction of the total input size.\nHence, we want this to be as low as possible, and ideally much smaller than 0.1\\%, say. \n\nSome results immediately stand out: the results from local differential privacy are much weaker, and frequently the error is sufficiently large that the line does not appear on the plots (similar results were seen for other choices of frequency oracle, such as direct encoding and unary encoding---we use the Hadamard encoding as it obtained the best accruacy for these experiments).  \nThis is consistent with our understanding of LDP, and further motivates the desire to achieve accuracy closer to the centralized case in federated settings. \nThe approach from the shuffle model, where each client adds Bernoulli noise to each cell of the histogram (i.e., for each cell they report a 1 value with some probability $q$) incurs higher error for small $\\epsilon$ (where more noise is added by the sampled clients). \nThe gap is larger as the size of the histogram increases, since there are more chances for cells to incur more noise.  \nMost intriguingly, the approach of adding Laplace noise, which is the gold standard in the centralized case, does not obtain the least error in this setting.  \nRather, the sample and threshold approach, which does not add explicit noise, but just removes small sampled counts, often achieves less error, particularly for small $\\epsilon$, where the magnitude of the Laplace noise is larger. \nThis is more pronounced for larger histograms. \nThe exception is for the Shakespeare data for larger histograms (Figures~\\ref{fig:acc:shake:1024} and~\\ref{fig:acc:shake:16384}). \nHere, the combination of skewed data, and smaller sample sizes for smaller $\\epsilon$, mean that only a small fraction of the histogram buckets pass the threshold (often, fewer than 10\\% of buckets). \nAlthough the contribution of the buckets to the distribution is small, this means that \nwhile the federated approach sample and threshold improves over shuffling, it does not reach the accuracy of centralized noise addition when there are many infrequent items. \n\nLast, we note that the magnitude of the error decreases as the histogram size increases. \nThis is in part since the magnitude of the bucket frequencies decreases, and we are showing the (mean) error per bucket. \nAs a sanity test, we also computed accuracy of the trivial approach of reporting zero for each bucket. \nThe error for this approach falls above the range of each graph plotted, giving reassurance that we are achieving non-trivial accuracy for the histogram problem. \n\n\\paragraph{Recall results.}\nTo better understand the ability of the different approaches to capture the high counts (as needed for finding heavy hitters), we measure the recall of the top-$k$ items, for $k = B\/10$. \nThat is, we test whether the histogram correctly reports the (true) top-10\\% of items among the 10\\% of largest items recovered. \nFigure~\\ref{fig:recall} shows the results across different datasets. \nFor moderate sized histograms ($B=2^{8}$), the sample and threshold approach achieves close to perfect recall for all datasets.  \nOther methods are comparable, but weaker for small $\\epsilon$.  \nAgain, it is the Shakespeare data for a larger histogram that presents the greatest challenge (in Figure~\\ref{fig:rec:shake:4096}).  \nHere, the same issue as above affects sample and threshold: \na large fraction of small frequencies mean that these do not meet the threshold for the sample size. \nAccepting a larger $\\epsilon$, or working over a larger population to obtain a larger sample size would be needed to improve the recall. \nHowever, it could be argued that items missed are not very significant: already at $\\epsilon = 0.6$, the threshold applied means that only items with frequency less than 0.1\\% are likely to be dropped. \n\\eat{\nOur results are shown in Figures~\\ref{fig:bin}, \\ref{fig:geom} and \\ref{fig:unif} for the Binomial, Geometric and Uniform data, respectively. \nIn each plot, Laplace noise addition is shown by the left bar, local DP noise by the central bar, and sample-and-threshold by the right bar. \nWe observe that the sample-and-threshold approach obtains strong accuracy across all these scenarios.  \nThe accuracy of the local DP approach is usually worse, sometimes by an order of magnitude or more, particularly in the high privacy ($\\epsilon = 0.1$) case.  \nThe federated approach of sample-and-threshold is also competitive with the approach of centralized noise addition, with many cases of improved accuracy. \nThe main exception to this trend is on uniform data (Figure~\\ref{fig:unif}), where the data is spread very thinly across the domain.  As a result, in the small $\\epsilon$ case (Figure~\\ref{fig:unif:eps0.1b100}) and the large domain case (Figures~\\ref{fig:unif:eps0.1b10000} and \\ref{fig:unif:eps1.0b10000}), sample-and-threshold obtains weaker accuracy, since fewer of the samples make it past the threshold $\\tau$. \nHowever, in the latter case the the absolute magnitude of the error is lower than in other cases, closer to 100, indicating that there is very little signal in this data to miss. \n\nThe setting of $B=$10,000, $\\epsilon=0.1$ may be considered as the ``hardest'' case, since it entails more noise over a larger number of histogram buckets. \nWe say that the top-$k$ accuracy for sample and threshold is weaker in this case for Binomial data (Figure~\\ref{fig:bin:eps0.1b10000}), although for the more sharply skewed geometric data, it is able to achieve improved accuracy (Figure~\\ref{fig:geom:eps0.1b10000}), indicating its suitability for the more realistic skewed data seen in practice.  \nIndeed, for the Geometric data, the sample-and-threshold method always obtains equal or better performance than Laplace noise addition, and always substantially better than the local DP noise. \nWe conclude that the sample-and-threshold technique is highly practical for realistic data analysis tasks, giving results in the federated model that are close to or improve on those in the central model. \n}\n\\section{Related work}\n\\paragraph{Histograms.}\nDue to their broad applications, histograms are one of the most heavily studied tasks in differential privacy (DP). \nOne of the first DP results is that a private histogram can be created by adding independent Laplace noise to each entry of the exact histogram~\\citep{Dwork06,DworkR14}.\nFor large domains, an $(\\epsilon, \\delta)$-DP guarantee can be obtained by applying a threshold to the noisy counts, and omitting any histogram entries whose (true) count is zero, which preserves the sparsity of the input~\\citep{Korolova:Kenthapadi:Mishra:Ntoulas:09,Bun:Nissim:Stemmer:18,Balcer:Vadhan:18}. \nFor multi-dimensional data, histograms of low-degree marginal distributions can be created via noise addition to the Hadamard transform of the data~\\citep{BarakCDKMT07}. \nThese results assume a given set of histogram bucket boundaries; \n\\cite{XuZXYY12}\nconsidered choosing bucket boundaries privately to minimize squared error. \nThe histogram problem has also been heavily studied in the local model of DP, where each individual adds noise to their input independently. \nHere, histograms are often implemented via `frequency oracles', and used to identify frequent items from the input~\\citep{bassilysmith}. \nOptimized constructions make use of hashing~\\citep{wangetal} and Hadamard transforms~\\citep{hadamardresponse} to minimize the variance of the estimate. \nMore recently, results are shown in the shuffle model, where messages from individuals are anonymized by a ``shuffler'', so the analyst sees only the multiset of messages received without attribution~\\citep{ESA20}. \nUnder shuffling, for a fixed privacy level $\\epsilon$, accuracy bounds closer to the central case are achievable via the introduction of small amounts of random noise from each participant~\\citep{BalcerCheu20,DUMP20}. \n\n\\paragraph{Heavy hitters.}\nThe problem of finding the most frequent items from a collection is a core analytics task that supports a range of objectives, from simple popularity charts, to instantiating complex language models.  \nDue to the sensitivity of data used within these applications, it is  necessary to apply strong privacy protections to the data. \nThere have been multiple efforts to address this problem in the Local DP setting~\\citep{bassilysmith,wangetal,wangConsistency, rappor, appledp, practicalhh} and shuffle model~\\citep{Ghazi21}.\nThe closest work to ours is recent work on Federated Heavy Hitters discovery~\\citep{TrieHH},\nwhich describes an $(\\epsilon, \\delta)$-DP algorithm to collect information from a set of distributed clients, who each hold a (private) item.  \nWe can treat these items as strings of characters over a fixed alphabet. \nThe algorithm proceeds in a series of $L$ rounds to build up a trie describing the frequent items among the client population.  \nIn each round, the server contacts a random sample of $m$ clients, and shares the current trie with them. \nEach client replies if its item extends the trie, and if so the client ``votes'' for the prefix that its item extends, along with the next character. \nThe server receives these votes, and tallies them.  \nPopular prefixes are added to the trie, and are candidates for further extension in the next round. \nThe procedure stops after the trie has been built out to $L$ levels, or if the trie cannot be extended beyond a certain level. \n\n\\paragraph{Quantiles and range queries.}\nThe quantiles of a distribution give a compact description of its (one-dimensional) CDF, generalizing the median. \nThe problem has also been studied in the central, local and shuffled models. \nMany solutions first solve range queries, then reduce quantile queries to  range queries. \n\\cite{XiaoWG10} propose using the Haar wavelet transform with noise, \nwhile \\cite{QardajiYL13} use hierarchical histograms. \n\\cite{CormodeKS19} compare both methods in the local setting and observe similar levels of accuracy. \nIn the shuffle model, quantiles are addressed via frequency histograms in the work of \\cite{Ghazi21}. \n\n\\paragraph{Sampling and DP.}\nIt is well-known that sampling can be used to amplify the guarantees of differential privacy when combined with a DP mechanism on the sample: \n\\cite{Li:Qardaji:Su:12} combine sampling with $k$-anonymization to achieve a DP guarantee,  \n\\cite{BalleBG20} show results for Poisson sampling, and fixed-size sampling with and without replacement,\nwhile \\cite{Imola21} study privacy amplification when sampling according to differentially private parameters. \nBy contrast, we consider mechanisms where sampling in isolation (with a threshold) provides the DP guarantee directly. \nThis idea is inspired by~\\cite{TrieHH}, which materializes a set of items based on sampling and thresholding. \nThe key advance in our work is to show that we can output the sampled frequencies as well as the sampled items, and hence produce private histograms. \nOur work complements other efforts in the federated setting to achieve privacy guarantees with a restricted set of operations---for instance, \\cite{KairouzM00TX21} seek to perform federated learning via noise addition \\textit{without} sampling. \n\n\n\\section{Concluding Remarks}\n\nIn this paper, we have shown how the sample-and-threshold approach can be applied to the fundamental problem of private histogram computation, and related tasks like heavy hitter and quantile estimation. \nThe key technical insight is that sampling a large enough number of indistinguishable examples introduces sufficient uncertainty to meet the differential privacy guarantee. \nAs with other works on private histograms, we assume that the bucket boundaries of the histogram are given. \nAdaptive division of the histogram buckets is possible, as seen in the TrieHH++ protocol. \nNevertheless, this approach can give poor results in extreme cases, such as when the bulk of the data resides in a very small fraction of the input domain. \n\nIt is natural to consider what other computations might benefit from this sample-and-threshold approach. \nDirect application of the technique makes sense when many users hold copies of the same value. \nHence, it is not well-suited to questions like finding sums and means of general distributions, unless we additionally apply some rounding and noise addition to input values first. \nThe approach may be of value for more complex computations, such as clustering or outlier removal, where dropping rare items is a benefit, or tasks where we seek to discover descriptions of patterns in the data that have large support, such as frequent itemsets. \n\n\n\n\n\n\\section{Analysis of original TrieHH protocol}\n\nWe adopt the algorithm and notation from the TrieHH paper~\\cite{TrieHH}. \n\nThe algorithm proceeds over $L$ levels. \nAt each level, we sample $m = \\gamma \\sqrt{n}$ clients to report on their items that extend the current trie. \nWe add items to the current trie based on an (absolute) threshold $\\tau$. \n\n\n\\subsection{Simplification of \\cite[Lemma 2]{TrieHH}}\n\nLemma 2 in~\\cite{TrieHH} considers the difference in probabilities between inputs $D$ and $D'$\nfor reporting a particular prefix.\nThe set up is that there are $k$ users in $D'$ who hold the prefix, but $k+1$ in $D$. \n\n\\begin{lemma}\nGiven the set up above, with $k+1 \\leq \\sqrt{n}\/\\gamma$, \nthe probability of reporting the prefix is at most\n$\\exp(-\\tau \\ln \\tau + \\tau - 1)$ for any $\\tau > 1$.\n\\label{lem:ptheta}\n\\end{lemma}\n\n\\begin{proof}\nThe algorithm will report the prefix if the number of sampled occurrences is at least $\\tau$. \nWe analyze the case of sampling the clients with replacement.  \nNote that this only increases the chances of this event. \n\nThe probability of picking the prefix in one sample is just\n$p:= (k+1)\/n \\leq 1\/m$.  \nLet $X$ denote the random variable with the number of successes (times the prefix is picked) out of the $m$ trials. \nWe have \n\\[ \\mathsf{E}[X] = \\frac{m(k+1)}{n} \\leq \\frac{m \\sqrt{n}}{\\gamma n} = 1\\] \nusing the bound on $k+1$, and the fact that $m = \\gamma \\sqrt{n}$. \nThen, \n$X$ is a sum of $m$ Bernoulli random variables with $p \\leq 1\/m$. \n\nWe then apply a Chernoff-Hoeffding bound to the mean of the $m$ trials: \n\n\\[ \\Pr[ X \\ge \\tau ] = \\Pr\\left[ \\frac{1}{m}X - \\frac{1}{m}\\mathsf{E}[X] \\ge \\left(\\frac{\\tau}{m} - \\frac{1}{m}\\right) \\right] \\le \n\\exp\\left(- D\\left( \\frac{\\tau}{m} \\middle\\| \\frac{1}{m}\\right)m\\right) . \n\\]\n\nHere, $D(p\\|q)$ denotes the K-L divergence (relative entropy) between the (Bernoulli) distributions with parameters $p$ and $q$. \nWe have\n\\begin{align*}\n-D(p\\|q) m  & = -{\\tau} \\ln\\left(\\frac{\\tau}{m} \\cdot \\frac{m}{1}\\right) - {m-\\tau}\\ln\\left(\\frac{m - \\tau}{m} \\cdot \\frac{m}{m-1}\\right)\n\\\\\n& =  {-\\tau} \\ln \\tau - (m-\\tau) \\ln\\left(1 - \\frac{\\tau-1}{m-1}\\right)\\\\\n& = -\\tau \\ln \\tau + (m-\\tau) \\ln\\left( \\frac{m-1}{m -\\tau}\\right) \\\\\n& = -\\tau \\ln \\tau + (m-\\tau) \\ln \\left(1 + \\frac{\\tau-1}{m-\\tau}\\right) \\\\\n& \\leq - \\tau \\ln\\tau + \\tau - 1\n\\end{align*}\n\n\\begin{equation}\n\\text{Hence, }\n  \\Pr[ X \\ge \\tau ] \\leq \\exp(-\\tau \\ln \\tau + \\tau - 1)\n\\end{equation}\n\\end{proof}\n\n\nIn contrast, the result proved in~\\cite{TrieHH} is a bound of \n$\\frac{\\tau-2}{\\tau-3} \\frac{1}{\\tau!}$. \nThe expression in Lemma~\\ref{lem:ptheta} is slightly looser. \n\nTo achieve a target error bound $\\delta$, we rearrange to obtain\n$\\frac{\\tau}{e} \\ln \\frac{\\tau}{e}  = \\frac{1}{e} \\ln(1\/e\\delta)$, and apply Lambert's W function. \nThis gives \n$\\frac{\\tau}{e} = W(\\frac{1}{e} \\ln(1\/e\\delta)$, i.e., \n$\\tau = e W(\\frac{1}{e} \\ln \\frac{1}{e\\delta})$. \n\n\n\\subsection{Short version of \\cite[Lemma 1]{TrieHH}}\n\nWe next give a short version of a bound on the ratio of probabilities of seeing the same output on neighboring inputs. \nThis follows the same proof outline as in the original paper. \n\n\\begin{proof}\nThe case to focus on is when input $D$ has one extra copy of a particular prefix compared to $D'$, at some intermediate stage of the algorithm. \nWithout loss of generality, we can imagine that this copy is held by the user with index $n$. \nFor notation, we will write \n$S_k(n, m, \\tau)$ to denote the number of ways to succeed in collecting at least $\\tau$ instances of the target prefix while picking $m$ items out of $n$, when there are $k$ total instances of the prefix. \n\nFor $D'$, we can break their successful instances into (a) those where they sample item $n$ and need to collect $\\tau$ instances with their remaining $m-1$ samples out of $n-1$ items, or (b) where they do not sample item $n$, and thus need to collect $\\tau$ instances from $m$ samples out of $n$. Thus, \n\\begin{equation}\nS_k(n, m, \\tau) = S_k(n-1, m, \\tau) + S_k(n-1, m-1, \\tau)\n\\label{eq:dprime}\n\\end{equation}\n\nFor $D$, we first derive a useful relationship that corresponds to the case that $D$ does not pick index $n$. \nThis is equivalent to considering their success on a smaller instance without the $n$'th item. \n$D$ will succeed if they pick exactly $\\tau$ instances of the target prefix from the $k$ instances\n(and $m-\\tau$ from any of the remaining $n-1-k$ items), or pick more than $\\tau$.\nWe can express this in our notation as\n\\[ S_{k}(n-1, m, \\tau) = {k \\choose \\tau} {n-k-1 \\choose m-\\tau} + S_k(n-1, m, \\tau+1)\n\\]\n\nRearranging, we obtain that\n\\[ {n-k-1 \\choose m - \\tau} \\leq  {k \\choose \\tau}^{-1} S_{k}(n-1, m, \\tau)\n\\]\n\nThe set of routes to success for $D$ look similar to those for $D'$, with one extra case: \n(a) $D$ does not pick item $n$, and succeeds with $m$ samples over the $n-1$ remaining items;\n(b) $D$ does pick $n$, and picks $\\tau$ or more items from the remaining $n-1$;\n(c) or else $D$ picks index $n$ and exactly $\\tau-1$ copies of the target prefix from among the \n$k$ copies, along with $m - \\tau$ from the remaining $(n-1) - k$ items. \nThis gives\n\\begin{align*}\n    S_{k+1}(n, m, \\tau) & =  S_k(n-1, m, \\tau) + S_k(n-1, m-1, \\tau) + {k \\choose \\tau-1} {n-k-1 \\choose m-\\tau} \\\\\n    & \\leq  S_k(n-1, m, \\tau) + S_k(n-1, m-1, \\tau) + {{k \\choose \\tau-1}\\over {k \\choose \\tau}} S_k(n-1, m, \\tau) \\\\\n    & =  S_k(n-1, m-1, \\tau) + \\left(1 + \\frac{\\tau}{k - \\tau +1}\\right)S_k(n-1, m, \\tau)\\\\\n    & \\leq  \\left(1 + \\frac{\\tau}{k-\\tau+1}\\right)S_k(n, m, \\tau) \n\\end{align*}\n\n\\noindent\nwhere the last line follows from \\eqref{eq:dprime}.\n\nThis determines the critical ratio $S_{k+1}(n,m, \\tau)\/S_k(n, m, \\tau) \\leq \\frac{k+1}{k-\\tau+1}$, as required, which is the ratio between the seeing the prefix appear in the output of $D$ versus that of $D'$. \n\\end{proof}\n\n\n\n\n\\subsection{Simplification of \\cite[Corollary 1]{TrieHH}}\n\nWe consider the proof of the main technical result in~\\cite{TrieHH}. \n\n\\cite[Lemma~1]{TrieHH} shows that the ratio of probabilities of seeing outputs at level $i$ is bounded by a factor of \n$(1 + \\frac{\\tau}{k - \\tau + 1})$, for $k \\geq \\tau$. \nThis is used to prove the main bound in \\cite[Theorem 1]{TrieHH}. \nThe proof of \\cite[Theorem 1]{TrieHH} proceeds by considering the accumulated probability of different outcomes, based on a case split \nfor when the probability of a certain event falls below $p_\\tau$. \nIn the proof, $p_\\tau$ is set to $\\frac{\\tau-2}{\\tau-3} \\frac{1}{\\tau!}$, to agree with the bound proved in \\cite[Lemma 2]{TrieHH}. \nHowever, there is nothing in the proof that requires this particular value of $p_\\tau$, so we can follow the same proof argument, and obtain a restatement of \\cite[Theorem 1]{TrieHH} as providing $(\\epsilon, \\delta)$-differential privacy\nwith $\\epsilon = L \\ln( 1 + \\frac{1}{\\frac{\\sqrt{n}}{\\gamma \\tau} - 1})$\nand $\\delta = \\exp(-\\tau \\ln \\tau)$. \n\nWe will provide a simplified bound on the expression of $\\epsilon$ to make it easier to work with.\n\n\nFirst, observe that, with the assumption that $k \\ge \\sqrt{n}{\\gamma} - 1 \\ge \\tau$ and $m = \\gamma n$,  \n\\begin{equation}\n\\label{eq:epsbound}    \n1 + \\frac{\\tau}{k-\\tau + 1}\n\\le 1 + \\frac{\\tau}{\\frac{n}{m} - \\tau}\n= 1 + \\frac{1}{\\frac{n}{m\\tau} - 1}\n = 1 + \\frac{m\\tau}{n - m\\tau} . \n\\end{equation}\n\nWe will assume that $\\gamma \\leq \\frac{\\sqrt{n}}{10\\tau}$. \nThis is a stronger assumption that\n\\cite{TrieHH}, which assumes only\n$\\gamma \\leq \\frac{\\sqrt{n}}{\\tau + 1}$. \nThe effect is to ensure that the sample size $m$ is a small fraction of $n$. \nSubstituting this assumption in \\eqref{eq:epsbound}, we conclude \n\\begin{equation}\n    \\label{eq:finalepsbound}\n    1 + \\frac{1}{\\frac{\\sqrt{n}}{\\gamma \\tau} - 1 }  \\leq 1 + \\frac{m\\tau}{n - n\/10}\n    = 1 + \\frac{10 m \\tau}{9n} \n\\end{equation}\n\nSince $\\frac{10m\\tau}{9n} \\leq 1$, we can bound\n$\\epsilon \\leq \\frac{10 L m \\tau}{9n}$. \n\nThis form is a bit looser than the result in~\\cite{TrieHH}.\nSpecifically, given a target $(\\epsilon, \\delta)$, we can \nfirst use Lemma~\\ref{lem:ptheta} to set\n$\\tau = e \\exp( W(\\frac{1}{e}\\ln(1\/e\\delta))$,\\footnote{Recall, $W(\\cdot)$ denotes the Lambert W-function.  \nSince we only need to consider integral values of $\\tau$, we can also find $\\tau$ from \n$\\delta$ with a small lookup table: \n$\\tau = 100$ gives $\\delta = 10^{-157}$, which is likely to be small enough for all purposes. } and \nhence \n$m = \\frac{9\\epsilon}{10 L \\tau} n$.\n\nInterpreting this expression, we can treat $\\tau$ effectively as a small constant, and the main dependence is on \n$\\epsilon\/L$.  \nThis means for that stronger privacy, we should pick a smaller sample (giving a lower chance to learn information about any particular client). \nAs we see below, this contrasts with the goal of accuracy, where we would prefer a larger sample to get more accurate results. \n\nConcretely, comparing to values computed in~\\cite{TrieHH}, \nfor $n = 10^{5}$, $\\epsilon = 2$, $L=10$, and $\\delta = 1\/n^2$, \nwe obtain\n$\\tau = \\lceil \\exp(W(\\ln(10^{-12}) \\rceil = 14$, and\n$m = (18\/1400) n = 0.01285n$. \nUsing the expression from~\\cite{TrieHH} of\n$m = \\frac{1}{\\tau}(1 - \\exp(-\\epsilon\/L)) n = 0.01293 n$, i.e., very close agreement. \n\n\n\n\n\\end{document}\n\\subsubsection*{\\bibname}}\n\n\n\\begin{document}\n\n\n\n\\onecolumn\n\\aistatstitle{Instructions for Paper Submissions to AISTATS 2022: \\\\\nSupplementary Materials}\n\n\\section{FORMATTING INSTRUCTIONS}\n\nTo prepare a supplementary pdf file, we ask the authors to use \\texttt{aistats2022.sty} as a style file and to follow the same formatting instructions as in the main paper.\nThe only difference is that the supplementary material must be in a \\emph{single-column} format.\nYou can use \\texttt{supplement.tex} in our starter pack as a starting point, or append the supplementary content to the main paper and split the final PDF into two separate files.\n\nNote that reviewers are under no obligation to examine your supplementary material.\n\n\\section{MISSING PROOFS}\n\nThe supplementary materials may contain detailed proofs of the results that are missing in the main paper.\n\n\\subsection{Proof of Lemma 3}\n\n\\textit{In this section, we present the detailed proof of Lemma 3 and then [ ... ]}\n\n\\section{ADDITIONAL EXPERIMENTS}\n\nIf you have additional experimental results, you may include them in the supplementary materials.\n\n\\subsection{The Effect of Regularization Parameter}\n\n\\textit{Our algorithm depends on the regularization parameter $\\lambda$. Figure 1 below illustrates the effect of this parameter on the performance of our algorithm. As we can see, [ ... ]}\n\n\\vfill\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\n\n\n\n\nA high level of intermittent wind generation may result in  high price volatility  in electricity markets \\cite{ketterer,wozabal,Woo}. In the long term, extreme levels of price volatility can  lead to undesirable consequences such as bankruptcy of retailers \\cite{deng} and market suspension. In a highly volatile electricity market, the participants, such as generators, utility companies and large industrial consumers, are exposed to a high level of financial risk as well as costly risk management strategies \\cite{volatility}. In some electricity markets, e.g., Australia's National Electricity Market (NEM), which has experienced high levels of price volatility \\cite{windTansu}, the market is suspended if the sum of spot prices over a certain period of time is more than cumulative price threshold (CPT). A highly volatile market is subject to frequent CPT breaches due to the low conventional capacity and high level of wind variability.\n\n\n\n\n\n\n\n\n\n\n\nThe current paper proposes a stochastic optimization framework for finding the required nodal storage capacities in electricity markets with high levels of wind penetration such that the price volatility in the market is kept below a certain level. \nThe contributions of this paper are summarized as follows: \n\n\\begin{enumerate}\n\\item  A bi-level optimization model is proposed to find the optimal nodal storage capacities required for avoiding the extreme price volatility levels in a nodal electricity market.\n\n\\item  In the upper level problem, the total storage capacities are minimized subject to a price volatility target constraint in each node and at each time. \n\n\\item  In the lower level problem, the non-cooperative interaction between generation, transmission and storage players in the market is modeled  as a stochastic Cournot-based game with an exponential inverse demand function. Note that the equilibrium prices at the lower level problem are functions of the storage capacities. The operation of storage devices at the lower level problem is modeled without introducing binary variables.\n\n\\item  The existence of Nash equilibrium  under the exponential inverse demand function is established for the lower level problem.\n\n\\end{enumerate}\n\n\nUnder the proposed framework, the size of storage devices at two nodes of South Australia (SA) and Victoria (VIC) in NEM is determined such that the market price volatility is kept below a desired level at all times. The desired level of price volatility  can be determined based on various  criteria such as net revenue earned by the market players, occurrence frequency of undesirable prices, number of CPT breaches, etc \\cite{AEMC}. \n\nThe proposed storage allocation framework allows the policy makers and market\/system operators to compute the required nodal storage capacities for managing the price volatility level in electricity markets.  Although the current cost of storage systems is relatively high, the support from governments (in the form of subsidies) and the eventual decline of the technology cost can lead to  large scale integration of storage systems in electricity markets.  \n\n\n\n\n\n\n\nThe rest of the paper is organized as follows. The existing related literature is discussed in Section \\ref{sec: related}. The system model and  the proposed bi-level optimization problem are formulated in Section \\ref{Sec: SM}. The equilibrium analysis of the lower level problem  and the solution method are presented in Section \\ref{sec: sol}. The simulation results are presented in Section \\ref{sec: sim}. The conclusion remarks are discussed in Section \\ref{sec: con}.\n\n\\vspace*{-.2 cm} \n\\section{Related Works} \\label{sec: related}\n\nThe problem of optimal storage operation or storage allocation  for facilitating the integration of intermittent renewable energy generators in electricity networks has been studied in \\cite{NanLi,VenkatKrishnan,AsmaeBerrada,WeiQi,MahdiSedghi,LeZheng,JunXiao,YuZheng}, with total cost minimization objective functions, and in \\cite{RahulWalawalkar,HamedMohsenian2,HosseinAkhavan,HamedMohsenian1,EhsanNasrolahpour}, with profit maximization goals. However, the price volatility management problem using optimal storage allocation has not been investigated in the literature.\n\n\n\n\n\nThe operation of a storage system is optimized, by minimizing the total operation costs in the network, to facilitate the integration of intermittent renewable resources in power systems in \\cite{NanLi}. \n Minimum (operational\/installation) cost storage allocation problem for renewable integrated power systems is studied in \\cite{VenkatKrishnan,AsmaeBerrada,WeiQi}  under deterministic wind models, and in \\cite{MahdiSedghi} under a stochastic wind model.    \nThe minimum-cost storage allocation problem is studied in a bi-level problem in \\cite{LeZheng,JunXiao}, with the upper and lower levels optimizing the allocation and the operation, respectively.  The paper \\cite{YuZheng} investigates the optimal sizing, siting, and operation strategies for a storage system to be installed in a distribution company controlled area. \nWe note that these works only study the minimum cost storage allocation or operation problems, and the interplay between the storage firms and other participants in the market has not been investigated in these works.\n\n\n The paper \\cite{RahulWalawalkar} studies the optimal operation of a storage unit, with a given capacity, which aims to maximize its profit in the market from energy arbitrage and provision of regulation and frequency response services. The paper \\cite{HamedMohsenian2} computes the optimal supply and demand bids of a storage unit so as to maximize the storage's profit from energy arbitrage in the day-ahead and the next 24 hour-ahead markets. The paper \\cite{HosseinAkhavan} investigates the profit maximization problem for a group of independently-operated investor-owned storage units which offer both energy and reserve  in  both day-ahead and  hour-ahead markets. In these works, the storage firm receives the market price as an exogenous input, i.e. the storage is modeled as a price taker firm due to its small capacity.\n\n  \n\nThe operation of a price maker storage device is optimized  using a bi-level stochastic optimization model, with the lower level clearing the market and the upper level maximizing the storage profit by bidding on price and charge\/discharge in \\cite{HamedMohsenian1}. The storage size in addition to its operation is optimized in the upper level problem in \\cite{EhsanNasrolahpour}  when the lower level problem clears the market. Note that the energy and price bids of market participants other than the storage firm are treated exogenously in these models.  \n\n\nIn \\cite{Ventosa,Schill}, the storage firms are modeled as strategic players in Cournot-based electricity markets. However, they do not study storage sizing problem and the effect of intermittent renewables on the market. Therefore, to the best of our knowledge, the problem of finding optimal storage capacity subject to a price volatility management target in electricity markets has not been  addressed before.    \n\n\n\n   \n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\\vspace*{-.2 cm} \n\\section{System Model}\\label{Sec: SM}\nConsider a nodal electricity market with ${I}$ nodes. Let $\\mathcal{N}^{\\rm cg}_i$ be the set of classical generators, such as coal and gas power plants, located in  node $i$ and $\\mathcal{N}^{\\rm wg}_i$ be the set of wind firms located in  node $i$. The set of neighboring nodes of node $i$ is denoted by $\\mathcal{N}_i$. Since the wind availability is a stochastic parameter, a scenario-based model,  with $N_{\\rm w}$ different scenarios, is considered to model the wind availability in the electricity network. \nThe nodal prices in our model are determined by solving a Cournot-based game among all market participants, that is, classical generators, wind firms, storage firms and transmission interconnectors which are introduced in detail in the lower level problem. More precisely, the market price in node $i$ at time $t$ under the wind availability scenario $w$ is given by an exponential function:\n\\begin{align}\\label{Eq: Price}\n& P_{itw}\\left(\\vec{q}_{itw}\\right)\\!=\\! \\alpha_{it} e^{\\!\\!-\\beta_{it} \\! \\left(q^{\\rm s}_{itw} + \\sum\\limits_{m \\in  \\mathcal{N}^{\\rm wg}_i} q^{\\rm wg}_{mitw} +\\sum\\limits_{n \\in \\mathcal{N}^{\\rm cg}_i } q^{\\rm cg}_{nitw}+ \\sum\\limits_{j\\in\\mathcal{N}_i}q^{\\rm tr}_{ijtw} \\right)} \n\\end{align}\n  where  $\\alpha_{it}, \\beta_{it}$  are positive real values in the inverse demand function, $q^{\\rm cg}_{nitw}$ is the generation strategy of the $n$th classical generator located in node $i$ at time $t$ under  scenario $w$, $q^{\\rm wg}_{mitw}$ is the generation strategy of the $m$th wind generator located in node $i$ at time $t$ under  scenario $w$, $q^{\\rm s}_{itw}$ is the charge\/discharge strategy of the storage firm in node $i$ at time $t$ under scenario $w$, $q^{\\rm tr}_{ijtw}$ is the strategy of transmission firm located between node $i$ and node $j$ at time $t$ under scenario $w$.  The collection of strategies of all firms located in node $i$ at time $t$ under the scenario $w$ is denoted by $\\vec{q}_{itw}$.\n  \n\t\n\n\tIn this paper, we propose a bi-level optimization approach for finding the minimum required total storage capacity in the market such that the market price volatility stays within a desired limit at each time.\n\n\n\n\\vspace*{-.2 cm}\t\n\t\n\\subsection{Upper-level Problem} \nIn the  upper-level optimization problem, we determine the nodal storage capacities such that a price volatility constraint is satisfied in each node at each time. In this paper, the variance of market price is considered as a measure of price volatility. The variance of the market price in node $i$ at time $t$, \\emph{i.e.} ${\\rm Var}\\bigl(P_{itw}\\bigr)$, can be written as: \n\t\\begin{align}\n\t&{\\rm Var}\\left(P_{itw} \\right)=\\mathsf{E}_w{\\left[\\left(P_{itw}\\left(\\vec{q}_{itw}\\right)\\right)^2\\right]}-\\left(\\mathsf{E}_w\\left[{P\\left(\\vec{q}_{itw}\\right)}\\right]\\right)^2\\nonumber \\\\\n\t&\\hspace{.3 cm}  =   \\sum_w \\bigg(P_{itw}\\left(\\vec{q}_{itw}\\right)\\bigg)^2 \\PRP{w} - \\bigg(  \\sum_w P_{itw}\\left(\\vec{q}_{itw}\\right) \\PRP{w}    \\bigg)^2  \\label{Var}\n\t\\end{align}\nwhere $\\Psi_w$ is the probability of scenario $w$.\n\nThe notion of variance quantifies the  \\emph{effective} variation range of random variables, i.e. a random variable with a small variance has a smaller effective range of variation when compared with a random variable with a large variance.\n\nGiven the price volatility relation (\\ref{Var}) based on the Nash Equilibrium (NE) strategy collection of all firms $\\vec{q}^{\\star}_{itw}$, the upper-level optimization problem is given by:\n\n\\begin{subequations} \\label{ISO}\n\\begin{align}\n&\\min_{\\left\\{Q_i^{\\rm s}\\right\\}_{i}} \\sum_{i=1}^I  Q^{\\rm s}_i  \\nonumber\\\\\n&\\text{s.t.}  \\nonumber \\\\\n& Q_i^{\\rm s} \\ge 0 \\quad \\forall i  \\label{ISO1}\\\\\n& {\\rm Var}\\left(P_{itw}\\left(\\vec{q}^\\star_{itw}\\right) \\right) \\le \\sigma_0^2 \\quad \\forall i,t \\label{ISO3} \n\\end{align}  \n\\end{subequations}\nwhere $Q^{\\rm s}_i$ is the storage capacity at node $i$, $P_{itw}\\left(\\vec{q}^\\star_{itw}\\right)$ is the market price at node $i$ at time $t$ under the wind availability scenario $w$, and $\\sigma_0^2$ is the  price volatility target. The price volatility of the market is defined as the maximum variance of market price, \\emph{i.e.} $\\max_{it} {\\rm Var}(P_{itw}(\\vec{q}^\\star_{itw}))$.\n\n\n\n\n\\vspace*{-.2 cm} \n\\subsection{Lower-level Problem}\nIn the lower-level problem, the nodal market prices and the NE strategies of firms are obtained by solving an extended stochastic Cournot game between wind generators, storage firms, transmission firms, and classical generators. In our formulation, storage and transmission firms can be either regulated or strategic players. \n\n\n\\begin{definition} \\label{def1}\nA strategic firm decides on its strategies over the operation horizon $\\{1,...,{N}_{\\rm{T}}\\}$ such that its aggregate expected profit, over the operation horizon, is maximized. On the other hand, a regulated firm aims to maximize the net market value, i.e. the social welfare. \n\\end{definition}\n\nIn what follows,  the variable $\\mu$ is used to indicate the associated Lagrange variable with its corresponding constraint in the model.\n\n\n\\subsubsection{Wind Generators}\nThe NE strategy of the $m$th wind generator in node $i$ is obtained by solving the following optimization problem: \n\\begin{subequations} \\label{Pwind}\n\t\\begin{align}\n\t&\\hspace{0cm}\\max_{\\left\\{q_{mitw}^{\\rm wg}\\right\\}_{tw} \\succeq 0} \\sum_w \\PRP{w} \\sum_{t=1}^{{N}_{\\rm T}} P_{itw}\\left(\\vec{q}_{itw}\\right)q^{\\rm wg}_{mitw}\\nonumber \\\\\n\t&{ \\rm s.t.} \\nonumber \\\\\n\t&  q^{\\rm wg}_{mitw} \\le {Q}_{mitw}^{ \\rm wg} \\quad : \\quad \\mu^{\\rm wg,max}_{mitw} \\quad \\forall t,w \\label{cons3}\n\t\\end{align}\n\\end{subequations}\nwhere $q^{\\rm wg}_{mitw}$ and $Q^{\\rm wg}_{mitw}$ are the generation level and the available wind capacity of the $m$th wind generator located in node $i$ at time $t$ under  scenario $w$. Note that the wind availability changes in time in a stochastic manner, and the wind firm's bids depend on the wind availability. As a result, the nodal prices and decisions of the other firms become stochastic in our model. \n\n\n\n\\subsubsection{Storage Firms} Storage firms benefit from price difference at different times to make profit, i.e. they sell the off-peak stored electricity at higher prices at peak times. The NE strategy of  storage firm located in node $i$ is determined by solving the following optimization problem:\n\\begin{subequations} \\label{Pst}\n\t\\begin{align}\n\t&\\max_{\\substack{\\left\\{q_{itw}^{\\rm dis},q_{itw}^{\\rm ch}\\right\\}_{tw} \\succeq 0\\\\,\\left\\{q_{itw}^{\\rm s}\\right\\}_{tw}}} \\sum_w \\PRP{w} \\sum_{t=1}^{{N}_{{\\rm T}}} P_{itw}\\left(\\vec{q}_{itw}\\right) q^{\\rm s}_{itw} -c_{i}^{\\rm s} \\left(q^{\\rm dis}_{itw}+ \\right.  \\nonumber \\\\ \n\t&\\hspace{1 cm} \\left. q^{\\rm ch}_{itw} \\right) -\\gamma^{{\\rm s}}_{i} \\left(P_{itw}\\left(\\vec{q}_{itw}\\right) q^{\\rm s}_{itw}+ \\frac{P_{itw}\\left(\\vec{q}_{itw}\\right)}{\\beta_{it}}\\right) \\label{storageU} \\\\\t\n\t&{\\rm s.t.} \\nonumber \\\\\n\t& q^{\\rm s}_{itw}=\\eta^{\\rm dis}_{i} q^{\\rm dis}_{itw} - \\frac{q^{\\rm ch}_{itw}}{\\eta^{\\rm ch}_{i}} \\quad : \\quad \\mu^{\\rm s}_{itw} \\quad \\forall t,w \\label{cons41}\\\\\n\t& q^{\\rm dis}_{itw} \\le \\zeta_i^{{\\rm dis}} Q^{\\rm s}_i \\quad : \\quad \\mu^{\\rm dis,max}_{itw} \\quad \\forall t,w \\label{cons42}\\\\\n\t& q^{\\rm ch}_{itw} \\le \\zeta_i^{{\\rm ch}} Q^{\\rm s}_i \\quad : \\quad \\mu^{\\rm ch,max}_{itw} \\quad \\forall t,w  \\label{cons43}\\\\\n\t& 0 \\le \\sum_{k=1}^{t} \\left(q^{\\rm ch}_{ikw}-q^{\\rm dis}_{ikw}\\right) \\Delta \\le Q^{\\rm s}_i \\smallskip : \\smallskip \\mu^{\\rm s,min}_{itw},\\mu^{\\rm s,max}_{itw} \\quad \\forall t,w \\label{cons44} \n\t\\end{align}\n\\end{subequations}\nwhere $q^{\\rm dis}_{itw}$ and $q^{\\rm ch}_{itw}$ are the discharge and charge levels of the storage firm in node $i$ at time $t$ under  scenario $w$, respectively, $c_{i}^{\\rm s}$ is the unit operation cost,  $\\eta^{\\rm ch}_{i}$ ,$\\eta^{\\rm dis}_{i}$ are the charging and discharging efficiencies, respectively, and $q^{\\rm s}_{itw}$ is the net supply\/demand of the storage firm in node $i$. The parameter $\\zeta_i^{{\\rm ch}}$ ($\\zeta_i^{{\\rm dis}}$) is the percentage of  storage capacity $Q^{\\rm s}_i$, which can be charged (discharged) during  time period $\\Delta$. It is assumed that the storage devices are initially fully discharged.\nThe energy  level of the storage device in node $i$ at each time is limited by its capacity $Q_i^{\\rm s}$. Note that the nodal market prices depend on the storage capacities, \\emph{i.e.} $Q_i^{\\rm s}$s, through the constraints \\eqref{cons42}-\\eqref{cons44}. This dependency allows the market operator to meet the volatility constraint using the optimal values of the storage capacities.  \n\n \nThe storage firm in node $i$ acts as a strategic firm in the market if $\\gamma^{{\\rm s}}_{i}$ is equal to zero and  acts as a regulated firm if $\\gamma^{{\\rm s}}_{i}$ is equal to one. The difference between regulated and strategic players corresponds to the strategic price impacting capability.   Note that the derivative of objective function of the regulated storage firm $i$ is proportional to $P(\\cdot)-c^{\\rm s}_i$. This intuitively suggests that a regulated storage firm prefers to reduce the market price  to its operation cost while it discharges.\n\n\t\n\n\n\\begin{proposition}\nAt the NE of the lower level game, each storage firm is either in the charge mode or discharge mode, i.e. the charge and discharge levels of each storage firm cannot be simultaneously positive at the NE.\n\\end{proposition}\n{\\it{Proof:}} See Appendix \\ref{App1}\n\n\\subsubsection{Classical Generators} \nClassical generators include coal, gas, and nuclear power plants. The NE strategy of $n$th classical generator located in node $i$ is determined by solving the following optimization problem:\n\\begin{subequations} \\label{Pcg}\n\\begin{align}\n&\\max_{\\left\\{q_{nitw}^{\\rm cg}\\right\\}_{tw}\\succeq 0} \\sum_w  \\PRP{w} \\sum_{t=1}^{{N}_{{\\rm T}}} \\left(P_{itw}\\left(\\vec{q}_{itw}\\right)-c_{ni}^{\\rm cg} \\right) q^{\\rm cg}_{nitw}\\nonumber  \\\\\n&{\\rm s.t.} \\nonumber \\\\\n& q^{\\rm cg}_{nitw} \\le {Q}_{ni}^{\\rm cg}  \\quad : \\quad \\mu^{\\rm cg,max}_{nitw} \\quad \\forall t,w  \\label{cons51}\\\\\n& q^{\\rm cg}_{nitw} - q^{\\rm cg}_{ni(t-1)w} \\le R^{\\rm up}_{ni} \\quad : \\quad \\mu^{\\rm cg,up}_{nitw} \\quad \\forall t,w \\label{cons52}\\\\\n& q^{\\rm cg}_{ni(t-1)w} - q^{\\rm cg}_{nitw} \\le R^{\\rm dn}_{ni} \\quad : \\quad \\mu^{\\rm cg,dn}_{nitw} \\quad \\forall t,w \\label{cons53}\n\\end{align}\n\\end{subequations} \nwhere $ q^{\\rm cg}_{nitw} $ is the generation level of the $n$th classical generator in node $i$ at time $t$ under  scenario $w$,  ${Q}_{ni}^{\\rm cg}$ and $c_{ni}$  are the capacity and the short term marginal cost of the $n$th classical generator in node $i$, respectively. The constraints \\eqref{cons52} and \\eqref{cons53} ensure that the ramping limitations of the $n$th classical generator in node $i$ are always met.      \n\\subsubsection{Transmission Firms} \n The NE strategy of the transmission firm between nodes $i$ and $j$ is determined by solving the following optimization problem:\n\\begin{subequations} \\label{Ptr}\n\t\\begin{align}\n\t&\\!\\!\\!\\max_{\\left\\{q^{\\rm tr}_{jitw},q^{\\rm tr}_{ijtw}\\right\\}_{tw}} \\!\\!\\sum_w \\PRP{w}\\sum_{t=1}^{{N}_{{\\rm T}}} \\left(P_{jtw}\\left(\\vec{q}_{jtw}\\right) q^{\\rm tr}_{jitw}\n\t \\!+\\! P_{itw}\\left(\\vec{q}_{itw}\\right)q^{\\rm tr}_{ijtw}\\right)\\nonumber\\\\\n\t&\\hspace{2cm} \\left(1-\\gamma^{{\\rm tr}}_{ij}\\right)\n\t+ \\gamma^{{\\rm tr}}_{ij}  \\left(  \\frac{P_{jtw}\\left(\\vec{q}_{jtw}\\right)}{-\\beta_{jt}}   +\\frac{P_{itw}\\left(\\vec{q}_{itw}\\right)}{-\\beta_{it}} \\right)     \\nonumber \\\\\t\t\n\t&{\\rm s.t.} \\nonumber \\\\\n\t& q^{\\rm tr}_{ijtw}=-q^{\\rm tr}_{jitw} \\quad :  \\quad \\mu^{\\rm tr}_{ijtw} \\quad \\forall t,w\\\\\n\t& -Q_{ij}^{\\rm tr} \\le q^{\\rm tr}_{ijtw} \\le Q_{ij}^{\\rm tr} \\quad : \\quad \\mu^{\\rm tr,min}_{ijtw},\\mu^{\\rm tr,max}_{ijtw} \\quad \\forall t,w \\label{cons61}\n\t\\end{align}\n\\end{subequations} \nwhere $q^{\\rm tr}_{ijtw}$ is the electricity exchange level between nodes $i$ and $j$ at time $t$ under  scenario $w$, and $ Q_{ij}^{\\rm tr}$ is the capacity of the transmission line between node $i$ and node $j$. The transmission firm between nodes $i$ and $j$ behaves as a strategic player when $\\gamma^{{\\rm tr}}_{ij}$ is equal to zero and behaves as a regulated player when $\\gamma^{{\\rm tr}}_{ij}$ is equal to one. Note that the term $P_{jtw}\\left(\\vec{q}_{jtw}\\right) q^{\\rm tr}_{jitw}\n+ P_{itw}\\left(\\vec{q}_{itw}\\right)q^{\\rm tr}_{ijtw}$ in the objective function of the transmission firm is equal to $\\left( P_{jtw}\\left(\\vec{q}_{jtw}\\right) - P_{itw}\\left(\\vec{q}_{itw}\\right) \\right) q^{\\rm tr}_{jitw}$ which implies that the transmission firm between two nodes  makes profit by transmitting electricity from the node with  lower market price to the node with  higher market price.\n\n\nTransmission lines or interconnectors are usually controlled by the market operator and are regulated to maximize the social welfare in the market. The markets with regulated transmission firms are discussed as  electricity markets with transmission constraints in the literature, e.g., see  \\cite{JudithCardel,WilliamHogan,EvaggelosKardakos}. However, some electricity markets allow the transmission lines to act strategically, i.e. to make revenue by trading electricity across the nodes \\cite{AEMO}.\n\n\\vspace*{-.2 cm} \n\n\\section{Solution Approach} \\label{sec: sol}\nIn this section, we first provide a game-theoretic analysis of the lower-level problem. Next, the bi-level price volatility management problem is transformed to a single optimization Mathematical Problem with Equilibrium Constraints (MPEC). \n \n\\vspace*{-.2 cm}  \n \n\\subsection{Game-theoretic Analysis of the Lower-level Problem}\nTo solve the lower-level problem, we need to study the best response functions of  firms participating in the market. Then, any intersection of the best response functions of all firms will be a NE. In this subsection, we first establish the existence of NE for the lower-level problem. Then, we provide the necessary and sufficient conditions which can be used to solve the lower-level problem. \n\nTo transform the bi-level price volatility management problem to a single level problem, we need to ensure that for every vector of storage capacities, \\emph{i.e.} $\\vec{Q}^{\\rm s}=\\left[Q_1^{\\rm s},\\cdots,Q_I^{\\rm s}\\right]^\\top\\geq\\vec{0}$, the lower-level problem admits  a NE. At the NE strategy of the lower-level problem, no single firm has any incentive to unilaterally deviate its strategy from its NE strategy.  Note that the objective function of each firm is quasi-concave in its strategy and constraint set of each firm is closed and bounded for all $\\vec{Q}^{\\rm s}=\\left[Q_1^{\\rm s},\\cdots,Q_I^{\\rm s}\\right]^\\top\\geq\\vec{0}$. Thus, the lower level game admits a NE. This result is formally stated in Proposition \\ref{pro2}.\n\n\n\\begin{proposition} \\label{pro2}\nFor any vector of storage capacities, $\\vec{Q}^{\\rm s}=\\left[Q_1^{\\rm s},\\cdots,Q_I^{\\rm s}\\right]^\\top\\geq\\vec{0}$, the lower level game admits a Nash Equilibrium.\n\n\\end{proposition} \n\n\\begin{IEEEproof}\n\tNote that the objective function of each firm is continuous and quasi-concave in its\n\tstrategy. Also, the strategy space is non-empty, compact and\n\tconvex. Therefore, according to Theorem 1.2 in \\cite{quasiconvex}, the lower level\n\tgame admits a NE.\n\\end{IEEEproof} \n\n\\subsubsection{Best responses of wind firm $mi$} \nLet $\\vec{q}_{-(mi)}$ be the strategies of all firms in the market except the wind generator $m$ located in node $i$.  Then, the best response of  the wind generator $m$ in node $i$ to $\\vec{q}_{-(mi)}$ satisfies the necessary and sufficient Karush-Kuhn-Tucker (KKT) conditions ($t \\in \\{1, ...,  N_{\\rm T}\\}; w \\in \\{1, ..., N_{\\rm w}\\}$):\n\\begin{subequations} \\label{KKTw}\n\t\\begin{align}\n& P_{itw}\\left(\\vec{q}_{itw}\\right)+\\frac{\\partial{P_{itw}\\!\\left(\\vec{q}_{itw}\\right)}}{\\partial{q^{\\rm wg}_{mitw}}}q^{\\rm wg}_{mitw}  - \\frac{\\mu^{\\rm wg,max}_{mitw}}{\\PRP{w}}\\le 0 \\perp q^{\\rm wg}_{mitw} \\ge 0   \\label{8LPs}\\\\\n&  q^{\\rm wg}_{mitw} \\le {Q}_{mitw}^{ \\rm wg} \\perp \\mu^{\\rm wg,max}_{mitw} \\ge 0 \\label{9s1}\n\t\\end{align}\n\\end{subequations} \nwhere the perpendicularity sign, $\\perp$, means that at least one of the adjacent inequalities must be satisfied as an equality \\cite{ferris}.\n\\subsubsection{Best responses of storage firm $i$} \nTo study the best response of the storage firm in node $i$, let $\\vec{q}_{-i}$ denote the collection of  strategies of all firms except the storage firm in node $i$. Then, the best response of the storage firm in node $i$ is obtained by solving the following KKT conditions ($t \\in \\{1, ...,  N_{\\rm T}\\}; w \\in \\{1, ..., N_{\\rm w}\\}$): \n\\begin{subequations} \\label{KKTst}\n\t\\begin{align}\n&  P_{itw} \\left(\\vec{q}_{itw}\\right)+(1-\\gamma^{{\\rm s}}_{i} )\\frac{\\partial{P_{itw} \\left(\\vec{q}_{itw}\\right) }}{\\partial{q^{\\rm s}_{itw}}}q^{\\rm s}_{itw}   +\\!\\frac{\\mu^{\\rm s}_{itw}}{\\PRP{w}}=0 \\label{9LPs}\\\\\n& \\frac{-\\!\\eta^{\\rm dis}_{i} \\mu^{\\rm s}_{itw}\\!-\\!\\mu^{\\rm dis,max}_{itw}\\!-\\!\\Delta\\!\\sum_{k=t}^{{{N}_{{\\rm T}}}}\\! \\left(\\!\\mu^{\\rm s,min}_{ikw}\\!-\\!\\mu^{\\rm s,max}_{ikw} \\!\\right)}{\\PRP{w}}-c_{i}^{\\rm s} \\le 0 \\nonumber\\\\\n& \\hspace{6cm} \\perp q^{\\rm dis}_{itw} \\ge 0 \\\\\n& \\frac{\\frac{\\mu^{\\rm s}_{itw}}{\\eta^{\\rm ch}_{i}}\\! +\\!\\mu^{\\rm ch,min}_{itw}\\!-\\!\\mu^{\\rm ch,max}_{itw}\\!+\\!\\Delta\\!\\sum_{k=t}^{{{N}_{{\\rm T}}}}\\! \\left(\\!\\mu^{\\rm s,min}_{ikw}\\!-\\!\\mu^{\\rm s,max}_{ikw}\\! \\right)}{\\PRP{w}} \\!-\\!c_{i}^{\\rm s} \\! \\le \\! 0 \\nonumber \\\\\n& \\hspace{6cm} \\perp q^{\\rm ch}_{itw} \\ge 0 \\\\\n& q^{\\rm s}_{itw}=\\eta^{\\rm dis}_{i} q^{\\rm dis}_{itw} - \\frac{q^{\\rm ch}_{itw}}{\\eta^{\\rm ch}_{i}} \\\\\n& q^{\\rm dis}_{itw} \\le \\zeta_i^{{\\rm dis}} Q^{\\rm s}_i \\perp \\mu^{\\rm dis,max}_{itw} \\ge 0\\\\\n& q^{\\rm ch}_{itw} \\le \\zeta_i^{{\\rm ch}} Q^{\\rm s}_i \\perp \\mu^{\\rm ch,max}_{itw} \\ge 0\\\\\n& 0 \\le \\sum_{k=1}^{t} \\left(q^{\\rm ch}_{ikw}-q^{\\rm dis}_{ikw}\\right) \\Delta  \\perp  \\mu^{\\rm s,min}_{itw} \\ge 0\\\\\n&  \\sum_{k=1}^{t} \\left(q^{\\rm ch}_{ikw}-q^{\\rm dis}_{ikw}\\right) \\Delta \\le Q^{\\rm s}_i \\perp  \\mu^{\\rm s,max}_{itw} \\ge 0 \\label{10s2}\n\t\\end{align}\n\\end{subequations}\n\n\\subsubsection{Best responses of classical generation firm $ni$} \nThe best response of the classical generator $n$ in node $i$ to $\\vec{q}_{-(ni)}$, i.e. the collection of strategies of all firms except the classical generator $n$ in node $i$, is obtained by solving the following KKT conditions ($t \\in \\{1, ...,  N_{\\rm T}\\}; w \\in \\{1, ..., N_{\\rm w}\\}$):\n\\begin{subequations} \\label{KKTcg}\n\t\\begin{align}\n&   P_{itw}\\left(\\vec{q}_{itw}\\right)\\!-\\!c_{ni}^{\\rm cg}\\!+\\! \\frac{\\partial{P_{itw}\\left(\\vec{q}_{itw}\\right)}}{\\partial{q^{\\rm cg}_{nitw}}}\\!q^{\\rm cg}_{nitw}    \\!+\\!\\frac{\\!-\\!\\mu^{\\rm cg,max}_{nitw}\\!+\\!\\mu^{\\rm cg,up}_{ni(t+1)w}}{\\PRP{w} } \\nonumber \\\\\n& +\\frac{-\\mu^{\\rm cg,up}_{nitw}+\\mu^{\\rm cg,dn}_{nitw}-\\mu^{\\rm cg,dn}_{ni(t+1)w}}{\\PRP{w} }\\le 0 \\perp q^{\\rm cg}_{nitw} \\ge 0 \\label{10LPs}\\\\\n&  q^{\\rm cg}_{nitw} \\le {Q}_{ni}^{\\rm cg} \\perp \\mu^{\\rm cg,max}_{nitw} \\\\\n& q^{\\rm cg}_{nitw} - q^{\\rm cg}_{ni(t-1)w} \\le R^{\\rm up}_{ni} \\perp \\mu^{\\rm cg,up}_{nitw} \\ge 0\\\\\n& q^{\\rm cg}_{ni(t-1)w} - q^{\\rm cg}_{nitw} \\le R^{\\rm dn}_{ni} \\perp \\mu^{\\rm cg,dn}_{nitw} \\ge 0 \\label{11s2}\n\t\\end{align}\n\\end{subequations}\n\n\\subsubsection{Best responses of transmission firm $ij$} \nFinally, the best response of the transmission firm between nodes $i$ and $j$, to $\\vec{q}_{-(ij)}$, i.e. the set of all firms' strategies except those of the transmission line between nodes $i$ and $j$, can be obtained  using the KKT conditions ($t \\in \\{1, ...,  N_{\\rm T}\\}; w \\in \\{1, ..., N_{\\rm w}\\}$):\n\\begin{subequations} \\label{KKTtr}\n\t\\begin{align}\t\n& P_{itw}\\left(\\vec{q}_{itw}\\right) +\\left(1-\\gamma^{{\\rm tr}}_{ij}\\right) \\frac{\\partial{P_{itw}\\left(\\vec{q}_{itw}\\right)}}{\\partial{q^{\\rm tr}_{ijtw}}}q^{\\rm tr}_{ijtw} +\n\\frac{\\mu^{\\rm tr}_{jitw}+\\mu^{\\rm tr}_{ijtw}}{\\PRP{w}} \\nonumber\\\\\n& \\hspace{3.5cm} +\\frac{\\mu^{\\rm tr,min}_{ijtw}-\\mu^{\\rm tr,max}_{ijtw}}{\\PRP{w}}=0  \\label{11LPs}\\\\\n& q^{\\rm tr}_{ijtw}=-q^{\\rm tr}_{jitw}\\\\\n& -Q_{ij}^{\\rm tr} \\le q^{\\rm tr}_{ijtw} \\perp \\mu^{\\rm tr,min}_{ijtw}\\ge 0\\\\\n& q^{\\rm tr}_{ijtw} \\le Q_{ij}^{\\rm tr} \\perp \\mu^{\\rm tr,max}_{ijtw}\\ge 0   \\label{12s1}\n\t\\end{align}\n\\end{subequations} \n\n\\vspace*{-.2 cm} \n\\subsection{The Equivalent Single-level Problem}\n\nHere, the bi-level price volatility management problem is transformed into a single-level MPEC. To this end, note that for every vector of storage capacities the market price can be obtained by solving the firms' KKT conditions. Thus, by imposing the KKT conditions of all firms as constraints in the optimization problem \\eqref{ISO}, the price volatility management problem can be written as the following single-level optimization problem:\n\t\\begin{align}\n\t& \\min \\sum_{i=1}^I  Q^{\\rm s}_i  \\label{optimization}\\\\\n\t&{\\rm s.t.} \\nonumber \\\\ \n\t& (\\ref{ISO1}-\\ref{ISO3}),(\\ref{8LPs}-\\ref{9s1}),(\\ref{9LPs}-\\ref{10s2}),(\\ref{10LPs}-\\ref{11s2}),(\\ref{11LPs}-\\ref{12s1})  \\nonumber\\\\\n\t& m \\in \\{1,..., N^{\\rm wg}_i\\} , n \\in \\{1, ..., N^{\\rm cg}_i\\}, i,j \\in \\{1,..., I\\}, \\nonumber \\\\\n\t& t \\in \\{1, ...,  N_{\\rm T}\\}; w \\in \\{1, ..., N_{\\rm w}\\}  \\nonumber\n\t\\end{align}\nwhere the optimization variables are the storage capacities, the bidding strategies of all firms and the set of all Lagrange multipliers. Because of the nonlinear complementary constraints, the feasible region is not necessarily convex or even connected. Therefore, increasing the storage capacities stepwise, we solve the lower level problem, which is convex. Once the price volatility constraint is addressed, the optimum solution is found.   \n\n\n\t\\begin{remark}\nIt is possible to convert the equivalent single level problem (\\ref{optimization}) to a Mixed-Integer Non-Linear Problem (MINLP). However, the large number of integer variables potentially makes the resulting MINLP computationally infeasible.   \n\t\\end{remark}\n\n\n\n\n\n\\vspace*{-.2 cm} \n \n\\section{Case Study and Simulation Results} \\label{sec: sim}\n\nIn this section, we study the impact of  storage installation on price volatility in two nodes of Australia's National Electricity Market (NEM): South Australia (SA) and Victoria (VIC). SA has a high level of wind penetration and VIC has high coal-fueled classical  generation.  Real data for price and demand from the year 2013\nis used to calibrate the inverse demand function in the model. Different types of generation firms, such as coal, gas, hydro, wind and biomass, with generation capacity (intermittent and dispatchable) of 3.7 GW and 11.3 GW were active in SA and VIC, respectively, in 2013.\nThe transmission line interconnecting SA and VIC, which is a regulated line, has the capacity of 680 MW but currently is working with just 70\\% of its capacity.  \nThe generation capacities in our numerical results are gathered from Australian Electricity Market Operator's (AEMO's) website (aemo.com.au) and all the prices are shown in Australian dollar.       \n\nSimilar to \\cite{Morales}, we consider a scenario based analysis wherein three scenarios, i.e. high wind scenario (with probability of 0.2), low wind scenario (with probability 0.2) and base wind scenario (with probability of 0.6), are defined  to capture the wind power availability.\n The base wind scenario indicates the available wind generation level for a day (24 hours), in each node,  averaged over a year \\cite{AEMOwind}. \nGiven that the wind turbines are dispersed over the whole region in each node, we assume that the wind power availability is often around its expected value, i.e. the base wind level. \nThe wind generation level at high wind and low wind scenarios are assumed to be $\\phi\\%$ above and below the wind generation level at the base wind scenario, respectively. Various levels of wind availability can be captured by changing the wind power fluctuation parameter $\\phi$ \\cite{Ding}.\n\n\n\n\n\nIn what follows, by price volatility we mean the maximum variance of market price, i.e. $\\max_{it}{\\rm Var}(P_{itw}(q^\\star_{itw}))$. Also, by square root of price volatility we mean the maximum standard deviation of market price, i.e. $\\max_{it}\\sqrt{{\\rm Var}(P_{itw}(q^\\star_{itw})}$.\n\n\n\n\n\\vspace*{-.2 cm} \n\n\\subsection{One-node model simulations in South Australia}\nIn this subsection, we first study the impacts of peak demand levels and supply capacity shortage on the electricity price in SA with no storage. Next, we study the effect of storage on price volatility in SA. Fig. \\ref{fig_price} shows the hourly prices for a day in SA (with no storage) for three different cases: $(i)$ a regular demand day, $(ii)$ a high demand day, $(iii)$ a high demand day with coal plants outage. An additional load of  1000 MW  is considered  in the high demand case during hours 16, 17 and \n18\nto study the joint effect of wind intermittency  and large demand variations on the price volatility. The additional loads are sometimes demanded in the market due to unexpected high temperatures happening  in the region. \nThe coal-plants outage case is motivated by the recent retirement of two coal plants in SA with total capacity of 770 MW \\cite{AER}. This allows us to investigate the joint impact of wind indeterminacy and low supply capacity on the price volatility. \n\n \n\n\n\n \nAccording to Fig. \\ref{fig_price}, in a regular demand day, wind power fluctuation with  $\\phi=50\\%$  does not create much price fluctuation. \nIn the regular demand day, the maximum price is equal to 194 \\$\/MWh in the base wind scenario  whereas it changes to 161 \\$\/MWh and 244 \\$\/MWh in the high wind and the low wind scenarios, respectively. The square root of the price volatility  in the regular demand day is equal to 26 \\$\/MWh.\nBased on Fig. \\ref{fig_price}, the maximum price in a high demand day in SA changes from 933 \\$\/MWh in the base wind scenario to 576 \\$\/MWh and 1837 \\$\/MWh in the high wind and the low wind scenarios, respectively. The square root of the price volatility in the high demand day is equal to 420 \\$\/MWh. The extra load at peak times and  the wind power fluctuation create a higher level of price volatility during a high demand day compared with a regular demand day.\n\nThe outage of coal plants in SA beside the extra load at peak hours increases the price volatility due to the wind power fluctuation. The maximum price during the high demand day with coal plants outage varies from 3446 \\$\/MWh to 1436 \\$\/MWh and 9634 \\$\/MWh in the high wind and the low wind scenarios, respectively. The square root of the price volatility  during the high demand day with coal plant outage is equal to 2787 \\$\/MWh. The square root of the price volatility during the high demand day with coal plant outage is almost 107 times more than the regular demand day due to the simultaneous variation in both supply and demand.\n\n\n\n\n\n\n\\begin{figure}[!htb] \n\n\t\\centering\n\t\\includegraphics[scale=.6]{price4.pdf} \n\t\\caption{Hourly wholesale electricity prices in SA with $\\phi=50\\% $ and no storage.} \t\t\t\n\t\\label{fig_price}\n\\end{figure}\n\n\nFig. \\ref{table1} shows the minimum required (strategic\/regulated) storage capacities for achieving various levels of price volatility in SA during a high demand day with coal plants outage. The minimum storage capacities  are calculated by solving the optimization problem (\\ref{optimization}) for the high demand day with coal-plants outage case with $\\phi=50\\%$.\nAccording to Fig. \\ref{table1}, a strategic storage firm requires a substantially larger capacity, compared with a regulated storage firm, to achieve a target price volatility level due to the selfish behavior of the storage firms.\nIn fact, the strategic storage firms may sometimes withhold their available capacities and do not participate in the price volatility reduction as they do not always benefit from reducing the price. \nThe price volatility in SA can be reduced by 80\\% using either 420 MWh strategic storage or 340 MWh regulated storage.\nNote that AEMO has forecasted about 500 $\\rm MWh$ battery storage to be installed in SA until 2035 \\cite{BattFuture}.\n\n\n\n\n\\begin{figure}[!htb] \n\n\t\\centering\n\t\\includegraphics[scale=.6]{table1_3.pdf} \n\t\\caption{Optimal strategic and regulated storage capacity for achieving different price volatility levels in SA node with $\\phi=50\\%$ for a high demand day with coal-plants outage.} \t\t\t\n\t\\label{table1}\n\\end{figure}      \n\n\nAccording to our numerical results, storage can displace the peaking generators, with high fuel costs and market power, which results in reducing the price level and the price volatility. A storage capacity of 500 MWh (or $\\frac{500}{2}$ MW given the discharge coefficient $\\eta^{\\rm dis}=\\frac{1}{2}$) reduces the square root of the price volatility from 2787 \\$\/MWh to 919 \\$\/MWh, almost 30\\% reduction, during a high demand day with coal-plant outage in SA.\n\n \n\n\n\n\n\n\n \n\n     \n\n\n\n\n\n\n\n\n \n\n \n\n\n\n\n\n\n\n\n\n\n\n\nThe behaviour of the peak and the daily average prices for the high demand day with coal plants outage in SA is illustrated in Fig. \\ref{fig_AP}. In this figure, the peak price represents the maximum of price over all scenarios during the day, i.e. $\\max_{t,w}P_{tw}(\\vec{q}^\\star_{tw})$ and the daily average price indicates the average of price over time and scenarios, i.e. $\\frac{1}{N_T}\\sum_{tw}P_{tw}(q^\\star_{tw})\\Psi_w$.\nSensitivity analysis of the peak and the daily average prices in SA with respect to storage capacity  indicates that high storage capacities lead to relatively low prices in the market. \nAt very high prices, demand is almost inelastic and a small amount of excess supply leads to a large amount of price reduction.\nAccording to Fig. \\ref{fig_AP}, the rate of price reduction decreases as the storage capacity increases since large storage capacities lead to relatively low peak prices which make the demand more elastic.\n\nBased on Fig. \\ref{fig_AP}, the impact of storage  on the daily average and peak prices depends on whether the storage firm is strategic or regulated. It can be observed  that the impacts of strategic and regulated storage firms on the daily peak\/average prices are almost similar for small storage capacities,\ni.e. when the storage capacity is smaller than 100 MWh (or $\\frac{100}{2}$ MW given $\\eta^{\\rm dis}=\\frac{1}{2}$). However, a regulated firm reduces both the peak and the average prices more efficiently compared with a strategic storage firm  as its capacity becomes large. \nA large strategic storage firm in SA does not use its excess capacity beyond  500 MWh to reduce the market price since it acts as  a strategic profit maximizer, but a regulated storage firm contributes to the price volatility reduction  as long as there is potential for price reduction by its operation.    \n\n\n\n\n\n\n\\begin{figure}[!htb] \n\n\t\\centering\n\t\\includegraphics[scale=0.6]{APprice.pdf} \n\t\\caption{Daily peak and average prices in SA versus storage capacity with wind power fluctuation parameter $\\phi=50\\%$ in a high demand day with coal-plant outage.} \t\t\t\n\t\\label{fig_AP}\n\\end{figure}\n\n\n\nFig. \\ref{fig_vol} depicts the square root of price volatility  in SA during the high demand day with coal plant outage for $\\phi=50\\%$ and $\\phi=40\\%$. \nAccording to this figure, the price volatility in the market decreases by installing either regulated or strategic storage devices.\nHowever, a strategic storage firm stops reducing the price volatility  when its capacity exceeds a threshold value. \n Moreover, the square root of price volatility, in all cases, diminishes almost by 30\\% as the wind\n power fluctuation parameter decreases from 50\\% to 40\\%. Note that,  as $\\phi$ becomes small, the wind generation level becomes less volatile which results in a relatively low price volatility.   \nThis observation  indicates that the required storage capacity  to ensure a price volatility reduction target decreases as the wind power fluctuation parameter becomes small. Note that wind power fluctuation parameter $\\phi$ can be reduced by improving the geographic diversity of wind farms in a region. \n\n\n\n\nBased on Fig. \\ref{fig_vol}, both price volatility and the required storage capacity for achieving a target price volatility become large as the wind power fluctuation parameter $\\phi$ increases. \n To reduce the square root of price volatility to 1200 \\$\/MWh, the required strategic capacity with $\\phi=40\\%$ and $\\phi=50\\%$ is 60 MWh and 70 MWh, respectively, more than that of a regulated storage.\nThis observation confirms that regulated storage firms are more efficient  than strategic firms in reducing the price volatility. Although storage alleviates the price volatility in the market, it is not capable to eliminate it completely.  \n\n\n\n\n\n\n\\begin{figure}[!htb]  \n\n\t\\centering\n\t\\includegraphics[scale=0.6]{volatility5.pdf} \n\t\\caption{Square root of price volatility in SA versus storage capacity  with  $\\phi$ equal to $40\\%$ and $50\\%$ during a high demand day with coal-plants outage.} \t\t\t\n\t\\label{fig_vol}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n  \n\n\n\n\n\n\n \n\n\n\n  \n\n\n \n\n\n\n\n\\vspace*{-.2 cm}\n\n\\subsection{Two-node model simulations in South Australia and Victoria}\n\nIn the previous subsection, we analysed the impact of storage on the price volatility in SA when the SA-VIC interconnector is not active. In this subsection, we first study the  effect of  the interconnector between  SA and VIC on the price volatility in the absence of  storage firms. Next, we investigate the impact of storage firms on the price volatility when the SA-VIC transmission line operates at various capacities.\nIn our numerical results, SA is connected to VIC using a 680 MW interconnector which is currently operating with  70\\% of its capacity, i.e. 30\\% of its capacity is under maintenance.\nThe numerical results in this subsection are based on the two-node model for a high demand day with coal plant outage in SA. To investigate the impact of transmission line on  price volatility, it is assumed that the SA-VIC interconnector  operates with 60\\% and 70\\% of its capacity.\n\nAccording to our numerical results, the peak price in SA and VIC is equal to 9634 \\$\/MWh when the SA-VIC interconnector is completely  in outage and  the wind power fluctuation parameter $\\phi$ is equal to $50\\%$. However, the peak price reduced to 1406 \\$\/MWh and 1114 \\$\/MWh when the interconnector operates at  60\\% and 70\\% of its capacity. The square root of price volatility is 2787 \\$\/MWh, 303 \\$\/MWh, and 219 \\$\/MWh when the capacity of the SA-VIC transmission line is equal to 0\\%, 60\\%, and 70\\%, respectively. \n\nSimulation results show that as long as the transmission line is not congested, the interconnector alleviates the price volatility phenomenon in SA by importing electricity from VIC to SA at peak times. Since the market in SA compared to VIC is much smaller, about three times, the price volatility abatement in SA after importing electricity from VIC is much higher than the price volatility increment in VIC. Moreover, the price volatility reduces as the capacity of transmission line increases. \n\nFig. \\ref{table2} shows the optimum storage capacity versus the percentage of price volatility reduction in the two-node market.  According to our numerical results,  storage is just located in SA, which witnesses a high level of  price volatility as the capacity of transmission line decreases. \nAccording to this figure, the optimum storage capacity becomes large as the capacity of transmission line decreases. Note that a sudden decrease of the transmission line capacity may result in a high level of price volatility in SA. However, based on Fig. \\ref{table2}, storage firms are capable of reducing the price volatility during the outage of the   interconnecting lines. \n\n\n   \n\n\\begin{figure}[!htb]  \n\n\t\\centering\n\t\\includegraphics[scale=0.6]{table2_3.pdf} \n\t\\caption{Optimal regulated storage capacity versus the percentage of  price volatility reduction in the two-node market with wind power fluctuation parameter $\\phi=50\\%$ in a high demand day with coal-plants outage in SA.} \t\t\t\n\t\\label{table2}\n\\end{figure}\n\n\n\n\n\\vspace*{-.2 cm}\n\n\\section{Conclusion} \\label{sec: con}\nHigh penetration of intermittent renewables, such as wind or solar farms, brings high levels of price volatility in electricity markets. Our study presents  an optimization model which decides on the minimum storage capacity required for achieving a price volatility  target in electricity markets. Based on our numerical results, the impact of storage on the  price volatility  in one-node electricity market of SA and two-node market of SA-VIC can be summarized as:\n\n\\begin{itemize}\n\\item Storage alleviates price volatility in the market due to the wind intermittency. However, storage does not remove price volatility completely, i.e. storage stops reducing the price volatility when it is not profitable.\n\n\\item The effect of a storage firm on  price volatility reduction depends on whether the firm  is  regulated or strategic. Both storage types have similar operation behaviour  and price reduction effects when they possess small capacities. For larger capacities, a strategic firm may under-utilize its available capacity and stop reducing the price level  due to its profit maximization strategy. On the other hand, a regulated storage firm is  more efficient  in price volatility reduction because of its social welfare maximization strategy. The price level and volatility reduction patterns observed when storage firms are regulated provide stronger incentives for the market operator to subsidize the storage technologies.\n\n\n\n\\item Both storage devices and transmission lines are capable of reducing the price volatility.  \nHigh levels of price volatility that may happen due to the line maintenance  can be alleviated by storage devices.         \n\n\n\n \n\\end{itemize}\n\nWe intend to study the impact of ancillary services markets \\cite{MulCommSpatial} and capacity markets \\cite{CapEnergy}  on the integration of storage systems  in electricity networks in our future work. We also plan to investigate the impacts of other factors in both demand and supply sides on the price volatility.\n\n\\appendices\n\n\n\n\\vspace*{-.2 cm} \n\\section{Charging\/Discharging} \\label{App1}\nIn this appendix, we show that the charge and discharge levels of any storage device  cannot be simultaneously positive at the NE of the lower game.\nConsider a strategy in which  both charge and discharge levels of storage device $i$ at time $t$ under  scenario $w$, \\emph{i.e.} $q^{\\rm dis}_{itw} ,q^{\\rm ch}_{itw}$, are both positive. We show that this strategy cannot be a NE strategy as follows. The net electricity flow of storage can be written as  $q^{\\rm s}_{itw}=\\eta^{\\rm dis}_{i} q^{\\rm dis}_{itw} - \\frac{q^{\\rm ch}_{itw}}{\\eta^{\\rm ch}_{i}}$. Let $\\bar{q}^{\\rm dis}_{itw}$ and $\\bar{q}^{\\rm ch}_{itw}$ be the new   discharge and charge levels of storage firm $i$ defined as $ \\Big \\{{{\\bar{q}}^{\\rm dis}_{itw}}=q^{\\rm dis}_{itw}-\\frac{q^{\\rm ch}_{itw}}{\\eta^{\\rm dis}_{i}\\eta^{\\rm ch}_{i}}, \\quad {{\\bar{q}}^{\\rm ch}_{itw}}=0 \\Big \\}$ if $ q^{\\rm s}_{itw}>0 $, or $\\Big \\{{{\\bar{q}}^{\\rm dis}_{itw}}=0, \\quad {{\\bar{q}}^{\\rm ch}_{itw}}=q^{\\rm ch}_{itw}-q^{\\rm dis}_{itw} \\eta^{\\rm dis}_{i}\\eta^{\\rm ch}_{i} \\Big \\}$ if $ q^{\\rm s}_{itw}  <0 $.\nThe new net flow of electricity can be written as  $\\bar{q}^{\\rm s}_{itw}=\\eta^{\\rm dis}_{i} {{\\bar{q}}^{\\rm dis}_{itw}}-\\frac{{{\\bar{q}}^{\\rm ch}_{itw}}}{\\eta^{\\rm ch}_{i}}$. Note that the new variables $\\bar{q}^{\\rm s}_{itw}$, ${{\\bar{q}}^{\\rm ch}_{itw}}$ and ${{\\bar{q}}^{\\rm dis}_{itw}}$  satisfy the constraints\n(\\ref{cons41}-\\ref{cons44}).\n\n \n\nConsidering the new charge and discharge strategies ${{\\bar{q}}^{\\rm dis}_{itw}}$ and ${{\\bar{q}}^{\\rm ch}_{itw}}$, instead of $q^{\\rm dis}_{itw}$ and $q^{\\rm ch}_{itw}$,  the nodal price and the net flow of storage device $i$ do not change. However, the charge\/discharge cost of the storage firm $i$, under the new strategy, is reduces by:\n\t\\begin{align}\n c_{i}^{\\rm s} \\left(q^{\\rm ch}_{itw}+q^{\\rm dis}_{itw} \\right) - c_{i}^{\\rm s} \\left({\\bar{q}}^{\\rm dis}_{itw}+{\\bar{q}}^{\\rm ch}_{itw}\\right) > 0  \\nonumber\n\t\\end{align}\nHence, any strategy in which the charge and discharge variables are simultaneously positive cannot be a NE, i.e. at the NE of the lower game each storage firm is either in the charge mode or discharge mode.\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}