diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzeyta" "b/data_all_eng_slimpj/shuffled/split2/finalzzeyta" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzeyta" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \\label{sec:intro}\n\n\nAmyotrophic lateral sclerosis (ALS) is a rare progressive neurological disorder resulting in the degeneration of both upper motor neurons of the cerebral cortex and lower motor neurons of the spinal cord and peripheral nervous system, with a very poor prognosis. Currently, there is no cure for ALS and clinical care is generally limited to treating secondary infections and palliative care, such as surgically inserting a percutaneous endogastrostomy (PEG) tube to provide enteral nutrition for individuals having difficulty swallowing \\citep{procaccini2008}. Our objective is to assess the effect of inserting a PEG feeding tube on preventing weight loss. PEG insertion is an individual decision and one that must be made while the individual is strong enough to proceed with surgery. Hence, a randomized controlled trial to study the effect of PEG would be implausible. We develop new methods to evaluate PEG using data from the Emory ALS Clinic registry.\n\nLet $T$ denote the continuously-defined time of PEG insertion for a randomly selected patient from the population. The observed outcome $Y$ is collected at or just after a fixed point at time $L$, which consequently restricts the time of PEG insertion. If subjects were randomly assigned to receive PEG prior to $L$ and randomly assigned to treatment times, then both treatment effect and dose-response curve could be estimated using standard methods. However, treatment assignment depends on patient characteristics and confounds the effect of treatment on outcome. To remove confounding associated with covariate imbalance among treatment levels, we rely on the general concept of the propensity score \\citep{rosenbaumandrubin1983,rubin1996}.\n\nWhen treatment assignment is binary, the propensity score \\citep{rosenbaumandrubin1983} is defined as probability of receiving a treatment given a set of observed variables. Generalizations of the propensity score as a balancing score have been investigated in various settings \\citep{hirano2004,imaiandvandyk2004,hansen2008prognostic,allen2011control,hu2014estimation}. For continuously-defined treatment levels, \\citet{hirano2004} proposed a direct translation of the propensity score by replacing the conditional probability mass function with the conditional density function of treatment assignment given covariates, known as a generalized propensity score (GPS); while this approach leads to as many propensity scores as there are levels of the treatment it uses only one single score at a time. Although \\citet{imaiandvandyk2004} similarly found that the conditional density function of treatment assignment given covariates could serve as a propensity score, they noted potential limitations of this approach and suggested instead using the linear predictor in regression models or other summary statistic that are of finite dimension. When treatment assignment occurs over time as in the case where an individual chooses to receive PEG insertion or not, we must allow for the possibility of time-dependent confounding. To this end,\nlet $X_t$ denote a set of $p$-dimensional time-dependent covariates at time $t$ and ${\\ensuremath{\\mathcal X}}_t=\\left\\{X_s,~0\\le s\\le t\\right\\}$ denote the history of covariates up to time $t$. Then, the probability of treatment assignment at time $t$ given the covariate history up to time $t$ is\n\\begin{equation}\\label{eq:density.tz}\nf(t\\mid{\\ensuremath{\\mathcal X}}_t) = \\lim_{\\epsilon\\rightarrow 0} \\epsilon^{-1} P\\left(t\\le T < t+\\epsilon \\mid {\\ensuremath{\\mathcal X}}_t\\right),\n\\end{equation}\nwhere $f(t\\mid {\\ensuremath{\\mathcal X}}_t) = h(t\\mid X_t)\\exp\\left\\{-\\int_0^t h(s\\mid X_s)\\, ds\\right\\}$\nand the hazard function is\n\\begin{equation}\\label{eq:haz.tz}\nh(t\\mid X_t) = \\lim_{\\epsilon\\rightarrow 0} \\epsilon^{-1} P\\left(t\\le T < t+\\epsilon \\mid T\\ge t, X_t\\right).\n\\end{equation}\nBecause $h(t\\mid X_t)$ uniquely parameterises $f(t\\mid \\cal{X}_t)$, either model~\\eqref{eq:density.tz} or model~\\eqref{eq:haz.tz} may be regarded as a legitimate treatment assignment model for continuous treatment with time-independent or time-dependent confounding \\citep{li2001,lu2005}.\n\nOf note, $f(t\\mid \\cal{X}_t)$ is a function of the entire covariate history ${\\ensuremath{\\mathcal X}}_t$, whereas the hazard function $h(t\\mid X_t)$ is a function of $X_t$ only. This subtle, yet important difference can lead to difficulties when extending methods proposed by \\cite{imaiandvandyk2004} and \\citet{hirano2004} to time-dependent confounding via standard hazard modeling. In addition, both \\citet{li2001} and \\citet{lu2005} used the hazard function $h(t\\mid X_t)$ as a GPS for matching which allows for balancing $X_t$ at the time of treatment in a matched set. However, they did not establish the strong ignorability of treatment assignment given their time-dependent GPS; this property does not hold if $Y$ is associated with ${\\ensuremath{\\mathcal X}}_t$ rather than just $X_t$, in which case their proposed procedures may not lead to valid causal inference. Additionally, their proposed methods are only applicable to studies with data routinely collected at regular intervals, which is often not true in clinical registries.\n\n\n\nWe propose the propensity process to correct for confounding in observational studies by balancing the covariate history ${\\ensuremath{\\mathcal X}}_t$. After the propensity process is estimated,\nbias-corrected data analyses can be achieved through matching or stratification \\citep{rosenbaumandrubin1983}. Establishing formally the theoretical properties of the propensity process for time-independent confounding requires different arguments than those presented in \\citet{imaiandvandyk2004}.\n\n\n\n\n\\section{Methods}\\label{sec:PS}\n\n\\subsection{Notation and Assumptions}\n\nOur framework is constructed through potential outcomes \\citep{rubin2005causal}. For $t\\in [0,L)$, we define $U_t=T\\wedge t$ as the treatment time restricted to time $t$ and $U=T\\wedge L$ as the treatment time restricted to time $L$, where $a \\wedge b$ denotes the minimum of $a$ and $b$. Let ${\\ensuremath{\\mathcal T}}_t=\\left\\{[0,t),t+\\right\\}$ define the set of potential treatment times restricted to $t$, $t\\in[0,L)$, where $t+$ means that a patient did not receive PEG treatment before $t$. Let $Y^*_t$ be the potential outcome if a subject received PEG treatment at time $t$, $t\\in[0,L)$, and $Y^*_{t+}$ the potential outcome if a subject did not receive PEG treatment in the interval $[0,t)$. It follows that $Y^*_{L+}$ denotes the potential outcome if a subject did not receive PEG treatment in the interval $[0,L)$. We also define the treatment-free potential covariate process ${\\ensuremath{\\mathcal X}}^*_t,~t\\le L$. Then, the set of potential outcomes and treatment-free potential covariate process for a randomly selected subject from the population is $\\{Y^*_s, {\\ensuremath{\\mathcal X}}^*_s,~s\\in {\\ensuremath{\\mathcal T}}_t\\}$ when treatment time is restricted at $t$, $t\\in[0,L)$. In contrast, the observed data are $(Y, U, {\\ensuremath{\\mathcal X}}_{U})$, where the observed outcome $Y=Y^{*}_{U}$, and the observed covariate history ${\\ensuremath{\\mathcal X}}_{U}={\\ensuremath{\\mathcal X}}^*_{U}$.\n\nGiven $\\theta_t=h(t\\mid X^*_t)$, we define the propensity process as the sample path of the hazard function from baseline to time $t$, i.e.,\n\\begin{equation}\\label{eq:pp}\n\\Theta_t = \\left\\{ \\theta_s=h(s\\mid X^*_s),~0\\le s\\le t \\right\\},\n\\end{equation}\nnoting that $\\Theta_t$ is dependent on ${\\ensuremath{\\mathcal X}}^*_t$. As ${\\ensuremath{\\mathcal X}}^*_t$ is observable only up to $U$, $\\Theta_t$ is estimable only up to $U$. While this concept seems similar to the propensity function \\citep{imaiandvandyk2004}, the distinguishing factor of the propensity process is that $\\Theta_t$ depends on $t$ and is of infinite dimension and $\\Theta_L$ cannot be fully estimated for subjects receiving PEG before $L$, whereas the propensity function in \\citet{imaiandvandyk2004} only allows for incorporation of time-independent covariates and can be estimated for all subjects.\n\nIn our framework, we make two assumptions.\n\\begin{assump} [Stable unit treatment value assumption] The distributions of potential outcomes for different subjects are independent of one another.\n\\end{assump}\n\\begin{assump}[Strong Ignorability] For every $t\\in [0, L)$, $\\mbox{pr}(U_t\\in{\\ensuremath{\\mathcal A}}\\mid Y^*_s,{\\ensuremath{\\mathcal X}}^*_t)=\\mbox{pr}(U_t\\in{\\ensuremath{\\mathcal A}}\\mid {\\ensuremath{\\mathcal X}}^*_t)$ and $\\mbox{pr}\\left(U_t\\in{\\ensuremath{\\mathcal A}}\\mid {\\ensuremath{\\mathcal X}}^*_t\\right)>0$ for all $s\\in {\\ensuremath{\\mathcal T}}_t$, ${\\ensuremath{\\mathcal X}}^*_t$, and ${\\ensuremath{\\mathcal A}}\\subseteq{\\ensuremath{\\mathcal T}}_t$.\n\\end{assump}\nAssumption~1 is a common assumption in causal inference. However, our Assumption~2 is defined for each time point $t$ and differs from the standard strong ignorability of treatment assignment assumption used in earlier work for balancing scores. One implication of Assumption~2 is that, conditional on the treatment-free history ${\\ensuremath{\\mathcal X}}^*_t$, receiving treatment at $t$ or not is independent of the set of potential outcomes, allowing us to model treatment assignment without conditioning on potential outcomes. \n\n\\subsection{Main results}\\label{theory}\n\nWe establish the large-sample results of the propensity process assuming that the true propensity process is known along the lines of \\citet{rosenbaumandrubin1983} and \\citet{imaiandvandyk2004}.\n\\begin{prop}\n\\label{prop1}\n$U$ is conditionally independent of treatment-free covariate history ${\\ensuremath{\\mathcal X}}^*_L$ given $\\Theta_L$, where ${\\ensuremath{\\mathcal X}}^*_L$ and $\\Theta_L$ are the entire treatment-free covariate history and propensity process, respectively.\n\\end{prop}\nProposition 1 establishes $\\Theta_L$ as a balancing functional that balances the entire covariate history. Proposition 1 requires that $\\Theta_L$ is known or can be estimated in the entire domain $[0,L)$. In practice, however, we can only observe the covariate process ${\\ensuremath{\\mathcal X}}^*_{U}$ and hence estimate $\\Theta_{U}$. Proposition 2 establishes the balancing property for every given time point $t$ in $[0,L)$.\n\n\\begin{prop}\\label{prop2}\nFor every $t\\in[0,L)$, $U_t$ is conditionally independent of treatment-free covariate history ${\\ensuremath{\\mathcal X}}^*_t$ given $\\Theta_t$, where ${\\ensuremath{\\mathcal X}}^*_t$ and $\\Theta_t$ are the treatment-free covariate history and propensity process through time $t$, respectively.\n\\end{prop}\nWhen $t=U$ in Proposition~\\ref{prop2}, we have that $U$ is independent of treatment-free covariate history ${\\ensuremath{\\mathcal X}}^*_U$ given $\\Theta_U$, where ${\\ensuremath{\\mathcal X}}^*_U={\\ensuremath{\\mathcal X}}_U$ is observable and hence $\\Theta_U$ is estimable.\n\\begin{thm}\\label{Thm1} For every $t\\in [0, L)$, $\\mbox{pr} \\left(U_t\\in{\\ensuremath{\\mathcal A}}\\mid Y^*_s,\\Theta_t\\right)=\\mbox{pr}(U_t\\in{\\ensuremath{\\mathcal A}}\\mid \\Theta_t)$ for all $s\\in {\\ensuremath{\\mathcal T}}_t$, $\\Theta_t$, and ${\\ensuremath{\\mathcal A}}\\subseteq{\\ensuremath{\\mathcal T}}_t$.\n\\end{thm}\nWhen $t=U$ in Theorem~\\ref{Thm1}, we have that $U$ is independent of potential outcomes given $\\Theta_U$, where $\\Theta_U$ is estimable. Several remarks are in order. First, in \\S~\\ref{ssec:pp} we suggest modeling the hazard function in~\\eqref{eq:haz.tz} through the proportional hazards model ~\\eqref{eq:PH}; one could also use other model formulations for~\\eqref{eq:haz.tz} and the results in Propositions~1--2 and Theorem~1 would still apply. Second, Proposition~\\ref{prop2} and Theorem~\\ref{Thm1} provide justifications for matching a subject treated at $t$ with an eligible control subject untreated at $t$ based on the propensity process up to $t$. It follows that each matched pair would have the same distribution for the covariate process up to $t$ and their potential outcomes are independent of their treatment assignments, allowing for valid causal inference. Third, our Proposition 2 is similar in spirit to Proposition 1 in \\citet{lu2005} but is more general in the sense that the propensity process balances the entire covariate history up to $t$ not just the covariates measured at $t$. In addition, \\citet{lu2005} did not establish the strong ignorability of treatment assignment given propensity scores similar to our Theorem 1. Proofs for Propositions~1--2 and Theorem~1 are given in the Appendix.\n\n\\section{Implementation and Practical Considerations} \\label{methods}\n\n\\subsection{Interpolated Propensity Processes}\\label{ssec:pp}\n\nIn practice, the propensity process $\\Theta_{U}$ must be estimated from the observed data. The challenge for estimating the propensity process is that we may not observe the complete treatment-free covariate process ${\\ensuremath{\\mathcal X}}^*_U$ on $[0,U]$; rather, we only get to observe the covariate process at a coarse set of discrete time points as is the case in the motivating ALS study. Here, we propose to borrow strength across subjects in the study sample by modeling each time-dependent covariate as a random curve over time via nonlinear mixed effects models. This allows a predictive curve to be estimated for the entire treatment-free covariate process for each subject.\n\nFirst, suppose we parameterize the hazard function in \\eqref{eq:haz.tz} through Cox's proportional hazards model and define the propensity process\nthrough the linear predictor,\n\\begin{align}\\label{eq:PH}\n&h(t\\mid X_t;\\beta)=h_{0}(t)\\exp(\\beta^{\\rm T}X_t),\n&&\\Theta_t = \\left\\{ \\theta_s=\\beta^{\\rm T}X^*_{s},~0\\le s\\le t \\right\\},\n\\end{align}\nwhere $h_{0}(t)$ is the unspecified baseline hazard function. Next, write the observed treatment-free covariate history for the $i$-th subject and $k$-th covariate as ${\\ensuremath{\\mathcal X}}_{ik}=\\left(X_{i1k},\\ldots,X_{im_ik}\\right)$, with time-dependent covariate $X_{ijk}$ measured at time $t_{ij}$. We note that the observation times $(t_{ij},~j=1,\\ldots,m_i)$ may be different for each subject but are assumed to be the same for all covariates within a subject. Then, for each time-dependent covariate, we fit the model,\n\\begin{eqnarray}\nX_{ijk} &=& b_k^{\\rm T}(t_{ij})\\gamma_k + b_k^{\\rm T}(t_{ij})\\alpha_{ik} + \\epsilon_{ijk},~(i=1,\\ldots,n;~j=1,\\ldots,m_i;k=1,\\ldots,p),\\label{eq:mm}\n\\end{eqnarray}\nwhere $\\epsilon_{ijk}$ are independent, mean-zero random errors. To provide greater flexibility in modeling the covariate process over time, we use spline-type models \\citep{ruppert2003semiparametric} in \\eqref{eq:mm} where $b(\\cdot)$ denotes a set of basis functions and $\\gamma_k$ and $\\alpha_{ik}$ are regression coefficients corresponding to the basis functions for the fixed and random effects, respectively. The interpolated treatment-free $\\widehat{{\\ensuremath{\\mathcal X}}}_t$ can be obtained from model~\\eqref{eq:mm} by replacing regression coefficients $\\gamma_k$ and $\\alpha_{ik}$ with their estimates $\\widehat\\gamma_k$ and $\\widehat\\alpha_{ik}$, respectively. Then the estimated propensity process $\\widehat\\Theta_{U}$ can be obtained from~\\eqref{eq:PH} by plugging in the interpolated $\\widehat{{\\ensuremath{\\mathcal X}}}_U$ and $\\widehat\\beta$, where $\\widehat\\beta$ is the estimated regression coefficient vector in the Cox proportional hazards model.\n\n\n\\subsection{Matching}\\label{subsec:matching}\n\n\nThe use of matched analyses based on propensity scores for testing causal null hypotheses has been advocated by several other authors; for example, see \\citet{rosenbaumandrubin1983}, \\citet{li2001} and \\citet{lu2005} and references therein. Matching can be performed by minimizing the integrated squared error between the estimated propensity process $\\widehat\\Theta_{t}$ of a subject who received PEG treatment at time $t$ and that of each eligible control with $U>t$. To accomplish this task, we implement a sequential matching algorithm. We start by ordering chronologically subjects according to their time of PEG treatment or censoring, namely $U$. Set the matched pair counter to $m=1$ and select the subject with the smallest time to PEG treatment, say subject $i_1$. Define the integrated squared difference in interpolated propensity processes between $i_1$ and $l$ as $Q(i_1,l) = I(T_{i_1}\\le L) \\int_0^{T_{i_{1}}} (\\widehat\\theta_{i_{1},t}-\\widehat\\theta_{l,t})^2\\, dt,$ for all subjects $l$ in the set of $n-1$ eligible controls $\\mathcal{C}_1=\\{l\\mid l=1,\\ldots,n,~l\\ne i\\}$. The matched control for $i_1$ is the nearest neighbor in interpolated propensity processes among eligible controls, i.e., $\\mbox{argmin}_{l\\in\\mathcal{C}_1} Q(i_1,l)$. Increment the matched pair counter by one to $m=2$ and select the subject with the smallest time to PEG treatment, say $i_2$, excluding the two subjects in the first matched pair. Therefore, the set of eligible controls, say $\\mathcal{C}_2$, contains $n-3$ subjects: all $n$ subjects less the two subjects in the first matched pair and $i_2$. The matched control for $i_2$ is the nearest neighbor in interpolated propensity processes among the set of eligible controls, $\\mbox{argmin}_{l\\in\\mathcal{C}_2} Q(i_2,l)$. Increment the matched pair counter by one and continue until all treated individuals are matched or until there are no suitable controls available for matching.\n\n\\section{Analysis of the ALS Registry Data} \\label{results}\n\nUsing a data set from the Emory ALS Registry, we assess the association of PEG treatment with the change in body mass index (BMI) from baseline to 18 months, i.e., $L$ = 18 months. The data set includes 240 patients who survived past $L$ and had at least one clinic visit between baseline and $L$. The patients who received PEG did so after their first clinic visit. The timing of recommending PEG by the physician involved many factors and the final decision to have PEG was made by each patient. We model treatment\nassignment through the proportional hazards model \\eqref{eq:PH} including the following covariates. The baseline risk factors are age at diagnosis, sex, site of onset of disease, negative inspiratory force, and time from diagnosis to the first clinic visit. Two time-varying covariates are forced vital capacity and body mass index, which may not be measured at every clinic visit for every patient. Each time-varying covariate is modeled over time using the mixed model~\\eqref{eq:mm}, where polynomial spline basis functions are used. The estimated curves are used to interpolate the covariate values needed for estimating the propensity process based on \\eqref{eq:PH}.\n\nWe compare three alternative approaches to the proposed propensity process. First, a na\\\"{i}ve analysis compares all treated individuals to those who are untreated prior to $L$. The second approach is the propensity function \\citep{imaiandvandyk2004} that uses baseline risk factors $X_0$ only in the treatment assignment model~\\eqref{eq:PH}, where $\\theta_0=\\beta^{\\rm T}X_0$ defines the propensity function. The third approach is the interpolated generalized propensity score, which uses the interpolated treatment-free $\\widehat{X}_t$ defined in \\S~\\ref{ssec:pp} to obtain the GPS for each subject in the spirit of \\citet{lu2005}, noting that $X_t$ may not be observed at time $U$ for a subject and its eligible controls as defined in \\S~\\ref{subsec:matching}. The same sequential matching algorithm in \\S~\\ref{subsec:matching} is used for all propensity score methods. Our matching algorithm resulted in $M=74$ pairs for the analysis using the propensity function and $M=76$ pairs for both analyses using the generalized propensity score and propensity process.\n\nFollowing \\citet{li2001} and \\citet{lu2005}, we assess balance of covariates by examining Type I errors from a log-rank test of the effect of the covariate on time to treatment, one covariate at a time. In the matched analyses, this model is stratified by the $M$ matched pairs. As shown in Table 1, prior to matching, balance is not achieved. While other methods improve covariate balance, they do not balance all covariates. However, matching using the propensity process results in balance across all covariates. This indicates that the propensity process outperforms the baseline propensity function or interpolated GPS in terms of balancing covariates and there may be residual confounding after matching by the other propensity score methods.\n\n\\begin{table}[h]\\label{tab:balance}\n \\centering\n \\caption{Covariate balance before and after matching}\n \\begin{tabular}{lcccc}\n \\hline\n & Prior to & Propensity & Generalized & Propensity \\\\\n Covariate & Matching & Function & Propensity Score & Process \\\\\\hline\n Body mass index & 0.277 & 0.245 & 0.986 & 0.991 \\\\\n Forced vital capacity & 0.764 & 0.539 & 0.201 & 0.317 \\\\\n Negative inspiratory force & 0.151 & 0.022 & 0.016 & 0.704 \\\\\n Age & 0.162 & 0.718 & 0.378 & 0.195 \\\\\n Sex & 0.577 & 0.695 & 0.002 & 0.706 \\\\\n Site & 0.001 & 0.003 & 1.000 & 0.341 \\\\\n Time from diagnosis & 0.676 & 0.633 & 0.033 & 0.854\\\\\n\\hline\n \\end{tabular}\n\\end{table}\n\nAfter matching, we test the causal null hypothesis that the mean potential outcome is the same whether a patient received PEG treatment at time $t$ versus PEG treatment at some time after $t$ or untreated by $L$, which can be written as $H_0: E(Y^*_t)=E(Y^*_s)$ for all $t 0$, we can find $\\delta > 0$ such that if $||(x,u)-(\\bar{x}_{N-1},\\bar{u}_{N-1})||_2 < \\delta$, then $||e_{N-1}||_2 < \\varepsilon$.\nWe can choose $\\varepsilon$ and $a_{N-1}$ such that any $(x,u)$ satisfying $x \\in \\tilde{Y}_{N-1}$ and Equation (\\ref{poly_u}) also satisfies $||(x,u)-(\\bar{x}_{N-1},\\bar{u}_{N-1})||_2 < \\delta$ and such that $e_{N-1} \\oplus G_N \\mathbb{P}_{N-1} \\subseteq G_N \\mathbb{P}_{N}$.\nThen $x^+ \\in \\tilde{Y}_N$.\nThe proof is complete by repeating the procedure for $i=N-2,\\ldots,0$.\n\\end{proof}\nThe Proposition says that if the system state is close enough to the nominal trajectory and if the nonlinear dynamics is Lipschitz, then we can use the {affine} control law (\\ref{poly_u}) for the nominal linearization of the trajectory to steer the nonlinear system to the target region for sure.\n{\nThis gives us stability guarantees near the vicinity of the nominal trajectory.\nIn practice, for any arbitrary system, we do not know how large its polytopic funnel can be, or how likely it is for the state to fall into the funnel.}\nWe want to be able to handle larger external disturbances during online execution.\nThis is what we are going to {address} in the next section.\n\n\\section{Online Execution}\nThe polytopic tree method in \\cite{sadraddini2019sampling} hopes to probabilistically cover the state space by polytopes by growing a single polytopic trajectory to an existing polytopic tree, similar in methodology to the growth of the LQR-trees \\cite{tedrake2010lqr}.\nThe method samples points in the state space, steers sample points to the current polytopic tree as well as building polytopes along the way by solving mixed-integer linear program (MILP), and enlarges the current tree by adding the new polytopes to the tree.\n\nIn practice, for example for the half-cylinder flipping experiment, there are several problems with the polytopic tree method.\nFirst, the volumes of the polytopes {can be} very small, and hence a state {may} never fall into any polytope.\nSecond, the polytopic tree method requires checking the closest polytope online, which is computationally inefficient in the naive implementation when there are large number of polytopes.\nThird, the polytopic tree method deals with PWA systems, but our system is nonlinear and computing trajectory from a sample point to the current tree is potentially an expensive nonlinear trajectory optimization problem which cannot be carried out online.\n\nTherefore, we propose the practical improvement of the polytopic tree method, with the sacrifice of stability guarantees when there are large deviations to the nominal trajectory.\nWe only keep one nominal trajectory, which is computed in Section \\ref{sec:trajectory}.\nWe build polytopes $Y_i = \\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}, i=0,\\ldots,N$ around the nominal linearization of the trajectory.\nThis amounts to solving an LP.\nDuring online execution, we compute the closest nominal state $\\bar{x}_i$ to the current state $x$ (with respect to some weighted $L_2$ norm) and determine the current contact mode $j$.\nIf $x$ is inside the polytope $\\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}$, the we use the corresponding control law $u_i(x) = \\bar{u}_i + \\theta_i p(x)$, where $x = \\bar{x}_i + G_i p(x)$. \nOtherwise we let the target index be $v = \\min\\{i+1,N\\}$ and solve the following LP to get control $u$:\n\\begin{align}\\label{lp1}\n &\\quad \\ \\min_{\\gamma,p,\\delta,u} \\ \\alpha^\\top \\gamma \\\\\n &\\text{subject to} \\ x_{v} + G_{v} p = h_{i,j}(x,u) + \\delta \\nonumber\\\\\n & \\quad \\quad \\quad \\quad \\ p \\in \\mathbb{P}, |\\delta_k| \\leq \\gamma_k, k=1,\\ldots,n \\nonumber\n\\end{align}\nwhere $\\alpha$ is some weight or cost vector.\nThis LP means we want the state to get to the polytope with index $v$ as close as possible.\nWhen, for example in the half-cylinder flipping experiment, the volumes of the polytopes are very small, we can directly solve the LP\n\\begin{align}\\label{lp2}\n &\\quad \\quad\\min_{\\gamma,\\delta,u} \\ \\alpha^\\top \\gamma \\\\\n &\\text{subject to} \\ x_{v} = h_{i,j}(x,u) + \\delta \\nonumber\\\\\n & \\quad \\quad \\quad \\quad \\ |\\delta_k| \\leq \\gamma_k, k=1,\\ldots,n \\nonumber\n\\end{align}\nwhich means we want the state to get to the nominal state with index $v$ as close as possible.\n\n\\begin{algorithm}\n\\caption{Stabilizing controller around nominal trajectory}\\label{alg:stabilizing_controller}\n\\hspace*{\\algorithmicindent} \\textbf{Input} Current state $x \\not\\in X_G$ \\\\\n\\hspace*{\\algorithmicindent} \\textbf{Output} Control $u$\n\\begin{algorithmic}[1] \n \\If {$x \\in \\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}, \\forall i \\in I$, for some index set $I$,} \\Return $u = \\bar{u}_{i_0} + \\theta_{i_0} p(x)$, where $x = \\bar{x}_{i_0} + G_{i_0} p(x)$, and ${i_0}$ is the largest element in $I$.\n \\EndIf\n \\State Find the closest nominal state $\\bar{x}_i$ to $x$, w.r.t. some weighted $L_2$ norm.\n \\State Determine the current contact mode $j$.\n \\State Let the target index be $v = \\min\\{i+1,N\\}$.\n \\State Solve LP (\\ref{lp1}) or (\\ref{lp2}), \\Return $u$.\n\\end{algorithmic}\n\\end{algorithm}\n\nThe procedure is summarized in Algorithm \\ref{alg:stabilizing_controller}.\nThere can be many variants to Steps 4 and 5.\nFor example, one might use MPC-style planning based on local PWA linearization.\nDuring offline phase, one samples states $\\tilde{x}_k$ not in the nominal mode and solve MICP to get to some target points $\\bar{x}_{v(k)}$ on the nominal trajectory, hence storing a list of samples $\\{($state $\\tilde{x}_k$, mode sequence to get to target $\\bar{x}_{v(k)})\\}_{k=1}^M$.\nThe target index $v(k)$ for each sample state $\\tilde{x}_k$ can be chosen by comparing the cost to get to all nominal states $\\bar{x}_i, i=0,\\ldots,N$.\nDuring online execution, one finds the closest sample $\\tilde{x}_k$ to the current state $x$ and solve QP or LP to get to $\\bar{x}_{v(k)}$ fixing the mode sequence as stored.\n\nWe find empirically that for the half-cylinder flipping experiment, solving LP like (\\ref{lp1}) or (\\ref{lp2}) to directly go to the nominal trajectory is more efficient than MPC-style planning which plans multiple steps to reach the nominal trajectory.\nThis might be because of our assumptions that the change of contact modes is only caused by the manipulator making and breaking contacts with the object and that the manipulator is fully actuated.\nSo instructing the manipulator to directly go back to the desired position works.\nAlso MPC-style planning on linearized PWA systems may accumulate linearization errors.\n\nDuring the online execution, we use $\\mathbb{P} = [-1,1]^n$ instead of $\\mathbb{P}_i$ in Proposition \\ref{prop1}, because it is more computationally efficient to use $[-1,1]^n$ and it is hard to compute $\\mathbb{P}_i$.\nSince $\\mathbb{P}_i \\subseteq \\mathbb{P}$, we know once $x$ happens to fall into $\\{\\bar{x}_i\\} \\oplus G_i \\mathbb{P}_i$, then the system is guaranteed to reach the target region.\n\n\\iffalse \nIn our hardware experiment, in order to simplify the linearized piecewise affine system, we assume there is no sliding.\nWe only distinguish between two objects in contact or not in contact. \nWhen in contact, the two objects are sticking and not sliding.\nThis assumption reduces the dimension of the states, and simplifies the linearized PWA system, making the system easier to control.\nIn practice this simplification works better.\n\\fi \n\\section{Related Work}\n\\textit{Model-based feedback control for manipulation}:\nLynch's group used feedback control for sliding \\cite{shi2017dynamic}, rolling \\cite{ryu2013control}, vibratory manipulation \\cite{umbanhowar2012effect,vose2012manipulation}, and hybrid manipulation using motion primitives \\cite{woodruff2017planning,dafle2014extrinsic}.\nThey designed specific manipulators for specific tasks.\nIn contrast, our algorithm is more general and can be carried out on common robot platforms.\nRodriguez's group used feedback control for planar pushing \\cite{hogan2018reactive}.\nThey linearize the nonlinear system and solve linear model predictive control (MPC) online to plan the trajectory.\n\n\\textit{Path planning}:\nThere are generally two categories of approaches to path planning.\nOne is motion planning algorithms \\cite{lavalle2006planning}. \nFor discretized configuration space, grid search algorithms such as A$^*$ and its variants \\cite{likhachev2005anytime} are widely used.\nSampling-based algorithms such as rapidly exploring random trees \\cite{shkolnik2010sample} are common approaches for continuous configuration space. \nThe other category is the trajectory optimization algorithms.\nA common approach in this category would be to formulate the problem as nonlinear optimization programs and solve it using off-the-shelf numerical solvers \\cite{posa2014direct,dai2014whole,mordatch2012discovery}.\nOther approaches in this category include augmented Lagrangian \\cite{toussaint2014novel}, mixed-integer convex optimization (MICP) \\cite{valenzuela2016mixed}, differential dynamic programming (DDP) \\cite{tassa2014control}, and iterative linear quadratic Gaussian (iLQG) \\cite{tassa2012synthesis}. \nA combination of these two methods have proved even more useful \\cite{dolgov2010path,zhang2018autonomous}. \nIn this work, we use trajectory optimization and use off-the-shelf numerical solvers to solve a nonlinear optimization program.\n\n\\textit{Local feedback controllers}: In control literature, it is quite standard to track a system trajectory using linear quadratic servo (LQ servo) or time-varying linear quadratic regulator (TVLQR) based on linearization of the system trajectory around nominal states \\cite{anderson2007optimal}.\nIn reinforcement learning, it is common to learn local linear models of the system and linear feedback control gains \\cite{levine2015learning,kumar2016optimal}.\nHowever, the local linear model does not fully capture the contact-rich nature of manipulation. \nRobot fingers may make and break multiple contacts with the object due to small disturbances. \nWhen contact modes change, the dynamics may change.\nSo it is more natural to model the local dynamics as a PWA system, i.e., a system whose state-input space is partitioned into several polytopic regions, with each region\nassociated with a different affine dynamics equation.\n\nHowever, stabilizing PWA systems alone is not an easy problem. \nExplicit solutions of optimal control for PWA systems can be computed offline by multi-parametric programming \\cite{baoti2006constrained, christophersen2005optimal, borrelli2005dynamic, baric2008efficient, bemporad2000optimal,bemporad2000piecewise,bemporad2002optimal,mayne2002optimal}.\nThe computational complexity of these methods grows exponentially with respect to the number of time steps.\nLyapunov-based approaches \\cite{rodrigues2002dynamic,lazar2006model,han2017feedback} and occupation measure approaches \\cite{zhao2017optimal,han2019controller} do not depend on the number of time steps, but are quite conservative and may not always find solutions.\nSampling-based methods \\cite{marcucci2017approximate,sadraddini2019sampling} suffer from the issue of scalability.\nIn this work, we consider manipulation problems in which there is only one rigid object, the manipulators are fully actuated, and the PWA dynamics is caused by manipulators making and breaking contacts with the object.\nWe track the system trajectory by solving LP online.\n\\subsection{Trajectory Optimization}\nWe plan the path using trajectory optimization methods.\nIn particular, we use direct transcription as in \\cite{dai2014whole}.\nThe continuous-time dynamics $\\dot{x} = f(x,u)$ is discretized into a discrete-time system $x^+ = \\phi(x,u)$ with sampling time $dt$.\nThe trajectory is discretized into $N$ time steps with $N\\cdot dt = T$, where $T$ is the time horizon:\n\\begin{align*}\n \\text{minimize}\\ & \\sum_{t = 0}^T L(x[t],u[t])\\\\\n \\text{subject to} \\ & m \\ddot{{r}}[t] = m{g} + \\sum_j {F}_j[t] \\\\\n & I \\ddot{\\theta}[t] = \\sum_j ({c}_j[k]-{r}[k])\\times {F}_j[t] \\\\\n & \\text{friction cone constraints}, \\text{contact constraints}, \\\\\n & \\text{kinematics constraints}, \\text{time integration constraints}\n\\end{align*}\nwhere ${r}[t]$ is the position of the center of mass, ${F}_j[t]$ are forces, ${c}_j[t]$ is the the contact position of the $j$-th force for each $j$, $L$ is the loss function, and the decision variables include states $x[t]$, controls $u[t]$, and variables in the contact constraints.\nThe variables ${r}[t]$ and ${c}_j[t]$ are part of the states $x[t]$, and the forces ${F}_j[t]$ are part of the controls $u[t]$.\n\nThe first two constraints are Newton-Euler equations.\nThe friction cone constraints for planar systems are\n $-\\mu F_{j,n} \\leq F_{j,t} \\leq \\mu F_{j,n}$,\nwhere $F_{j,n}$ is the normal force and $F_{j,t}$ is the frictional force.\nFor 3-dimensional systems, we can use a polyhedral cone to approximate the friction cone \\cite{stewart2000implicit}:\n ${F}_j[t] = \\sum_{i} \\beta_{ij} {w}_{ij}, \\beta_{ij} \\geq 0$,\nwhere ${w}_{ij}$'s are the spanning vectors of the polyhedral cone.\nFor some contacts, we formulate the contact constraints as linear complementarity problems (LCP) \\cite{posa2014direct} to fully characterize all possible contact modes -- sticking, sliding, or breaking contacts.\nWe use IPOPT \\cite{wachter2006implementation} to solve the trajectory optimization offline.\nSince IPOPT is an interior point method solver, the LCP constraints $P(x)^\\top Q(x) = 0, P(x) \\geq 0,Q(x) \\geq 0$, can be replaced by the equivalent constraints $P(x)^\\top Q(x) \\leq 0, P(x) \\geq 0, Q(x) \\geq 0$, and further be relaxed as $P(x)^\\top Q(x) \\leq \\epsilon, P(x) \\geq 0, Q(x) \\geq 0$ for small $\\epsilon > 0$.\n\n\n\\subsection{Force as Control Input}\nIn the trajectory planning described in the previous subsection, we borrowed the idea of zero-moment point (ZMP) for bipedal footstep planning in the humanoid robot literature \\cite{kajita2014introduction, kajita2003biped}. \nFor a bipedal robot walking on the ground, ZMP is by definition a point on the ground where the sum of all the tangential moments equals zero.\nAlthough many humanoid robots have pressure sensors on the feet, because of lack of reliability (in the case of Atlas), researchers do not measure ZMP directly. What they do is to plan a joint ZMP and center of mass (CoM) trajectory, and then only track the CoM trajectory during online execution \\cite{kuindersma2016optimization}.\n\nSimilarly, in the trajectory optimization formulation, we use forces as part of the control input. \nIn the manipulation context, the forces are those between the gripper and the object, and those between the object and the environment, e.g. the table.\nAlthough the hardware we are using cannot directly measure the force it applies to the object, we still use forces as control variables to help plan the CoM trajectory of the object as well as the pose trajectory of the gripper.\nDuring online execution, we only track the trajectory of the CoM of the object and that of the gripper.\nWe find in practice this approach works well.\n\n\\iffalse \n\\subsection{Path Planning on Simplified States}\nWhen the state space is very complicated, for example due to the existence of obstacles or due to the dramatic change in dynamics, directly formulating and solving the trajectory optimization can be hard.\nHere we describe a practical extension to the single trajectory planning.\n\nThe idea is to concatenate several trajectories and finally reach the goal state.\nThe user can define some waypoints in the state space, and compute trajectories between each waypoints.\nHowever, specifying a point in the entire state space might be tricky, so we introduce the \\textit{simplified states} or \\textit{equilibrium states}, where the velocities and the accelerations are all 0, and everything is static.\nMore formally, if the state is $[\\mathbf{q}_1, \\dot{\\mathbf{q}}_1,\\mathbf{q}_2]$, where $\\mathbf{q}_1, \\mathbf{q}_2$ are vectors, then the simplified state representation is a vector consisting of some entries in the vector $[\\mathbf{q}_1, \\mathbf{q}_2]$. \nFor example, if the carrot state is $[x,y,\\theta,\\dot{x},\\dot{y},\\dot{\\theta}]$, then the simplified state can be $[x,y,\\theta]$ or even just $\\theta$, while $\\dot{x},\\dot{y},\\dot{\\theta}$ are assumed to be 0.\n\nIt is much more computationally efficient and more user-friendly to specify waypoints or do RRT-style search on simplified states.\nFor example, if we want to flip the carrot $90$ degrees, we can plan to first rotate $45$ degrees, and then another $45$ degrees without caring about the pose of the gripper as long as it is feasible.\nIf we want to push a block through a maze, we specify static waypoints as 2D coordinates in the plane, without worrying about whether the pusher is sticking or sliding on the surface of the block at those equilibrium states.\nThis idea of planning in simplified states is also exploited in autonomous car path planning \\cite{zhang2018autonomous,dolgov2010path}. \n\\fi \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nMisinformation is the act of accidentally spreading false or inaccurate information \\cite{online2019}. Some cases of Misinformation can be false rumors and pranks. Contrary to this, Disinformation describes the dissemination of malicious content, like, but not limited to, hoaxes, spear-phishing, and propaganda \\cite{wooley}.\n\nThe problem of the quality of information and how it reaches the European citizens has become enormous nowadays. It is important to note that a false expressed opinion by anybody, even without an intention to manipulate facts, could potentially fuel disinformation. An even more significant fact, is that the structure of the internet allows for a snowballing effect, potentially reaching a huge audience \\cite{renda}.\n\nThe phenomenon of Misinformation is emerging across Europe and has many manifestations. An insight into whether and to what extent European citizens trust their national information network comes from the fact that 23 out of the 28 EU states (82\\%) have at least a medium level of trust in the information provided by their national media \\cite{ebu}. Another statistic that should worry us regarding the confidence of European citizens in the media is that Social networks are the least trusted media in 29 out of 33 of the countries surveyed (88\\%). Citizens of the EU are more likely to trust radio and television over the internet and social media \\cite{ebu}. \n\nNot enough in-depth research has been conducted on networking patterns, which are a very valuable tool \\cite{koulas2}. On a more specific level, few studies have been carried out about Misinformation and fake news spreading. Still, they primarily measured the level of the phenomenon and the corresponding impact of it on the political and social environment. In the domain our research was held, there aren't any relevant studies dealing with the correlation among the websites containing misinformation content. \n\nNonetheless, few studies have focused on the analysis of Misinformation through Europe. A limited number of systematic reviews have examined the key actors that cause and encourage the spread of Misinformation. This study provides insights about the role of the European Governmental Organizations (GOs), Non-Governmental Organizations (NGOs), International Organizations (IOs), and International Institutes (IIs) in the spreading of that phenomenon. The method we used in order to construct our website collection sheet was by searching some key phrases about Misinformation on the search engines Google and Yahoo. On Bing, we faced some difficulties due to the lack of websites when we searched about specific key phrases. We collected and analyzed 49 seed websites with the criterion of the high ranking in the search engine. \n\nOur study provides a detailed webometric network analysis, based on the seed websites enlisted as Misinformation in Europe. It fills a gap in the literature for reviews of the correlation between sites. This study aims to empirically and methodologically assist in combating Misinformation both in Europe and the Global level. \n\nThe remainder of the paper is structured as follows. In Section 2, we perform an extensive literature review, comprising of the misinformation and disinformation effects in Europe, related webometric studies and action taken by European stakeholder on the issue. In Section 3, we formulate our research questions and analyze the procedure we follow regarding the selection of the seed sites and their analysis. Results of the webometrics analysis follow at Section 4. Finally, in Section 5 we discuss our findings and we provide our concluding remarks in Section 6.\n\n\\section{Literature Review}\n\n\\subsection{Misinformation and Disinformation effects in Europe}\nMisinformation and Disinformation are both phenomena that physically exist in everyday life almost since always. It is their digital nature that is novel nowadays and occurred a few years ago. \n\nRecently, the misinformation effect has been the source of significant concern in the Member States and has provoked discussions and investigations on this issue. The European Parliament, in June 2017, required, the European Commission analyzed thoroughly misinformation. Furthermore, it enquired from the Commission to formulating strategies for the effective mitigation of the problem. The Parliament even considered the possibility of legislative intervention to accomplish that mitigation, using as justification that fake news and Disinformation consist a considerable threat to the freedom of opinion, expression and democracy, which are of paramount importance under the European Union's Charter of Fundamental Rights. In a 2018 research conducted by the Robert Schuman Foundation, 68\\% of European citizens claim they encounter fake news at least once a week, while at the same time, 37\\% claim they encounter fake news daily \\cite{schuman}.\n\nThe European Commission, in order to address the issue, on the 26th of April 2018, proceeded with the publication of a communication titled \"Communication on tackling Disinformation: A European approach\" \\cite{commission}. Within this communication self-regulatory tools, which constitute the first step into countering effectively online disinformation in Europe, are contained.\n\nPresident Juncker, in his speech on the State of Union, presented in detail the actions that will take place based on this Communication and noted that he would do everything within his power to protect the civil rights and democracy. The purpose of this communication is to ensure that all citizens can have access to objective, quality information.\n\nThe Communications work plan was a series of actions that lead to the creation of a robust mechanism that prevents fake news and wrong information to spread. A team of fact-checkers was created, who reviewed information coming from reputable public sources and evaluated them. Another task was to promote non-toxic and quality journalism and punish any news media channel that doesn't provide valid information to the public. \n\nAt the same time, actions to educate people about choosing the right online and media information sources were taken in order to raise awareness for the issues of Misinformation and Disinformation and how these can affect the people. \n\nThe most significant success of this Communication was the acceptance and adaptation of the Code of good practice to fight online disinformation \\cite{commission1}, on the 26th of September 2018, which represents all the points and goals of the Communication. This Code successfully classifies all the prerequisites for a trustworthy only campaign, while at the same time enhancing fundamental principles, like the freedom of expression and media pluralism.\n\nMisinformation can be tackled, and its effects can be mitigated through ICT and monitored via the use of the internet. Internet is a useful tool to accomplish that feat, and ICT is the medium that gives the capability to collect, accumulate, interpret, and show data in order to make crucial decisions and formulate strategies. And as the initiator of Europe's democratic system Robert Schuman has said: \"Technology is changing, but our fundamental values remain. A citizen that is equipped with the necessary skills and that can listen, watch, and read critically is a prior condition for the success of these values\".\n\n\\subsection{Webometrics related studies}\nThe application of bibliometric and infometric approaches to study the web, its information resources, structures, and technologies, is known as webometrics. The term webometrics is a coinage from the English word \"web\" and the andicet Greek \"metric\", which means to measure. Since the term was coined in 1997 by Tomas Almind and Peter Ingwersen, the value of webometrics quickly became established through the Web Impact Factor, the critical metric for measuring and analyzing website hyperlinks \\cite{thelwall12}. If we would like to specify that methodology, an excellent definition to show would be the one by Mike Thelwall in 2004 \"\u2026 the study of web-based content with primarily quantitative methods for social science research goals using techniques that are not specific to one field of study\". The purpose of this alternative definition was not to replace the description within Information Science \\cite{thelwall09}. The actual use of the first definition is to support the publishing of appropriate methods out of the scope of Information Science \\cite{kunosic}.\n\nSince the emergence of webometrics, this tool has become a useful methodology that applies in many fields such as the ranking of universities and scientometric evaluations or investigations of research areas. In order for the effective analysis of data for webometrics usage, it is of paramount importance to use known credible sources, for every category of webometrics.\n\nThe study from Roghaye Tafaroji, Iman Tahamtan, Masoud Roudbari, and Shahram Sedghi, which was conducted in 2012, aimed to present the findings of a webometric analysis of web sites of medical universities of Iran. They tried to investigate the Webometric ranking of Iranian Universities of Medical Sciences. In comparison to other similar studies that were conducted before and used inlinks and size as the main webometric criteria. The authors of this study examined rich text format files (doc, pdf, etc.) as a webometric indicator, and measured the impact of this new indicator on webometric ranks. The main findings indicate that Iranian Medical Universities performed poorly in regard to number of webpages, external links, and rich files. This observation is very useful because it can explain the anemic presence of these universities on the web \\cite{tafaroji}.\n\nAnother study presented a ranking of Alternative Search Engines (ASEs). Using indicators to evaluate a large amount of data that can be retrieved effortlessly and effectively, Bernd Markscheffel and Bastian Eine managed to create a picture of the most popular ASEs currently available. This approach allows further investigation for other studies, exploring the dynamics of the search engine market, while at the same time assessing the categories of ASEs \\cite{markscheffel}.\n\nA recent study conducted by The University of Burdwan measured and gave a clear idea about the information provided by the websites of the 25 High courts using this time just Google Search engine in contrast with the previous studies. This paper highlighted the various web impact factors, scores, and ranking of the websites of high courts in India and the final results that gave did open the door to further studies of other new areas of the webometric analysis \\cite{mahji}.\n\nAnother study also examined information originating from the website of 71 universities in Bangladesh. The results given indicate that the universities of Bangladesh do not have greater web visibility and, it is clear that these universities need to focus on several issues to increase the visibility of their websites \\cite{islam}. \n\nFurthermore, Webometrics Analysis was also used to measure Web Impact Factor (WIF) of 8 National Libraries' websites in South Asian Countries. WIF provides tools for quantitative research for several categories, like ranking, evaluation, categorization and comparison of websites, both on top-level and sub-level domains. The results visualized that the National Library of India leads with the highest Domain Authority and Page Authority, and it is followed by the National Library of Sri Lanka and National Library of Bhutan among the other National Libraries websites. Users visit the websites of the National Libraries for their information needs \\cite{verma}. \n\nLast but not least, one of the significant studies regarding webometrics is \"Open Data in Nepal: A Webometric Analysis\", which measures the impact of Open Data in the Nepalese cyber domain \\cite{acharya}. Acharya and Park's study serves as a guide for this study.\n\nTaking in mind the related studies above it is clear that Webometrics is a tool used in many studies to examine the World Wide Web and give specific results about the construction and use of information resources. This is the reason why we decided to use Webometrics analysis in order to search the web and examine Misinformation in Europe.\n\n\\subsection{Anti-misinformation stakeholders inside and outside Europe}\nAs the volume of information flowing on the internet snowballs, the phenomenon of Misinformation is becoming more and more intense. For this reason, in recent years, many Governmental Organizations (GOs), Non-Governmental Organizations (NGOs) and International Organizations (IOs), inside and outside of the European border, have been mobilized to deal with Misinformation.\n\n\\subsubsection{European Level}\nThe anti-misinformation concern of the European Commission increased after some cases of intense manipulation of the public opinion on political issues. These phenomena occurred during the U.S and French presidential elections (2016-2017), as well as the Italian constitutional referendum (2016) \\cite{service}. After a two-day conference in Brussels, the European Commission concluded that expert's help is vital in order to combat Misinformation \\cite{commission17}. \n\nFinally, a High-Level Expert Group (HELG) on Artificial Intelligence was formed in January 2018 to reduce Misinformation, fake news, and Disinformation at any level within Europe. A report containing the best strategies and solutions about every disinformation issue depended on the same set of fundamental principles, which was the main deliverable of the HELG \\cite{connect}.\n\n\\subsubsection{Member states of European Union}\n\nBesides the European Commission, there are also many mechanisms within the states of Europe, that try to combat Misinformation.\n\nIn Germany and Croatia, strict laws about Misinformation and hate speech were applied. Websites that would not comply with the law within a specific period after the warning would pay a considerable fine \\cite{bbc1, vintof}. \n\nIn Italy, a portal where people could report to the authorities, fake news occurrences, was set, but unfortunately, it didn't work rationally because of the lack of knowledge about fake news \\cite{commissario}. However, when a man was sent to prison for using a false identity in TripAdvisor reviews, the government's intentions were very clear. After that, the country's communications authority released a report on Misinformation \\cite{agcom}.\n\nSweden, Denmark, Belgium, and the Netherlands launched campaigns on websites and social media in order to inform the citizens about Misinformation and fake news, on the initiative of their governments (2018-2019) \\cite{funke}. \n\nIn Spain, when Russia was blamed for spreading Misinformation concerning the Catalan referendum by national authorities, an agreement was signed between the two of them, to create together a cybersecurity team to prevent Misinformation \\cite{funke}. Moreover, a task force of about 100 officials was activated during the 2019 elections, with the aim to combat fake political posts, especially on social media.\n\nIn France, a very innovative law was set on the press, which gives the power to the authorities to do whatever they must do with sites that illegally use fake news and Misinformation, enabling them to shut them down. However, this measure was criticized especially from opposition senators and journalists because according to what they say, it is against the principle of proportional justice and press freedom \\cite{ricci}.\n\nIn Greece, there are many NGOs nowadays, dealing with the refugees coming from regions where the situation is turbulent. Because of the significant number of migrants, this situation has become very sensitive. As a result, Misinformation and fake news are spread both from people and the media very fast. In order to help the migrant, many mechanisms collaborated to protect refugees' rights and fight Misinformation about this topic. They also take care of their housing and education. These mechanisms are the \"United Nations High Commissioner for Refugees (UNHCR)\", together with the \"Emergency Support to Integration and Accommodation (ESTIA)\" program, funded by the European Union Civil Protection and Humanitarian Aid, and some other local NGOs.\n\nIn the United Kingdom, a parliamentary report was published with the purpose to enforce citizens to avoid fake news and misinformation spreading, because the country suffers from a democracy crisis since the idea of Brexit came to the surface \\cite{waterson}. Furthermore, the National Security Communications Unit was launched with the task to fight Disinformation from authority people and others, after Russia got involved in Brexit by spreading fake news \\cite{bbc2}.\n\n\\subsubsection{NGOs}\nAt the same time, even though the mobilization of governmental organizations is essential, the contribution of non-governmental organizations to combating misinformation is just as remarkable. An international NGO, \"Reporters Without Borders (RSF)\", launched an innovative media to deal with Disinformation online \\cite{boulay}. It is called the \"Journalism Trust Initiative (JTI)\" and is designed to encourage journalism by applying some agreed transparency standards to protect information and combat misinformation.\n\nMoreover, charity NGOs, who are dealing either with refugees or with citizens that need help, are those who try besides their actions to protect people's rights and to publish only the real thing about the issues in which they are involved. \"ActionAid\" is one fascinating example of international NGO which has already offered very much in this sector.\n\n\\subsubsection{Anti-misinformation stakeholders outside Europe}\n\nIn addition to the efforts made to combat Misinformation within the European Union, efforts are also being made from countries outside of Europe. In some cases, Europe is firmly connected and immediately affected by the efforts made to fight Misinformation outside of Europe.\n\nRussia has been one very important player in spreading fake news in recent years across Europe and the whole world, at both a political and social level. Especially in the US presidential elections, many campaigns where bots were used, were set in order to manipulate the result. For this reason, media continuously publishing fake news and misinformation are punished. For a start with a fine, if they would not conform with the law the people accountable for this would go to prison, and if that was not enough, their website would shut down \\cite{stanglin}.\n\nIn the Americas, fines are the most widely used method of combating fake news and misinformation. In the USA, Brazil, and Chile, when someone is found to be disseminating fake news, the responsible party will be fined, and may even face prison time. This applies to everyone whether they are citizens writing on the net, whether they are journalists, webpages or even politicians who spread fake news \\cite{funke}.\n\nIn Asia, many countries have adopted strict laws to deal with misinformation. In China and Malaysia spreading fake news is considered as a crime. Those who rumor fake news that can be harmful in public order are punished by the law with up to seven years in prison or public surveillance \\cite{zhang, leong}. Moreover, south Asia countries, Thailand and Indonesia have also enforced laws in cases of misinformation detection. Many people were arrested for fabricating fake news, especially on social media, and many others paid huge fines \\cite{funke}.\n\n\n\\section{Methods}\n\\subsection{Goals and Research Questions}\nThe rise of the internet and computational power has allowed for the disproportional growth of misinformation phenomena in the last years. In this study we want to discuss the measures taken by stakeholders in Europe to tackle those incidents and assess their effectiveness. For this, we formulated two research questions: (RQ1) Which are the key stakeholders and how do they fight the phenomenon of misinformation? (RQ2) Do the actions of the key stakeholders have a palpable impact? The first question aims to mapping the key stakeholders, as well as, assess their actions and cooperation. The second question tries to showcase the impact, if any, those actions have.\n\n\\subsection{Data Collection}\nOur method for gathering information and seed sites regarding Misinformation was Google's search engine, as well as, Bing and Yahoo search engines. Trying to have a variety in our results and have the whole idea of Misinformation, we used different keyword combinations to get more accurate results. These are the search queries we used: \n\n\\begin{itemize}\n\\item Misinformation in Europe\n\\item Role of NGOs in Europe to tackle Misinformation\n\\item Governmental Organizations tackling Misinformation in the Europe Area\n\\item Expanding Misinformation in society\n\\item The Consequences of Misinformation\n\\item Fighting Misinformation\n\\item Misinformation in Belgium and\n\\item Misinformation Tackling\n\\end{itemize}\n\nThe search results included most of European and Global NGOs, IOs, GOs, and IIs. The homepages of these organizations were visited and read individually to assess the importance of researching the phenomenon of Misinformation in Europe. We opted to include Belgium as a separate search query, due to the fact that the European Union's present there has led to the creation of a variety of think tanks, NGOs and corporations.\n\nSince the exposure to a large scale of disinformation is proliferating, fighting misinformation is a top priority for the European Commission. Therefore we emphasize identifying the key factors that encourage the spread of this phenomenon throughout Europe. It is highly essential to understand the role of all the European Stakeholders and Institutions, such as the European Commission, Council of Europe, etc., that are responsible for measuring all the appropriate criteria that give us a more analytic point of view about the issue. \n\nFurthermore, it is significant to measure the impact and the consequences of Misinformation to find new and more efficient ways to counter misleading information in the European Area. Subsequently, we should emphasize on social-economic elements that will be able to lead us to determine a general framework for the protection of information throughout Europe, in the same way, the General Data Protection Regulation has been defined.\n\nWe choose to collect data from three Top Level Domain Categories and the specific eight Country Code Top Level Domains. In this context, to make it more understandable, we must observe the following hierarchy tree: \n\n\\begin{figure}[h]\n\\caption{Hierarchy tree}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{tree.png}\n\\end{figure}\n\nFrom the hierarchy tree in Figure 1, we notice the Top-Level Domain categories that there will be shown in the webometrics analysis and the following Country Code Top Level Domains. It's time to perceive the provenance of each domain. Firstly, .com derives from commercial, indicating its original intended purpose for domains registered by commercial organizations. Secondly, .org is truncated from organizations. Following the same way of thinking, we recognize that all Country Code Top Level Domains reserved for specific countries. In our case, we ended up with seedsites having the country codes of the European Union, Germany, Belgium, and Luxembourg. Finally, it's meaningful to recognize the role of Sponsored Top Level Domains. This category of the domain name is supported by a community or organization and considered to have the strictest application policies of all TLDs, as it implies that the holder is a subject of international law.\n\nTable 2 lists the organizations, their established date, the sector in which they belong, their website address, and their URL.\n\n\\subsection{Process and Assessment}\nThe above websites are analyzed using Webometric Analyst 2.0 (http:\/\/lexiurl. wlv.ac.uk) and the Bing Application Program Interface, which is capable of carrying out advanced Boolean searches and tracking external websites linked to URL citations under study. Thus, the lists of external sites corresponding to the base query, i.e., the websites mentioned above, were obtained. \n\nInterlinkage and co-mention, as explained in Table 1, will be used for the data analysis.\n\n\\begin{table}[]\n\\caption{Analytical techniques and concepts of Webometrics\\cite{acharya}}\n\\label{tab:tools}\n\\begin{tabular}{p{0.2\\textwidth}p{0.8\\textwidth}}\n\\hline\nInter-mention network analysis & Network diagrams illustrate the accompanying networks of the communication strength of the provided pairs of websites. It is the indicators based on asymmetric (directed) inter-mention counts between a pair of websites. A diagram illustrates the pattern of interconnectivity between collections of sites. This analysis gives a proxy for the hyperlinks between the websites under study \\\\\n\\\\\nCo-mention network analysis & Network diagrams and their indicators based on the number of external sites referring to a pair of target sites. The co-mentions show something important in common but are not directly related to each other. The competitors who are also considered as stakeholders show a different pattern in the webometric analysis. Co-mention does not have a direction \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nFor the assessment we are going to use the inlink degree, the outlink degree and the betweenness centrality. The inlink degree shows how many links from the network are inbound for each specific node, while the outdegree shows the outbound links towards other nodes of the network. Betweenness centrality shows the sum of times any particular node is found on the shortest path between different nodes of the network \\cite{disney, valente, friedkin}. The higher the metrics, the more influential a node, thus an organization, is in the network. \n\\section{Results}\n Figure 2 and Tables 2 and 3 answer our first research question. Figure 3 and Table 4 answer our second research question. Figure 2 depicts a network diagram that demonstrates the inter-mention between websites conclusively. The red nodes and arrows show the linkage between the websites, whereas the green nodes indicate that there is no connectivity with any of the seed sites.\n \nEvery website domain and URL was converted accordingly to meet the requirements for Bing classification so that Bing can make the connections between the URLs and seed websites. According to the results below, websites are firmly connected and are the core of our findings, www.theverge.com, www.nytimes.com, www.cnet.com, www.washingtonpost.com, www.reuters.com, and www.bloomberg.com. As you see, the core URLs are oriented to the private company sector, and they have .com TLD. Most of the government based websites are connected, having a substantial presence on the web, for example, www.coe.int and www.poynter.org. \n\n\\begin{figure}[h]\n\\caption{Inter-mention Network Diagram}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{mininfo-interlinks.png}\n\\end{figure}\n\nWe also observe that www.goethe.de (which is a Nonprofit organization), www.zeit.de, and www.welt.de, which are both POs, are strongly connected to www.dw.com which is the most centralized website from all NPO's. Another seed site that has many connections to the central nodes of the diagram and vice versa is www.technologyreview.com and www.newscientist.com; both of them have a .com TLD. Additionally, we see that from websites with TLD .be and .lu only today.rtl.lu is not connected with any website. On the contrary, www.lesoir.be and www.knack.be are connected between them and also to the most central websites.\n\nFurthermore, the interlinkage was investigated for the purpose of analyzing the online networking patterns in different networking scenarios. The values that we have observed are the networking density value for the directed network, which is 0,0791, and the value for the undirected network which is 0,1182, increased in comparison to the direct network. Density is calculated by diving the number of relations by the maximum number of possible relations. Density means an average value of entire cell blocks, when we refer to a network matrix that is weighted and valued. Next, 'degree' and 'betweenness' network centrality values are calculated. The term degree centrality refers to the amount of ties that are immediately connected to a node (i.e., website), rather than indirect ties to all others in the network\\cite{acharya}. The two-degree centrality, specifically indegree, and outdegree, is calculated by the direction of the connection between two nodes. On the other hand, betweenness centrality measures how important a node is in the network. This is calculated by the effectiveness that a specific node plays as a broker, while connecting a pair of nodes. In this instance, the number of the shortest paths via the node is considered. Our network metrics were calculated utilizing built-in functions within Webometric Analyst.\n\n\\begingroup\n\\footnotesize\n\\begin{longtable}{p{0.1\\textwidth}p{0.3\\textwidth}p{0.1\\textwidth}p{0.1\\textwidth}p{0.3\\textwidth}}\n\\caption{Description of the Selected Seed Sites}\n\\label{tab:seedsites}\\\\\n\\hline\nNo. & Organization Name & Est. date & Sector & URL \\\\ \\hline\n\\endfirsthead\n\\multicolumn{5}{c}%\n{{\\bfseries Table \\thetable\\ continued from previous page}} \\\\\n\\hline\nNo. & Organization Name & Est. date & Sector & URL \\\\ \\hline\n\\endhead\n\\hline\n\\endfoot\n\\endlastfoot\n1 & Council of Europe & 1999 & GOV & coe.int \\\\\n2 & European Commission & 1958 & GOV & ec.europa.eu \\\\\n3 & Public Data Lab & 2017 & IO & publicdatalab.org \\\\\n4 & Reporters Without Borders & 1985 & IO & rsf.org \\\\\n5 & European External Action Service's East StratCom Task Force & 2015 & NGO & euvsdisinfo.eu \\\\\n6 & Center for Strategic and International Studies (CSIS) & 1962 & Non-profit Organization & csis.org \\\\\n7 & Investigate Europe & 2014 & NGO & investigate-europe.eu \\\\\n8 & Journalismfund.eu & 2008 & NGO & journalismfund.eu \\\\\n9 & CNN Digital & 1980 & Private Company & edition.cnn.com \\\\\n10 & European Data Journalism Network (EDJNet) & 2017 & Private Organization & europeandatajournalism.eu \\\\\n11 & European Centre for Press and Media Freedom (ECPMF) & 2009 & NGO & ecpmf.eu \\\\\n12 & Poynter Institute & 1975 & GOV & poynter.org \\\\\n13 & Social Observatory for Disinformation and Social Media Analysis (SOMA) & 2018 & Project & disinfobservatory.org \\\\\n14 & Media Freedom Resource Centre & 2015 & NGO & rcmediafreedom.eu \\\\\n15 & WAN-IFRA - World Association of News Publishers & 1948 & ORG & wan-ifra.org \\\\\n16 & Parliament of Europe & 1952 & Int'l Institution & europarl.europa.eu \\\\\n17 & EU Open Data Portal & 2012 & Portal & data.europa.eu \\\\\n18 & Euractiv & 1999 & Network of Media & www.euractiv.com \\\\\n19 & Fandango Project & 2018 & Project & fandango-project.eu \\\\\n20 & The Guardian & 2011 & Private Company & www.theguardian.com \\\\\n21 & Sunlight Foundation & 2006 & Non-profit Organization & sunlightfoundation.com \\\\\n22 & New Scientist & 1956 & NGO & newscientist.com \\\\\n23 & Fipp & 1920 & Private Company & fipp.com \\\\\n24 & The New York Times & 1851 & Private Company & nytimes.com \\\\\n25 & Washington Post & 1877 & Private Company & washingtonpost.com \\\\\n26 & Euronews & 1993 & Portal & euronews.com \\\\\n27 & MIT Technology Review & 1899 & Private Company & technologyreview.com \\\\\n28 & Media Frenzy Global & 2006 & Private Company & mediafrenzyglobal.com \\\\\n29 & Singularity University & 2008 & Univer-sity & singularityhub.com \\\\\n30 & The Atlantic & 1857 & Private Organization & theatlantic.com \\\\\n31 & The Verge & 2011 & NGO & theverge.com \\\\\n32 & Civil Beat & 2010 & NGO & https:\/\/www.civilbeat.org\/ \\\\\n33 & The World Bank Group & 1944 & NGO & blogs.worldbank.org \\\\\n34 & The Social Science Research Council (SSRC) & 1923 & NGO & ssrc.org \\\\\n35 & Bloomberg & 1981 & Private Company & bloomberg.com\/europe \\\\\n36 & Star Tribune & 1897 & Private Organization & startribune.com \\\\\n37 & Science Direct & 1997 & Private Organization & sciencedirect.com \\\\\n38 & NetGov & 2008 & GOV & nextgov.com \\\\\n39 & Lesoir & 1928 & Private Organization & lesoir.be \\\\\n40 & Knack & 1971 & Private Organization & knack.be \\\\\n41 & Deutsche Welle & 1953 & Non-profit Organization & dw.com \\\\\n42 & Zeit & 1946 & Private Organization & zeit.de \\\\\n43 & Welt & 1946 & Private Organization & welt.de \\\\\n44 & Goethe Institute & 1951 & Non-profit & goethe.de \\\\\n45 & RTL & 1929 & Private Organization & today.rtl.lu \\\\\n46 & American Press Institute & 1946 & Institute & americanpressinstitute.org \\\\\n47 & Reuters & 1851 & Private Organization & reuters.com \\\\\n48 & Computer Network & 1994 & Private Organization & cnet.com \\\\\n49 & Pew Research Center & 2004 & Research Center & pewresearch.org \\\\ \\hline\n\\end{longtable}\n\\endgroup\n\nParticularly, the 14 websites of our seed sites with the highest indegree and outdegree centrality are presented in Table 3. The Reuters (www.reuters.com) has the highest indegree centrality (74), and a Private Company 'The Guardian' (www.theguardian.com) has the highest outdegree centrality (66). The balance between big organizations, like the European Parliament, and case specific seed sites like the EUvsDisinfo, ensure that the metrics are accurately depicting connectivity around misinformation.\n\n\\begin{table}[]\n\\caption{Top 14 websites with the highest indegree and outdegree centralities.}\n\\label{tab:centralities}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{lllllll}\n\\cline{1-7}\nOrganization & Sector & Indegree & & Organization & Sector & Outdegree \\\\ \\hline\nReuters & Private Company & 74 & & The Guardian & Private Company & 66 \\\\\nWashington Post & Private Company & 54 & & The New York Times & Private Company & 60 \\\\\nThe Atlantic & Private Organisation & 50 & & Washington Post & Private Company & 56 \\\\\nThe Guardian & Private Company & 44 & & The Atlantic & Private Organization & 28 \\\\\nBloomberg & Private Company & 36 & & Reuters news agency & Private Company & 24 \\\\\nThe New York Times & Private Company & 26 & & European Commission & GOV & 24 \\\\\nThe Verge & NGO & 24 & & Science Direct & Private Organisation & 24 \\\\\nPew Research Center & Research Center & 20 & & The Verge & NGO & 20 \\\\\nDeutsche Welle & Non-profit Organisation & 16 & & CNET (Computer Network) & Private Company & 20 \\\\\nCNET (Computer Network) & Private Company & 14 & & Deutsche Welle & Non-profit Organisation & 14 \\\\\nMIT Technology Review & Private Company & 14 & & Poynter & GOV & 14 \\\\\nNew Scientist & NGO & 14 & & CNN World News & Private Company & 14 \\\\\nThe Star Tribune & Online Media Company & 10 & & MIT Technology Review & Private Company & 12 \\\\\nEuronews & Portal & 10 & & The Star Tribune & Online media Company & 10 \\\\ \\hline\n\\end{tabular}%\n}\n\\end{table}\n\nConcerning the Private Organizations, like theguardian.com, theatlantic.com, washingtonpost.com, and reuters.com, they have high betweenness centrality (248, 185, 184, 180, 137). The local NGO www.theverge.com has the highest betweenness centrality among all NGOs (11,217). The Government Organization ec.europa.eu has a high betweenness centrality (113,867). The Nonprofit Organization Deutsche Welle (www.dw.com) have 42,35 betweenness centrality. The betweenness centrality of a website shows the amount of control that this website exerts over the interactions of other websites in the network. The Pew Research Center is a research center that has the minimum betweenness centrality (0,2). Thus, it is noticeable that the Private Organizations have the highest betweenness centrality from NGOs.\nIn addition to the above conclusions, we see many websites with a weak connection between them such as www.disinfobservatory.org which is a technology-based company, also www.publicdatalab.org the only IO which has no connection. We have 6 NGO's websites with a minor presence on the web and no connection at all with the rest of the websites. Finally, www.fipp.com and www.mediafrenzyglobal.com are two private companies with no connection between them.\n\nFigure 3 shows the co-mention links of the websites. All of the nodes are colored red because all websites have at least one co-mention with another website. Each line's width is proportionately calculated and drawn based on the number of co-linking websites.\n\nThe paramount importance of the issues related to Misinformation is highlighted by the vast co-mentions among the websites analyzed. It is observed that the European Commission spearheads the efforts for tackling Misinformation.\n\nThe European Commission's role in promoting effectively European initiatives for tackling Misinformation can be shown by the fact that ec.europa.eu and www.disinfobservatory.org are co-mentioned 15 times, and ec.europa.eu and euvsdisinfo.eu are co-mentioned 185 times.\n\n\\begin{figure}[h]\n\\caption{Co-mention Network Diagram}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{misinfo_comention.png}\n\\end{figure}\n\n\\begin{table}[]\n\\caption{Seed site calculation (N=49). Data values are defined as the total number of counts of TLDs citing seed sites.}\n\\label{tab:domains}\n\\begin{tabular}{llll}\n\\hline\nNo & Seed site TLDs\/GTLDs\/CCTLDs & Total Number of Seed Sites & Percentage (\\%) \\\\ \\hline\n1 & .com & 21 & 42.85\\% \\\\\n2 & .org & 11 & 22.44\\% \\\\\n3 & .eu & 10 & 20.40\\% \\\\\n4 & .de & 3 & 6.12\\% \\\\\n5 & .be & 2 & 4.08\\% \\\\\n6 & .int & 1 & 2.04\\% \\\\\n7 & .lu & 1 & 2.04\\% \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nAt the same time, the Commission is used as a reference alongside the world's most influential think tanks, as it is observed that ec.europa.eu and www.theatlantic.com are co-mentioned 305 times; ec.europa.eu and www.csis.org are co-mentioned 359 times; ec.europa.eu and pewresearch.org are co-mentioned 541 times.\n\nLastly, the Commission is used to set a paradigm with other major organizations since ec.europa.eu and www.wan-ifra.org are co-mentioned 180 times; ec.europa.eu and www.rsf.org are co-mentioned 479 times, and ec.europa.eu and www.coe.int are co-mentioned 721 times.\n\nFurthermore, it is observed that resource produced by European initiatives is often used alongside resources produced by leading think tanks. For example, euvsdisinfo.eu and www.csis.org are co-mentioned 49 times; euvsdisinfo.eu and www.pewresearch.org are co-mentioned 81 times, and euvsdisinfo.eu and www.theatlantic.com are co-mentioned 178 times. \n\nThere are, however, initiatives that fall behind when it comes to European Projects. For instance, www.disinfobservatory.org and www.theatlantic.com are co-mentioned once, while www.disinfobservatory.org and www.csis.org are co-mentioned once, and www.disinfobservatory.org and pewresearch.org are co-mentioned twice.\n\nLast but not least, European NGOs play an important role in the efforts to tackle misinformation since www.investigate-europe.eu and ec.europa.eu are co-mentioned 33 times; www.investigate-europe.eu and www.theatlantic.com are co-mentioned 661 times; www.investigate-europe.eu and www.csis.org are co-mentioned 841 times; www.investigate-europe.eu and www.pewresearch.org are co-mentioned 845 times; www.investigate-europe.eu and www.rsf.org are co-mentioned 849 times; and www.investigate-europe.eu and fandango-project.eu are co-mentioned 979 times.\n\nAll those findings show the massive effort, and the resources pooled from various stakeholders to tackle Misinformation.\n\n\\section{Discussion}\n\nMisinformation is a constantly evolving threat that requires rigorous checks and balances in order to address it. The results can be categorized as those that stem from the theoretical part of the paper and those that derive from the webometrics analysis.\n\nOn the theoretical part, it is an interesting fact that in many cases, Webometrics is used as an evaluation system of a wide range of universities in the world. This system is known as a \"ranking\" system, where ranking describes a process where the position of the elements in a group regarding its entirety is defined by the relevance between the elements. The ranking process appears in many areas besides academics. For example, there was also a study on the ranking of Alternative Search Engines (ASEs) \\cite{markscheffel}. \n\nWebometrics is a tool used in many studies to examine the World Wide Web for different reasons. In this project, we chose to use this tool to explore a totally different issue that also has an enormous impact on people and tries to interpret the data exported from the analysis. \n\nFrom the research on combating Misinformation at the state level, it can be concluded that the use of strict laws or regulations, to punish the people accountable for this phenomenon , is not an effective strategy. Inform people how to detect Misinformation, thus preventing them from reproducing it, is much more effective. Democratic societies ought to help their citizens learn how to acquire their information only from proven reputable sources, question what they read, examine its accuracy, avoid reading only the headlines of articles, and in case something is fake to avoid sharing it.\n\nLast but not least, we observe the involvement of International Organizations in the effort to tackle Misinformation and Disinformation whose original mandate was totally irrelevant. For instance, the Council of Europe is heavily investing in the cyberspace domain, to remain relevant in a shifting world \\cite{koulas}, while this change in perspective can foster meaningful collaborations \\cite{brass}.\n\nIn the webometrics area, this study aims to investigate on a webometric dimension the role of all public and private sectors in Europe, as well as on the international level, that have taken actions as stakeholders to tackle the misinformation effect in Europe. We explore the structure of these stakeholders' portals and websites, their source, the organization's vision, methods of gathering and crosschecking data and information, and actions on the matter of Misinformation in Europe. We searched the seed sites countrywide and on an international level, for URL- based hyperlinks, title mentions and external links that refer to the seed sites of the stakeholders. According to the results, European governmental websites and portals are cooperating with the concerned NGOs inside Europe, but not as much as IOs are among themselves. In the co-mention network, all European portals show a strong connection with international websites and the other way around. Also, news organizations among them and other organizations of the same work- nature appear to have a very strong co-mention relationship. Besides, diverse organizations are also well co-mentioned. This could be because of the severe and sensitive nature of the issue and the urgency to counter it. Overall, European IOs and Governmental portals seem to have the most interlinkage and co-mentions as legal bodies that officially take actions to counter the Misinformation in Europe on a national governmental level with legislation, new laws and various efforts to raise awareness.\n\nIn our study we have identified 49 different stakeholder that took action in the fight against misinformation in Europe, as it is shown in Table 2. We found out that these efforts had some success, in terms of networking patterns. The limitations we faced are similar to those Acharya and Park \\cite{acharya} face. Firstly, from a total of 71 possible websites, we manually selected the ones we deemed more important and more relevant to our study and used them as seed sites. Early tries to use all 71 of the original findings resulted in messy diagrams and due to the high number of irrelevant sites, the different metrics, especially the betweenness centrality, were boosted, without actually ensuring the misinformation focus we opted for. Secondly, many of our seed sites are not solely focused on Misinformation and Misinformation tackling alone but have made contributions towards that field. Thirdly, the seed site analysis was conducted with third party software and the search engine www.bing.com.\n\n\\section{Conclusions}\nMisinformation is one of the significant challenges that modern societies need to address effectively because it severely impacts multiple aspects of our lives. We observe that there have been attempts to tackle Misinformation, with mixed results. The European Union spearheads these efforts, in cooperation with other organizations, but there it is possible to further enhance and improve those efforts. The ease world wide web provides for the spreading of Misinformation is a significant factor that increases the complexity of the situation. Further study should focus on the limitations of this study, as well as the usage of new technologies, like artificial intelligence, in order to process more data and yield better results.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{introduction}\n\nWhen an individual has to pay for the consumption of goods or services then there is a natural incentive to moderate this consumption. But there are cases when the relationship between the consumption and the emerging cost is more subtle because both actions are made by a group of persons. For example, in a university dormitory or a shared household each member can use electricity or water freely, but all members share the related bills equally. This collective responsibility may induce less trustworthy individual consumption because members may easily consume more than they would if the personal consumption is recorded and the related bill is paid separately. As a result, they can easily become careless because the extra use is paid collectively. Notably, the reverse is also true: an individual saving of usage would result in just a modest decrease in an individual bill because others will also benefit from a responsible act. On the other side, if all participants behave trustworthily and consume just the necessary resource then they would pay the minimal cost. In parallel, they could still enjoy the benefit of a collective venture, like maintaining a less expensive joint infrastructure. This establishes a dilemma between individual and collective interests when players are motivated to defect and consume more than they would individually. A similar dilemma occurs at a larger scale when countries use the same natural resources, like the oceans or the atmosphere~\\cite{milinski_pnas08,hilbe_prsb10,pacheco_plrev14,sun_ww_is21}.\n\nThe conflict of individual and collective interests, often called as a social dilemma, is the central problem of several game-theoretical models, including prisoner's dilemma game \\cite{szabo1998evolutionary,takeshue_epl19,amaral_pre21,zhu_pc_epjb21}, snowdrift game \\cite{hauert2004spatial,chen_xj_epl10,graeser_njp11}, stag-hunt game \\cite{skyrms2004stag,starnini_jsm11,deng_ys_epjb22}, ultimatum game \\cite{page2000spatial,sinatra_jstat09,szolnoki_prl12,chen_w_pa19}, trust game \\cite{berg1995trust,chica_srep19,zheng_lp_pa21}, donation game \\cite{nowak2006evolutionary,li_xy_epjb20,yang_hx_pa20}, or recently proposed involution game \\cite{wang2021replicator,wang2022modeling,wang_cq_amc22}, including the $N$-person versions of these games \\cite{zheng2007cooperative,souza2009evolution,li_k_csf21,pacheco2009evolutionary,luo2021evolutionary,chen_w_srep17,chica_cnsns19}. The common feature of all related situations, which are captured by the mentioned mathematical games, is that cooperation would provide the highest collective benefit, but players can gain more individually if they decide to defect and exploit others \\cite{sigmund2010calculus,hilbe_n18,couto_njp22}. However, from the viewpoint of our present work, the most important version is the so-called public goods game (PGG) played by several players simultaneously \\cite{perc2013evolutionary,alvarez2021evolutionary,lv_amc22,kang_hw_pla21,shen_y_pla22,liu_lj_rspa22}. In this framework, group members decide independently whether they want to contribute to a common pool or not. The accumulated contributions are then enhanced by an $r>1$ productivity factor which expresses the synergy among collaborators: their collective efforts are more effective than a simple sum of their contributions. Finally, the enhanced sum is distributed among all group members equally regardless of whether they contributed to the common pool earlier. It is worth noting that the original public good dilemma can be framed from different aspects: we may establish or maintain a common pool where both goals generate the same individual dilemma \\cite{gaechter17}.\n\nIn the original PGG, the contribution, which involves direct negative income to a participant, is optional, while the benefit is universal for all group members. From this viewpoint, the situation we described earlier is the opposite: consumption, which means a positive personal income, is optional while the contribution to the resulting cost is mandatory. Put differently, in a PGG the negative part is optional and the positive part is universal. In contrast, in our present case the positive part is optional and the negative part is universal. This is why we can call this setup a {\\it reversed} public goods game (R-PGG) where cooperation means to consume the minimal from the common resource, while a defector chooses to consume the pool extravagantly and wastefully. \n\nIn this work, we explore the proposed R-PGG model and reveal its relation to the original PGG. We therefore study the equilibrium and evolutionary behavior of competing strategies and investigate how it is possible to reduce the undesired overuse of common resources. Evidently, this is the core question of evolutionary game theory which targets to understand the evolution of cooperation among self-interested competitors \\cite{nowak2004emergence,quan_j_pa21,su2018understanding,fu_mj_pa21,ohdaira_srep22}.\nBesides a well-mixed population, studied by replicator dynamics \\cite{schuster1983replicator,duong_dga20,wang_q_amc18,liang_rh_pre22}, we also consider structured populations, which could be a decisive factor in how evolution proceeds \\cite{nowak1992evolutionary,ohtsuki2006simple,perc2017statistical,allen2017evolutionary,flores_jtb21,quan_j_c19,wei_x_epjb21}. Notably, we not only check the consequences when players are arranged on a lattice topology, but more realistic topologies, including small-world and scale-free features are also considered.\n\nMoreover, we also assume more complex conditions, including the heterogeneity of key parameters, and explore their impacts on the evolutionary outcome. Indeed, heterogeneity in various forms is not simply a more realistic approach, but could also be a decisive factor in how cooperation evolves \\cite{wang_cq_amc22,perc_pre08,santos_n08}. For the public goods game, the heterogeneity of allocation \\cite{lei2010heterogeneity,szolnoki_amc20,lee_hw_pa21}, productivity \\cite{liu_jz_epjb21,rong_zh_c19,deng_ys_pa21}, input \\cite{yuan2014role,huang2015effect,weng2021heterogeneous,ma2021effect}, and their combinations have been studied intensively \\cite{zhang2017impact,fan2017promotion,liu2021effects}. In particular, Hauser {\\it et al.} \\cite{hauser2019social} systematically analyzed public goods games among unequals by equilibrium calculation, evolutionary analysis, and human experiments, where heterogeneous game parameters depict unequal individuals. While heterogeneity can sometimes be a cooperator-supporting condition, it could also lead to inequality \\cite{mcavoy2020social}.\n\nThe remaining of our paper is organized as follows. In the next section we define the R-PGG model and discuss its extension to heterogeneous situations. Section~\\ref{equivalence} compares the original and reverse homogeneous models for well-mixed and spatial structured populations where their equivalence is demonstrated. In Sec.~\\ref{difference} we present the heterogeneous case where the surprising difference between the models is revealed. We here also present the mechanisms which explain the equivalence and difference between their behaviors. In the last section we discuss our findings and provide some potential applications to alternative real-life situations. Our arguments are greatly supported by an intensive Appendix where all complementary observations are presented.\n\n\\section{Model}\n\\label{model}\n\nWe here define the R-PGG model in analogy with the original PGG. First, we consider a specific case, where players are homogeneous (Sec.~\\ref{homomodel}), then we extend our model to a more general case, where the players are not necessarily equal (Sec.~\\ref{hetemodel}).\n\n\\subsection{The underlying model}\n\\label{homomodel}\n\nSimilarly to PGG, in an R-PGG a group is formed by $g$ members. A player is free to require goods, but all involved members should cover the emerging cost of the goods equally. For simplicity, we assume binary strategies: (i) not requiring goods (cooperation, $C$); (ii) requiring goods (defection, $D$). Later we explain why these strategies can be considered as cooperation or defection. If a player defects and requires goods, she receives goods valued $m$ ($m>0$). Otherwise, she receives nothing. When all group members made their independent decisions about their consumptions, each player equally pays a cost to cover the goods required by the whole group. The payoff $\\pi_C$ and $\\pi_D$ for a cooperative or for a defective player can be calculated as \n\\begin{subequations}\\label{payoffCD}\n\t\\begin{align}\n\t\t\\pi_C&=-\\frac{g_D m\\lambda}{g} \\label{payoffC}\\\\\n\t\t\\pi_D&=m-\\frac{(g_D+1) m\\lambda}{g} \\label{payoffD}\\,,\n\t\\end{align}\n\\end{subequations}\nwhere there are $g_D$ other defectors among the neighbors of the focal player. Here $\\lambda > 1$ parameter denotes the waste factor associated with requiring public goods. The larger the waste factor, the greater the loss caused by the consumption. If $\\lambda0$), with its waste factor $\\lambda _i$. In the extended version, we introduce a continuous strategy space, hence the strategy of player $i$ can be denoted by a proportion $y_i$ ($0\\leq y_i\\leq 1$) of goods $m_i$, where $y_i=0$ means full cooperation and $y_i=1$ is full defection. If not mentioned otherwise, we use the pure strategy set, $y_i \\in \\{ 0,1\\}$, in this work. The payoff $\\pi_i$ of player $i$ can be calculated as\n\\begin{equation} \\label{unequalpayoff}\n\t\\pi_i=y_i m_i-\\frac{1}{g}\\sum_{j=1}^g y_j m_j\\lambda_j\\,,\n\\end{equation}\nwhere $j$ goes through every player in the group, including player $i$. \n\nIn this extension, $m_i$ measures the full consumption capacity of player $i$. In real-world scenarios, an individual's needs and capacity may vary. Secondly, $\\lambda_i$ characterizes the ``effectiveness\" of player $i$ in spoiling the required public goods. Note that players with the same consumption capacity and strategy may generate different final cost values due to individual efficiency.\n\n\\section{Equivalence between R-PGG \\& PGG}\n\\label{equivalence}\n\nWe first demonstrate that the proposed R-PGG model is equivalent to the PGG in many aspects. The static analysis shows the equivalence in general homogeneous cases. From the perspective of evolutionary dynamics, the equivalence of the underlying homogeneous model also holds both in well-mixed and structured populations.\n\n\\subsection{Static analysis among unequals}\n\\label{unequals}\n\nHauser {\\it et al.}~\\cite{hauser2019social} proposed a general framework for the PGG among unequal participants (i.e., players with heterogeneous game parameters). We here show that our R-PGG model fits this general framework. In the following, we perform an equilibrium analysis.\n\nAccording to the general heterogeneous case defined by Eq.~(\\ref{unequalpayoff}), we denote the requirement vector by $\\mathbf{y}=(y_1, y_2, ..., y_g)$ in a group of $g$ players, and assume that $\\mathbf{y}>\\mathbf{y}'$ if $y_i>y'_i$ for all $i \\in \\{1,\\dots g\\}$ group members. We also denote the full consumption or desire profile by vector $\\mathbf{m}=(m_1, m_2,\\dots, m_g)$ and the waste vector $\\boldsymbol{\\lambda}=(\\lambda_1, \\lambda_2,\\dots, \\lambda_g)$ for the actual group. These $\\mathbf{m}$, $\\boldsymbol{\\lambda}$, and $\\mathbf{y}$ vectors determine the payoff values $\\pi(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\in \\mathbb R^g$ \nfor all group members. \nIn addition, we denote the accumulated group desire or maximal group consumption by $M\\coloneqq \\sum_{i=1}^g m_i$, and the total payoff of all involved players by $U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\coloneqq \\sum_{i=1}^g \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})$.\n\nIn Ref.~\\cite{hauser2019social}, the authors introduced a so-called ``cooperation vector\", denoted by $\\mathbf{x}=(x_1, x_2,\\dots, x_g)$, which characterizes the cooperation profile of all involved participants. In our R-PGG model, consumption is the focus, therefore it is appropriate to use a ``defection vector\" which practically determines the individual aims to consume their maximal capacities via $\\mathbf{y}=(y_1, y_2, \\dots, y_g)$ requirement vector. Since cooperation and defection complement each other, the sum of these vectors is always constant and gives a unit value for every $i$ component. Namely, $\\mathbf{y}+\\mathbf{x}=\\mathbf{1}$.\nIn strong analogy with the heterogeneous PGG model~\\cite{hauser2019social}, the following properties are valid here, which describe the conflict between the collective and individual incentives on how to utilize public resources:\n\\begin{itemize}\n\t\\item\n\t\\textbf{Continuity:} \n\tThe payoff function $\\pi(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})$ is continuous in both the arguments $\\mathbf{m}$ and $\\mathbf{y}$.\n\t\\item \n\t\\textbf{Negative externalities:}\n\tIf $\\mathbf{y}$ and $\\mathbf{y}'$ are two requirement vectors such that $y_i=y'_i$ and $y_j\\geq y'_j$ for $\\forall j\\neq i$, then $\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\leq \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ for $\\forall \\mathbf{m}$. The inequality $\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})< \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ holds iff $\\exists j$ such that $m_j>0, y_j>y'_j$.\n\t\\item \n\t\\textbf{Incentives to free-require:} \n\tIf $\\mathbf{y}$ and $\\mathbf{y}'$ are two requirement vectors such that $y_i>y'_i$ and $y_j=y'_j$ for $\\forall j\\neq i$, then $\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\geq \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ for $\\forall \\mathbf{m}$. The inequality $\\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})> \\pi_i(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ holds iff the required goods are positive, $m_i>0$.\n\t\\item \n\t\\textbf{Non-optimality of defection:}\n\tIf $\\mathbf{y}$ and $\\mathbf{y}'$ are two requiring vectors such that $\\mathbf{y}\\leq \\mathbf{y}'$, then $U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})\\geq U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$. The inequality $U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y})> U(\\mathbf{m},\\boldsymbol{\\lambda},\\mathbf{y'})$ holds iff $\\exists i$ such that $m_i>0, y_{i}0$) is considered as the strength of selection. A stronger selection strength $\\omega$ means that players are more sensitive to the payoff difference. The higher the payoff of player ${i'}$, the higher probability is that player $i$ imitates the strategy of player ${i'}$.\n\nWithout loss of generality, we first study the evolutionary dynamics of R-PGG in a well-mixed population of homogeneous players. \nIn Appendix~A1 we first apply deterministic dynamics and find two solutions to the replicator equation. They are full defector and full cooperator states. Their stability depends on the sign of the payoff difference between the competing strategies. To compare the results with the PGG case, we obtain equivalent expressions if we replace the original $r$ and $c$ parameters with $\\lambda$ and $m$. Hence the homogeneous R-PGG is equivalent to homogeneous PGG in the framework of replicator dynamics in an infinite and unstructured population.\n\nWe can also consider stochastic dynamics in a finite population of size $N$, as specified in Appendix~A2. By calculating the expected payoff values for both strategies, the transition matrix of the Markov process can be determined. Similarly to the above-discussed approach, the process has two absorbing states. In the $\\omega \\to 0$ weak selection limit, the fixation probability agrees with the one obtained for the original PGG if we replace $\\lambda$ and $m$ parameters with $r$ and $c$. Therefore, we can conclude that homogeneous R-PGG \\& PGG are also equivalent in the framework of stochastic dynamics in a finite and unstructured population.\n\n\\subsection{Evolutionary dynamics in structured populations (the homogeneous case)}\n\\label{struc}\n\nNow, we consider R-PGG on a graph $\\mathcal{G}=(\\mathcal{N},\\mathcal{L})$ of a finite and structured population $N$. The graph $\\mathcal{G}$ is unweighted and undirected. $\\mathcal{N}$ is the node set of the graph, $\\mathcal{N}=\\{1,2,\\dots,N\\}$ and each node represents a player. $\\mathcal{L}$ is the link set of the graph, $\\mathcal{L}=\\{l_1,l_2,\\dots,l_{|\\mathcal{L}|}\\}$. We denote by $l_{i_1 i_2}$ a link with order 2, connecting two nodes $i_1$ and $i_2$. \nThe size of $\\mathcal{L}$ varies with specific graphs (e.g., for a complete graph, the size of $\\mathcal{L}$ is $|\\mathcal{L}|=N(N-1)\/2$). The degree of node $i$ (i.e., the number of link(s) containing $i$) is denoted by $k_i$. Self-loop is not allowed.\n\nAt each elementary Monte Carlo (MC) step, a random node $i\\in \\mathcal{N}$ is selected. If $k_i=0$, nothing happens. If $k_i\\neq 0$, then a random link $l_{ii'}$ is selected between the focal player $i$ and a neighbor $i'$. We then calculate their payoff values. Player $i$ participates in $k_i+1$ R-PGGs centered on herself and all her neighbor(s). First, the contribution is calculated from the $g_i=k_i+1$-size group centered on player $i$. After, we calculate the income from each group centered on an $i$'s neighbor(s). Last, player $i$ accumulates these payoff values to reach the total sum\n\\begin{align}\\label{graphpayoff}\n\t\\pi_i=&~\\frac{1}{k_i+1}\\left[ y_im_i-\\frac{1}{g_i}\\left(y_im_i\\lambda_i+\\sum_{l_{ij_1}\\in\\mathcal{L}}y_{j_1}m_{j_1}\\lambda_{j_1}\\right)\\right.\\nonumber\n\t\\\\\n\t&\\left.+\\sum_{l_{ij_1}\\in \\mathcal{L}}\\left(y_i m_i-\\frac{1}{g_{j_1}}\\left(y_im_i\\lambda_i+\\sum_{l_{j_1j_2}\\in \\mathcal{L}} y_{j_2} m_{j_2}\\lambda_{j_2}\\right)\\right)\\right]\\,.\n\\end{align}\n\nThe $\\pi_{i'}$ payoff of player $i'$ can be calculated in a similar way. Then, player $i$ adopts the strategy of model player $i'$ with the probability $Q(\\pi_{i}\\gets\\pi_{i'})$ defined by Eq.~\\ref{fermi}. Each full time step contains $N$ elementary MC steps specified above.\n\nIn the weak selection limit, we can determine the $\\lambda^\\star$ threshold value for the success of cooperation through the identity-by-descent method introduced by Allen and Nowak \\cite{allen2014games}. The details can be found in \nAppendix~B. This calculation suggests that the threshold $\\lambda^\\star$ in R-PGG is exactly the same as $r^\\star$ for the original PGG (see Refs.~\\cite{su2018understanding,su2019spatial} for the deduction of $r^\\star$ in the original PGG). In other words, homogeneous R-PGG \\& PGG predict equivalent threshold values for the emergence of cooperation in the weak selection limit.\n\nFor non-transitive graphs, which may lead to varying $k_i$ degree among different players $i$, the explicit threshold of cooperation emergence in multiplayer games remains unsolved (only the results in two-player games have been solved \\cite{allen2017evolutionary}). In this case, we may use the average degree $\\langle k\\rangle=(\\sum_{i=1}^N k_i)\/N$ to estimate the threshold value. In this case, the average number of players in each group is $\\langle g\\rangle =\\langle k\\rangle+1$.\n\n\\begin{itemize}\n\t\\item \n\t\\textbf{Monte Carlo simulations}\n\\end{itemize}\n\nTo extend our study, we also apply numerical simulations which can be done for arbitrary selection strength. Now we choose $\\omega=10$ (i.e., $1\/\\omega=0.1$) and $N=40000$. This intermediate level of selection strength was frequently applied by previous works \\cite{lv_amc22,kang_hw_pla21,shen_y_pla22,deng_ys_pa21,szolnoki_pre09c,gracia-lazaro_pre14,yang_hx_pa19,meloni_rsos17,szolnoki_epl10}.\nAt the beginning, we randomly assign each agent's strategy by cooperation or defection. Then, we simulate for $15000$ MC steps, and measure the $\\langle x \\rangle$ stationary portion of cooperators. This can be done by averaging $x_i$ values over the last $5000$ steps of the simulations.\n\nTo cover all significantly different interaction topologies, we apply both regular and heterogeneous graphs. In the former case we use an $L \\times L$ square lattice with periodic boundary conditions where players have four nearest neighbors, hence they form 5-member groups. The typical linear system size was $L=200$. Furthermore, we also use Watts-Strogatz-type (WS) small-world topology where the graph is generated from a ring of $N$ nodes having $k=4$ bonds for each player \\cite{watts1998collective}. Then, each link is rewired randomly with probability $p=0.5$, where self-loop and double links are forbidden. Finally, we also use Barab{\\'a}si-Albert-type (BA) scale-free graphs where the graph is generated starting from a 3-node complete graph and new nodes are attached to the network with two new links \\cite{barabasi1999emergence}. In this way we keep $\\langle k\\rangle=4$ average degree, which makes the results comparable for all cases.\n\nOur observations are summarized in Fig.~\\ref{graph r} where we present the $\\langle x\\rangle$ cooperation level both for PGG and R-PGG models. Evidently, as we pointed out earlier, parameters $r$ and $\\lambda$ serve as control parameters in the former and latter cases, respectively. The overlap between the results of the models is convincing. Evidently, there are differences between the applied topologies, but both PGG and R-PGG change identically. A similar conclusion can be drawn about the influence of cost $c$ and consumption $m$ parameters. If one can require more goods (by increasing $m$), then the cooperation proportion is more sensitive to the waste factor (i.e., increasing with $\\lambda$ more gradually). On the one hand, this leads to the emergence of cooperation, $\\langle x\\rangle>0$, at a smaller $\\lambda$ value. On the other hand, it also leads to the dominance of cooperation, $\\langle x\\rangle=1$, at a greater $\\lambda$. The latter effect is more evident on the small-world graph than on the square lattice graph, and is most evident on the scale-free graph.\n\\begin{figure}[h!]\n\t\\centering\n\t\\makebox[\\textwidth][c]{\\includegraphics[width=14.5cm]{Fig1.eps}}\\\\\n\t\\caption{Cooperation level as a function of normalized synergy factor $r\/\\langle g\\rangle$ and waste factor $\\lambda\/\\langle g\\rangle$. Panels show the results on different graphs, as indicated in the titles. In addition, we mark by arrows the threshold level of the control parameter favoring cooperation under weak selection.}\\label{graph r}\n\\end{figure}\n\nAccording to Eq.~(\\ref{graphpayoff}), $m$ can be extracted as a common factor in every $\\pi_i$ for the homogeneous case. Then, $m$ plays the same role as $\\omega$ in the Fermi-function defined by Eq.~(\\ref{fermi}). In other words, the selection strength is also directly measured by $m$. The same inference holds in PGG, where the selection strength can be measured by $c$. Taking $c=m=10^{-2}$, we have the results valid under a weak selection in numerical simulations. Using $N=40000$ and $k=4$ in Eq.~(\\ref{emergelimit}), we have $\\lambda^\\star\\approx 5$ or $\\lambda^\\star\/\\langle g\\rangle \\approx 1$. This weak selection threshold is consistent with the cases of $c=m=10^{-2}$ in Fig.~\\ref{graph r}.\n\nFor completeness, we also measured how the cooperation level changes by varying the $c$ contribution (in PGG) or $m$ requirement (in R-PGG). Our results are shown in Fig.~\\ref{graph c} in Appendix~C for different topologies. The comparison supports our original observation about the equivalence of homogeneous PGG and R-PGG, no matter whether we consider spatially structured populations.\n\nSumming up, both $r$ $\\&$ $\\lambda$ and $c$ $\\&$ $m$ dependencies demonstrate nicely that R-PGG and PGG models are equivalent if applying the appropriate $\\lambda-m$ or $r-c$ parameter pairs, no matter we have unstructured or structured populations with different interaction topologies or how intensive the selection strength is. The only crucial condition, as we will demonstrate in the next section, is that the population should be homogeneous where all players can be characterized by the same parameter value of the game.\n\n\\section{Difference between R-PGG \\& PGG}\n\\label{difference}\n\nOur abovementioned conclusion becomes invalid if the population is more realistic and players are not uniform anymore. Because of realistic conditions, we here focus on structured populations where interactions are limited and fixedpermanent. The introduction of heterogeneity reveals the difference between PGG and R-PGG models, as we illustrate in Sec.~\\ref{hetestruc}. By means of the translation of payoff functions we will argue that R-PGG is not a simple transformation of the original PGG, but instead a reversed form of it. \n\n\\subsection{Evolutionary dynamics in structured populations (the heterogeneous case)}\n\\label{hetestruc}\n\nTo generalize our model, we here assume that players can be heterogeneous hence they may not participate in the same way in the game. This can be done by introducing player-specific parameter values. Accordingly, we assume uniform random distribution for each player's parameters $\\lambda_i$ and $m_i$. For each player $i$, we randomly generate two numbers between 0 and 1: $\\chi_i^{(\\lambda)}\\in [0,1)$, $\\chi_i^{(m)}\\in [0,1)$. Then, we set the parameters $\\lambda_i$ and $m_i$ for player $i$ as\n\\begin{subequations}\\label{etalambda}\n\t\\begin{align}\n\t\\lambda_i&=\\lambda+(-2\\chi_i^{(\\lambda)}+1)\\eta_{\\lambda}\\,,\\\\\n\tm_i&=m+(-2\\chi_i^{(m)}+1)\\eta_m\\,,\n\t\\end{align}\n\\end{subequations}\nwhere $\\eta_{\\lambda}$ ($\\eta_{\\lambda}\\geq 0$) measures the heterogeneity of the waste factor $\\lambda_i$, and $\\eta_m$ ($\\eta_m\\geq 0$) measures the heterogeneity of the requirement $m_i$. In this way, $\\lambda_i$ and $m_i$ are selected randomly from $[\\lambda-\\eta_{\\lambda},\\lambda+\\eta_{\\lambda})$ and $[m-\\eta_m,m+\\eta_m)$ intervals where $\\lambda$ and $m$ values act as the ``baselines\": the expected waste factor and requirement over all players are still $\\int_{0}^{1}\\lambda_i~\\mathrm{d}\\chi_i^{(\\lambda)}=\\lambda$ and $\\int_{0}^{1}m_i~\\mathrm{d}\\chi_i^{(m)}=m$. In addition, by taking $\\eta_{\\lambda}=0$, $\\eta_m=0$, we return to the homogeneous case where $\\lambda_i=\\lambda$ and $m_i=m$ for all players.\n\nFor PGG, we introduce the heterogeneity in the same way. Here, the two key parameters are $\\eta_r$ and $\\eta_c$, which make $r$ and $c$ player-specific in the extended case. As we demonstrated in the previous section, $\\lambda$ \\& $r$, furthermore $m$ \\& $c$ parameters are identical in the original and reversed games, therefore the $\\eta_\\lambda$ \\& $\\eta_r$ and $\\eta_m$ \\& $\\eta_c$ amplitudes have similar roles in characterizing heterogeneity in the extended versions. \n\nOur key findings are summarized in Fig.~\\ref{graph hetero r}, where we plot the $\\langle x\\rangle$ cooperation level in dependence of $\\eta_r$ or $\\eta_\\lambda$. While they are still equivalent when $\\eta_\\lambda=\\eta_r=0$ (in the homogeneous case), their difference becomes striking as we increase the heterogeneity of the systems. In agreement with previous observations about the principal role of heterogeneity \\cite{perc_pre08,santos_n08}, the cooperation level increases as we increase $\\eta_r$ in PGG for all interaction graphs. But the opposite is true for R-PGG where the extension to a heterogeneous population has the reversed consequence for the cooperation level. While increasing the heterogeneity in the synergy factor usually promotes cooperation in PGG, the opposite is observed for unequal waste factors in R-PGG. This is generally true independently of the applied topology or the dilemma strength.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=14.5cm]{Fig2.eps}\\\\\n\t\\caption{The cooperation level in dependence on the heterogeneity of the synergy factor or the waste factor in structured populations. Panels show different topologies, as indicated. When $\\eta_r=0$ \\& $\\eta_\\lambda=0$, PGG \\& R-PGG are equivalent. But their difference becomes crucial as we increase $\\eta_r$ and in parallel $\\eta_\\lambda$. We fix $c=m=1$.}\\label{graph hetero r}\n\\end{figure}\n\nOur observations can be explained as follows. In PGG, those with a higher synergy factor tend to have a higher payoff, for themselves and their co-players, which leads to an environment beneficial to cooperation. The strategy of cooperation, hence, is favored to reproduce. In R-PGG, however, if we transform the payoff function (see Eq.~(\\ref{rpgg1}) for details), we can observe a redundant item $-m\\lambda$ compared with the payoff function of PGG. Originally, those with a higher waste factor in R-PGG could have the chance of a higher payoff. Nevertheless, the item $-m\\lambda$ tends to bring them a lower payoff. As a result, the same effect in PGG is inhibited in R-PGG, and the heterogeneity of the waste factor usually hinders cooperation in R-PGG.\n\nEvidently, we can introduce heterogeneity in an alternative way when a cooperator's contribution (in PGG) or a defector's requirement (in R-PGG) becomes player-specific. Accordingly, we use individual $c_i$ or $m_i$ values as it is introduced in Eq.~\\ref{etalambda}b. Our results are summarized in Fig.~\\ref{graph hetero c} of Appendix~D. The comparison of PGG and R-PGG confirms what we found previously. Namely, these systems show similar cooperation levels only if the heterogeneity is mild. But their difference becomes striking if we increase the amplitude of $\\eta_c$ and $\\eta_m$.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=13.5cm]{Fig3.eps}\\\\\n\t\\caption{(a) The heat map of cooperation level on the parameter plane of normalized synergy factor $r\/g$ and heterogeneous $r_i$. (b) The same plot in dependence of normalized waste factor $\\lambda\/g$ and heterogeneous $\\lambda_i$. The difference between PGG and R-PGG is more striking at low $r$ and low $\\lambda$ region when there is a proper social dilemma and less noticeable for high $r$ and high $\\lambda$ values. But it still exists if the individual heterogeneity is large enough. The simulations are performed on square lattice topology. We fix $c=m=1$.}\\label{graph r hetero r}\n\\end{figure}\n\nA more comprehensive overview of the system behavior can be obtained if we plot the cooperation level not just at a specific $r$ or $\\lambda$, but for arbitrarily large average values. For simplicity, we only show the results obtained on square lattice topology. The results can be seen in Fig.~\\ref{graph r hetero r} where we present the cooperation level as a heat map on a two-dimensional parameter plane. Notably, Fig.~\\ref{graph hetero r} can be considered a cross-section of this new diagram. Our first qualitative observation is the ``red'' area, which represents the highest cooperation level in the population, is significantly larger for PGG than for R-PGG, which highlights that heterogeneity is generally more beneficial for the PGG system. If we compare the heat maps more carefully then we can see that the difference between PGG and R-PGG is more striking in the low $r$ and low $\\lambda$ region when there is a proper dilemma situation. In this parameter region, even minimal cooperation is challenging for the R-PGG model if the average $\\lambda$ value remains below $\\lambda\/g \\approx 0.75$. For large $r$ and large $\\lambda$ values the difference is less visible between the models. But it still exists if the amplitude of individual heterogeneity is large enough. Hence, we can generally conclude that the heterogeneous synergy factor in PGG can promote cooperation while the heterogeneous waste factor in R-PGG cannot.\n\nWe have also compared the cooperation levels by means of heat maps when heterogeneity was introduced via individual $c_i$ and $m_i$ values. The results are shown in Fig.~\\ref{graph r hetero c} in Appendix~D. The difference between PGG and R-PGG can also be revealed, showing that heterogeneity is a general condition to make a difference between these games. Summing up, we can conclude that introducing the same level of heterogeneity has significantly different consequences on PGG and R-PGG systems. This effect is robust and remains valid no matter how we introduce the distinction among players.\n\n\\subsection{Discussion on the translation of payoff function}\n\\label{translation}\n\nTo explain why R-PGG and PGG become different when heterogeneous players are present, we study the translation of payoff functions. \nAs Equation~(\\ref{payoffCD}) describes, a focal player's income from a group venture can always be expressed as a function of $g_C$ which is the number of cooperators among group neighbors. In the following, we consider four different translations of PGG. The details of these games are as follows.\n\\begin{itemize}\n\\item (i) {\\bf PGG}\nIn the most well-known and widely accepted version of PGG a cooperator contributes a $c$ amount to the common pool, while a defector player does not. The sum of contributions is enlarged and distributed among all group members. The corresponding payoff values of the focal player playing either $C$ or $D$ in the group are\n\\begin{align}\\label{pgg1}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\\pi_C=\\frac{(g_C+1) rc}{g}-c}\\\\\n\t\t\\displaystyle{\n\t\t\\pi_D=\\frac{g_C rc}{g} } \\,.\n\t\\end{cases}\n\\end{align}\n\n\\item (ii) {\\bf Alternative PGG}\nIn an alternative form, all players have an initial endowment $c$ \\cite{hauser2019social}. A cooperator player invests this whole amount to the common pool, while a defector player keeps it. As previously, the contributions are summed, enhanced and distributed among all competitors. The resulting payoff values are \n\\begin{align}\\label{pgg2}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=\\frac{(g_C+1) rc}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=\\frac{g_C rc}{g}+c } \\,.\n\t\\end{cases}\n\\end{align}\n\n\\item (iii) {\\bf R-PGG}\nAccording to our proposal, in an R-PGG a defector requires $m$ from a common resource, while a cooperator player does not. The sum of required goods is wasted at a certain rate, which means an enlarged, and equally shared cost to everyone in the group, yielding the payoff values \n\\begin{align}\\label{rpgg1}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=-\\frac{g_D m\\lambda}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=m-\\frac{(g_D+1) m\\lambda}{g} }\n\t\\end{cases}\n\t\t\\stackrel{g_C+g_D=g-1}{\\Longleftrightarrow}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=-m\\lambda +\\frac{(g_C+1) m\\lambda}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=-m\\lambda +m+\\frac{g_C m\\lambda}{g} } \\,.\n\t\\end{cases}\n\\end{align}\n\n\\item (iv) {\\bf Alternative R-PGG}\nIn the alternative version, each player has an initial $m\\lambda$ deposit. Additionally, a defector requires $m$ from a common resource, and related (enlarged) cost is shared among everyone in the group. Therefore, the payoff values are \n\\begin{align}\\label{rpgg2}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=m\\lambda-\\frac{g_D m\\lambda}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=m\\lambda+m-\\frac{(g_D+1) m\\lambda}{g} }\n\t\\end{cases}\n\t\\stackrel{g_C+g_D=g-1}{\\Longleftrightarrow}\n\t\\begin{cases}\n\t\t\\displaystyle{\n\t\t\t\\pi_C=\\frac{(g_C+1) m\\lambda}{g}}\\\\\n\t\t\\displaystyle{\n\t\t\t\\pi_D=m+\\frac{g_C m\\lambda}{g} }\\,.\n\t\\end{cases}\n\\end{align}\n\\end{itemize}\nNote that in Eq.~(\\ref{rpgg1}) and in Eq.~(\\ref{rpgg2}) we have used $g_D=g-g_C-1$ notation to obtain a consistent $g_C$-dependent form of payoff values for all cases. The connections between the above-specified cases are summarized in Fig.~\\ref{figureasymmetric}, where the curves represent the payoff of a focal player employing either cooperation or defection.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=13cm]{Fig4.eps}\\\\\n\t\\caption{Payoff value for a focal cooperator or defector player as a function of $g_C$, denoting the number of cooperators among neighbors. The four panels show the cases as indicated in the titles. They are equivalent in the payoff gap because the translational items in $\\pi_D-\\pi_C$ can be subtracted. They are different in $\\pi_C(g_C=0)$, $\\pi_C(g_C=g-1)$, $\\pi_D(g_C=0)$, and $\\pi_D(g_C=g-1)$, which means that their translational items in $\\pi_D-\\pi_C$ cannot be subtracted when parameters become heterogeneous. The green arrows indicate that the given game models are equivalent in the homogeneous case. The red arrows mean the two models are different in the heterogeneous case. The yellow arrow means the payoff functions of the models are seemingly equivalent, but later we will show they are actually different. $g=5$, $r=\\lambda=1.5$, and $c=m=1$ parameters were used.}\\label{figureasymmetric}\n\\end{figure}\n\nThe key assumption, which explains the equivalence, is the linear dependence on the $g_C$ value. We can see that with the same parameters, the slope of payoff functions is constant everywhere. Also, the difference between $\\pi_D$ and $\\pi_C$ (i.e., $\\pi_D-\\pi_C$) is constant at any $g_C$ value. Therefore, despite different interpretations, we can say that all four translations expressed in panels~(a1),~(a2),~(b1), and (b2) represent conceptually similar public goods games. \n\nThe equivalence in the homogeneous case is clear because the results depend only on $\\pi_{i'}-\\pi_i$. Here the translational items in $\\pi_{i'}-\\pi_i$ can be subtracted, which leads to the equivalence among the four interpretations. \nHowever, the absolute values of payoff functions vary from case to case, as indicated by the value of $\\pi_C(g_C=0)$, $\\pi_C(g_C=g-1)$, $\\pi_D(g_C=0)$, and $\\pi_D(g_C=g-1)$ in Fig.~\\ref{figureasymmetric}. This leads to a significant difference when introducing heterogeneity. As an example, take alternative PGG and original R-PGG versions. We can see that R-PGG has a redundant item $-m\\lambda$ compared with alternative PGG. As mentioned earlier, in a homogeneous case, such an item can be subtracted when calculating $\\pi_{i'}-\\pi_i$ for strategy update. However, in a heterogeneous case, such an item varies from player to player, $-m_i\\lambda_i\\neq -m_{i'}\\lambda_{i'}$, which cannot be subtracted anymore. \n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=13cm]{Fig5.eps}\\\\\n\t\\caption{Payoff values for a cooperator (open symbols) and for a defector (solid symbols) player in dependence on the number of other cooperative players in the group. Panels show different models, as indicated. While $g=5$, $c=m=1$ values are fixed, the three curves for each $\\pi_C$ or $\\pi_D$ are plotted by taking $r_i=\\lambda_i=1.5-2$, $1.5$, $1.5+2$ ($r=\\lambda=1.5$, $\\eta_r=\\eta_{\\lambda}=2$). The shaded areas mark the traces of curves when $r_i$ or $\\lambda_i$ changes in the mentioned interval bordered by $r\\pm\\eta_r$ ($\\lambda\\pm\\eta_{\\lambda}$). These areas are marked by different colors for competing strategies (green for $D$ and yellow for $C$). The meaning of color arrows are similar to those we used in Fig.~\\ref{figureasymmetric}.}\\label{figureasymmetric heter}\n\\end{figure}\n\nTo visualize this effect in a more intuitive way, we present Fig.~\\ref{figureasymmetric heter} which shows $\\pi_C$ and $\\pi_D$ as a function of $g_C$ in the mentioned cases when heterogeneous $r$ or $\\lambda$ are introduced. The curves represent three specific average values of key parameters, but we also mark the location of payoff values for intermediate values by shaded areas. In this way the conceptual difference between different cases becomes clear. Staying at Fig.~\\ref{figureasymmetric heter}, the shaded areas in panel~(a1) and in panel~(b1) are completely different because the difference in the payoff values leads to unequal outcomes in the presence of heterogeneity. \n\nNotably, the shape of the shaded area remains intact if we change PGG, shown in panel Fig.~\\ref{figureasymmetric heter}(a1), to the alternative PGG, presented in panel Fig.~\\ref{figureasymmetric heter}(a2). This is because their coefficients in front of $r$ are equal: $(g_C+1)c\/g$ in $\\pi_C$, and $g_C c\/g$ in $\\pi_D$. Therefore, the same change in $r$ leads to the same change in the payoff function. Although they are still different in their vertical positions, the identical shape confirms their equivalence in the stationary cooperation level (the reader may compare Fig.~\\ref{figurealter r}(a) in Appendix~D and Fig.~\\ref{graph r hetero r}(a)). Furthermore, if we compare the case of alternative PGG (Fig.~\\ref{figureasymmetric heter}(a2)) and the case of alternative R-PGG (Fig.~\\ref{figureasymmetric heter}(b2)) then we can see that they show not only the same shape of shaded areas, but also the same absolute payoff values. Their cooperation levels, however, as we show in Fig.~\\ref{figurealter r} in Appendix~D, are different. This fact is a direct consequence of the conceptual difference between PGG $\\&$ R-PGG, as we will discuss later.\n\nFor completeness, we also studied the consequence of heterogeneity in $c$ or $m$ parameter on the payoff functions. The results are presented in Fig.~\\ref{figureasymmetric hetec} in Appendix~D. It shows that the same level of change in individual $c$ leads to a varying change in their payoff functions, resulting in different shapes of shaded areas. We, however, stress that alone the payoff functions are not able to reveal all differences among the cases we discussed above. For example, similar shapes of shaded areas can produce unequal heat maps of cooperation levels on the parameter plane, as shown in Appendix~D.\n\nTo give a deeper insight into the origin of varying behavior in heterogeneous games, let us turn back to the payoff functions. When we used the term $g_C=g-g_D-1$ to replace $g_D$ with $g_C$ in R-PGG (as well as in alternative R-PGG), we implicitly assumed homogeneous payoff functions. Therefore, the individual parameter values have no importance. However, if we assume distinct players and consider player-specific payoff values via Eq.~(\\ref{unequalpayoff}), then we can rewrite the payoff function for the heterogeneous cases. For example, the payoff for the original PGG can be written as\n\\begin{equation}\n\t\\pi_i=\\frac{1}{g}\\sum_{j=1}^g x_j c_j r_j-x_i c_i\\,,\n\\end{equation}\nand for the original R-PGG it is\n\\begin{align} \\label{transRPGG}\n\t\\pi_i&=y_i m_i-\\frac{1}{g}\\sum_{j=1}^g y_j m_j\\lambda_j\\nonumber\\\\\n\t&=(1-x_i) m_i-\\frac{1}{g}\\sum_{j=1}^g (1-x_j) m_j\\lambda_j\\nonumber\\\\\n\t&=-\\frac{\\sum_{j=1}^g m_j\\lambda_j}{g}+\\frac{1}{g}\\sum_{j=1}^g x_j m_j\\lambda_j+(1-x_i) m_i\\,.\n\\end{align}\nNow we can see that the term $-\\sum_{j=1}^g m_j\\lambda_j\/g$ is related to the waste factor of all players in the group. Importantly, we can write the payoff for alternative R-PGG as\n\\begin{align} \\label{transalterRPGG}\n\t\\pi_i&=m_i\\lambda_i+y_i m_i-\\frac{1}{g}\\sum_{j=1}^g y_j m_j\\lambda_j\\nonumber\\\\\n\t&=m_i\\lambda_i+(1-x_i) m_i-\\frac{1}{g}\\sum_{j=1}^g (1-x_j) m_j\\lambda_j\\nonumber\\\\\n\t&=m_i\\lambda_i-\\frac{\\sum_{j=1}^g m_j\\lambda_j}{g}+\\frac{1}{g}\\sum_{j=1}^g x_j m_j\\lambda_j+(1-x_i) m_i\\,,\n\\end{align}\nwhile for alternative PGG it is\n\\begin{equation} \\label{alterPGG}\n\t\\pi_i=\\frac{1}{g}\\sum_{j=1}^g x_j c_j r_j+(1-x_i) c_i\\,.\n\\end{equation}\n\nBy comparing Eq.~(\\ref{transalterRPGG}) and Eq.~(\\ref{alterPGG}), we can identify a non-zero redundant item $m_i\\lambda_i-(\\sum_{j=1}^g m_j\\lambda_j)\/g$ in R-PGG. This is the reason behind the observed difference in cooperation levels when we change the independent variable from $g_D$ to $g_C$. In particular, if we apply identical extension to PGG and alternative PGG, then we have no such difficulties because cooperators take the key role hence crucial game parameters belonging to this strategy in both model versions. A similar argument can be raised when we compare R-PGG and alternative R-PGG, because the decisive game parameters determine the payoff values that belong to defectors in both models. This explains why we cannot transform the family of PGGs into the family of R-PGGs by simply switching cooperators \\& defectors. \n\nThe conceptual difference between the models can be termed a ``reversed effect'' and its essence is explained in the following way. The game parameters working in PGG \\& R-PGG belong to different strategies. In PGG, the personal features of cooperator players count: if a player $i$ cooperates, she contributes $c_i$ and enlarges the contribution by $r_i$; meanwhile, a defective player $j$ has no chance to ``activate'' her specific $c_j$ and $r_j$ values. But she only receives the public goods of $r_i c_i$ produced by a cooperator. On the contrary, in R-PGG, the diversity of defectors players becomes essential: if a player $j$ defects, she receives $m_j$ which is multiplied by $\\lambda_j$ factor as the public cost. Meanwhile, a cooperative player $i$ never has a chance to validate her $m_i$ and $\\lambda_i$ parameter values. Her role is limited to sharing the public cost of $m_j \\lambda_j$ of a defector player.\n\nIn this way, the role of defectors is essential in R-PGG which is irreplaceable in a more complex environment. Hence, we can conclude that R-PGG is not a simple translational transformation of the original PGG, but rather a reversed form of it.\n \n\\section{Conclusion}\n\\label{conclusion}\n\nGenerally, people make logical, and frequently economical decisions when their consumption and its cost directly relate. However, if this connection is less clear, because the emerging cost is shared among a group of people, then we face a dilemma situation: we may require more goods from a common source than it is absolutely necessary because the related extra expense will be covered collectively. The reverse is also true: our economic behavior by consuming fewer resources will not necessarily be awarded by a proportionally smaller bill in the mentioned situation because all the others will also benefit from it. One might think this social dilemma is just a simple transformation of the original PGG where players contribute freely to a common pool and receive goods equally. In the reversed case described above, players receive goods from a common resource freely and should contribute equally. But the relation between the original PGG and our present R-PGG is more subtle.\n\nIn the case of a homogeneous population, where potential contributions and demands are constant, R-PGG is equivalent to PGG. This fact can be supported by means of a static calculation, deterministic and stochastic evolutionary analysis in a well-mixed population, and Monte Carlo simulations in structured populations. In evolutionary game dynamics, the requirement $m$ parameter in R-PGG plays the same role as the cooperator contribution $c$ does in PGG. Furthermore, the waste factor $\\lambda$ in R-PGG has the same duty as the synergy factor $r$ has in PGG. By transforming these parameters between R-PGG and PGG, we can obtain exactly the same system behaviors.\n\nThe network reciprocity reveals the following conclusions in R-PGG. If requiring goods leads to more cost to the group, then players tend to require fewer goods. Intriguingly, if players are allowed to require more goods, then fewer players may do it. The opposite behavior, which is consistent with our intuition, can only be detected clearly on small-world and scale-free graphs under the weak dilemma limit.\n\nAlthough the underlying R-PGG model seems equivalent to PGG, the agreement diminishes when we introduce more realistic conditions, like supposing heterogeneous players with diverse contribution capacities or unequal requirement levels for common resources. In the mentioned cases, R-PGG and PGG show major differences and produce highly different cooperation levels even if we apply the same average values of key parameters with equally strong amplitudes of diversity. The key observation is that while heterogeneity can generally support cooperation in PGG, the mentioned condition impedes the evolution of cooperation in R-PGG.\n\nTo understand the deeper origin of these diverse behaviors, we studied how payoff functions vary in response to the heterogeneity condition for different cases. In the case of equivalent models, the mentioned function should vary similarly if we apply varying average or growing diversity of key parameters. This requirement is justified between alternative forms of the original PGG where cooperators play the decisive role in collective income. This is also true between the alternative versions of R-PGG where the individual defector's profile becomes crucial. In this way, the equivalence cannot be held between PGG and R-PGG where the heterogeneity reveals the conceptual difference between the model families. Hence, we can say that R-PGG is not a trivial transformation of the original PGG, but can be considered a reversed version of it.\n\nOur present work focused on the difference in structured populations, but of course, similar study can also be done in a well-mixed system. Hopefully, our observations will stimulate forthcoming research. When the applied microscopic rule leads to the heterogeneity of game parameters by any means, one can study the consequence of extended parameters on PGGs and R-PGGs separately. The different results under the same topic (e.g., environmental protection) can reveal how the introduced rule works in PGGs (e.g., players donate to environmental protection), and how it works in corresponding R-PGGs (e.g., players litter for convenience). Aside from heterogeneity, the potential difference between R-PGG and PGG can be further investigated.\n\n\\section*{Acknowledgements}\nA.S. was supported by the National Research, Development and Innovation Office (NKFIH) under Grant No. K142948.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Sec:Introduction}\nThe problem of characterizing the capacity region of a broadcast channel\\footnote{Throughout this article we restrict attention to memoryless channel. Please refer to \\cite{1998ADITI_Mas} or \\cite{2006EIT_CovTho} for a definition.} (BC) was proposed by Cover \\cite{197201TIT_Cov} in 1972, and he introduced a novel coding technique to derive achievable rate regions for particular degraded BCs. In a seminal work aimed at deriving an achievable rate region for the general degraded BC,\nBergmans \\cite{197303TIT_Ber} generalized Cover's technique into what is currently referred to as superposition coding. Gallager \\cite{197403PPI_Gal} and Bergmans \\cite{197403TIT_Ber} concurrently and independently proved optimality of\nsuperposition coding for the class of degraded BCs. This in particular yielded capacity region for\nthe scalar additive Gaussian BC. However, the case of general discrete BC (DBC) remained open. This (problem) led to the discovery of another ingenious coding technique\\footnote{We remark that general DBC is richer in terms of the strategies it permits.}. Gelfand \\cite{197707PIT_Gel} proposed an ingenious coding technique for a particular 2-DBC. In 1979, Marton \\cite{197905TIT_Mar} proposed the technique of binning. In conjunction with superposition, she derived the largest known achievable rate region for the general two user DBC (2-DBC). A generalization \\cite[p.391 Problem 10(c)]{CK-IT2011} of superposition and binning to incorporate\na common message is the largest known achievable rate region for the general 2-DBC and it's capacity is yet unknown.\\footnote{It is of interest to note that though superposition and binning were known in particular settings \\cite{197201TIT_Cov}, \\cite{197707PIT_Gel}, it's generalization led to fundamentally new ideas. For example, the description of a rate region using an auxiliary random variable \\cite{197303TIT_Ber} and the technique of binning have proved to be invaluable in\nderiving information theoretic achievable rate regions.}\n\nThough the capacity region has been found for many interesting classes of BCs \\cite{197201TIT_Cov,197507TIT_Cov,197303TIT_Ber,197403TIT_Ber, 197511TIT_AhlKor,197701TIT_KorMar,197903TIT_Elg, \n1977MMISIT_Mar,197804PPI_Pin,197905TIT_Mar,1980MMPPI_GelPin,198001TIT_Elg,198101TIT_GamMeu,197503TIT_Meu,197901TIT_HajPur}, the question of whether the\nrate region derived by Marton is optimal for the general DBC has remained open for over thirty years. Following a period of reduced activity, there has been renewed interest \\cite{201006ISIT_GenNaiShaWan,200910TIT_NaiElg} in settling this question. Gohari and Anantharam \\cite{201202TIT_GohAna} have proved computability of Marton's rate region. This enabled them identify a class of binary 2-DBCs for which Marton's \\cite{197905TIT_Mar} rate region when computed is strictly smaller than the tightest known outer bound \\cite{197805TIT_Sat,200701TIT_NaiGam}, which is due to Nair\nand El Gamal. On the other hand, Weingarten, Steinberg and Shamai \\cite{200609TIT_WeiSteSha} have proved Marton's binning (also referred to, in the Gaussian setting, as Costa's dirty paper coding \\cite{198305TIT_Cos}) to be optimal for Gaussian MIMO BC, and thereby characterized capacity region for the particular class of Gaussian vector BCs. We remark \\cite{200609TIT_WeiSteSha} proves optimality of Marton's binning technique for Gaussian vector BCs with arbitrary number of receivers.\n\nIn this article, (1) we propose a coding technique based on nested coset codes, an ensemble of codes endowed with algebraic structure, that enables us (2) present an achievable rate region for the general three user DBC (3-DBC), and thereby (3) strictly enlarge the current known largest achievable rate region\\footnote{The largest known achievable rate region for the general 3-DBC is the natural extension of Marton's rate region for the 2-DBC. We henceforth refer to this as Marton's rate region for 3-DBC.}. We characterize \\3To1BC DBCs, a class of 3-DBCs for which Marton's coding technique is proved to be strictly sub-optimal. In particular, we identify a novel \\3To1BC DBC for which the proposed coding technique yields achievable, a rate triple not contained within Marton's rate region. We remark that the \\3To1BC DBC presented herein is the first known BC for which the natural generalization of Marton's coding technique is strictly sub-optimal. Indeed, one of the key elements of our work is an analytical proof of sub-optimality of Marton's rate region for this 3-DBC. Our findings emphasize the need for codes endowed with algebraic structure for communicating over a general multi-terminal communication setting.\n\nWhy do codes endowed with algebraic structure outperform traditional independent unstructured codes for a BC? The central aspect of a coding technique designed for a BC is interference management. Marton's coding incorporates two techniques - superposition and binning - for tackling interference. Superposition enables each user decode a \\textit{univariate} component of the other user's signal and thus subtract it off. Binning enables the encoder counter the component of each user's interfering signal not decoded by the other, by precoding for the same. Except for particular cases, the most popular being dirty paper coding, precoding results in a rate loss, and is therefore less efficient than decoding the interfering signal at the decoder. The presence of a rate loss motivates each decoder to decode as large a part of interference as possible.\\footnote{For the Gaussian case, there is no rate loss. Thus the encoder can precode all the interference. Indeed, the optimal strategy does not require any user to decode a part of signal not intended for\nit. This explains why lattices are not necessary to achieve capacity of Gaussian vector BC.} However decoding a large part of the interference constrains the individual rates. In a three user BC, each user's reception is plagued by interference caused by signals intended for the other two users. The interference is in general a bi-variate function of signals intended for the other users. If the signals of the two users are endowed with structure that can help compress the range of this bi-variate function when applied to all possible signals, then the receivers can decode a large part of the interfering signal. This minimizes the component of the interference precoded, and therefore the rate loss.\\footnote{For the Gaussian case, precoding suffers no rate loss and hence no part of the interference needs to be decoded. Thus constraining interference patterns is superfluous. } This is where codebooks endowed with algebraic structure outperform unstructured independent codebooks. Indeed, linear codes constrain the interference pattern to an affine subspace if the interference is the sum of user 2 and 3's signals. It is our belief that additional degrees of freedom prevalent in a three user information theoretic problem can be harnessed with codebooks endowed with algebraic structure. Whether structure in codebooks can be exploited for a two user problem remains open.\n\nThis article is organized as follows. We begin with preliminaries in section \\ref{Sec:BroadcastChannelDefinitionsMartonRateRegion}. In particular, we provide relevant definition for a DBC and state Marton's achievable rate region. In section \\ref{Sec:3To1BCAndTheNeedForStructuredCodes}, describe a 3-DBC for which Marton's coding technique is strictly suboptimal and generalize this example into a class of 3-DBCs called \\3To1BC DBCs. We present our achievable rate region for the general 3-DBC in three steps. In the first two steps, presented in section \\ref{Sec:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes}, we present achievable rate regions for a \\3To1BC DBC. The final step, where we derive an achievable rate region for a general 3-DBC, is presented in section \\ref{Sec:AchievableRateRegionFor3BCUsingNestedCosetCodes}. The proof of strict sub-optimality of Marton's coding technique for the \\3To1BC DBC is detailed in section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, appendices \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx}, \\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss}. In section \\ref{Sec:EnlargingMarton'sRateRegionUsingNestedCosetCodes}, we indicate structure of a coding technique that yields the largest known achievable rate region for the 3-DBC which subsumes all of the known achievable rate regions. We conclude in section \\ref{Sec:ConcludingRemarks} by pointing at fundamental connection between the several layers of coding in a three user communication problem and common parts of a triple of random variables, thereby arguing the inherent role played by structured codes in information theory. \n\nThe reader solely interested in proof of strict sub-optimality of Marton's coding technique may read through sections \\ref{Sec:BroadcastChannelDefinitionsMartonRateRegion}, \\ref{SubSec:LimitationsOfUnstructuredCodes}, \\ref{SubSec:TheThreeUserBroadcastChannel} and skip over to section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique} without missing out on any necessary elements. However, we encourage the reader to peruse the proposed coding technique based on nested coset codes.\n\nWhile sections \\ref{Sec:BroadcastChannelDefinitionsMartonRateRegion}-\\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique} contain all the technical details presented in this article, we encourage the reader to read through section \\ref{Sec:RoleOfStructuredCodesInNetworkInformationTheory} for an account of the role of structured codes in network information theory.\n\n\\section{The role of structured codes in network information theory}\n\\label{Sec:RoleOfStructuredCodesInNetworkInformationTheory}\n\nIn this article, we demonstrate the role of algebraic structure through the simplest example of structured codes - linear codes - that contain a bivariate function - sum - of transmitted codewords to an affine space, thereby leaving sufficient room for generalization using other algebraic objects. In particular, we employ the ensemble of nested coset codes \\cite{201108ISIT_PadPra} to derive a new achievable rate region for the \\textit{general} 3-DBC. Indeed, analyzing the proposed coding technique for a \\textit{general} 3-DBC requires joint typical encoding and decoding. One of the key elements of our study is therefore the interplay of joint typical encoding and decoding of statistically correlated codebooks resulting in new proof techniques. While we do not provide proofs in this article, we refer the curious reader to \\cite[proof of theorem 2]{201301arXivMACDSTx_PadPra}, \\cite[proof of theorem 1]{201108ISIT_PadPra} that contain several of these techniques.\n\nThe astute reader will question the case when the\nbi-variate function is not a sum. Towards answering this question, we consider a natural\ngeneralization of linear codes to sets with looser algebraic structure such as groups. Our\ninvestigation of group codes, kernels of group homomorphisms, to improve achievable rate\nregions for information theoretic problems has been pursued in a concurrent research\nthread \\cite{201108ISIT_SahPra}.\\footnote{We also bring to the attention of the interested\nreader, our investigation \\cite{201207ISIT_SahPra} of pseudo group codes. While linear\ncodes are completely {\\it compressive} under the operation of addition, and unstructured\nindependent codes are completely {\\it explosive}, pseudo group codes lie in between. In\nother words, when two pseudo group codes of rate $R$ are operated under the group\noperation, the range of the resulting codebook lies between $R$ and $2R$. Pseudo group\ncodes are of interest since they outperform group codes for point to point communication.} Containing the sum of transmitted codewords using linear codes is just the first step, and we envision an achievable rate region involving a union over all algebraic objects pertaining to the several bivariate functions.\n\nThe role of structured codes for improving information theoretic rate regions began with the ingenious technique of K\\\"orner and Marton \\cite{197903TIT_KorMar} proposed for the source coding problem of reconstructing modulo$-2$ sum of distributed binary sources. Han and Kobayashi \\cite{198701TIT_HanKob} categorized a class of function reconstruction problems for which K\\\"orner and Marton's technique provided strict gains over the largest known rate regions using unstructured codes. Ahlswede and Han \\cite{198305TIT_AhlHan} proposed a universal coding technique that brings together coding techniques based on unstructured and structured codes\\footnote{Indeed, the coding techniques based on structured codes do not substitute for coding techniques based on unstructured codes. For example, lossless reconstruction of a pair of sources with two source codes that are partitioned using a common channel code can be strictly sub optimal. Similarly, the technique of partitioning independent source codes using independent channel codes is sub optimal for the problem of lossless reconstruction of modulo$-2$ sum of a class of binary sources.}. More recently, there is a wider interest \\cite{200710TIT_NazGas,200912arXiv_CadJaf} in developing coding techniques for particular problem instances that perform better than the best known techniques based on unstructured codes. Philosof and Zamir \\cite{200906TIT_PhiZam} employ nested linear codes to communicate over a particular binary doubly dirty multiple access channel (MAC) and analytically prove strict sub-optimality of best known coding technique based on unstructured independent codes. Bresler, Parekh and Tse \\cite{201009TIT_BreParTse} study the problem of characterizing approximate capacity of many-to-one Gaussian interference channels and demonstrate the need for lattice codes. Sridharan et. al. \\cite{200809Allerton_SriJafVisJafSha} employ lattice based scheme to prove achievability of rates not contained within the largest known achievable rate region using unstructured independent codes.\n\nIt was shown in \\cite{198701TIT_HanKob}, in the setting of distributed source coding that for every any non-trivial and truly bi-variate function, there exists at least one source distribution for which linear codes outperform random codes, Even then, it was largely believed that codebooks possessing algebraic structure can be exploited only for modulo additive channel and source coding problems. Indeed, linear codes were known to be sub optimal for communicating over arbitrary point to point channels \\cite{197102TAMS_Ahl} (and similarly for lossy compression of sources subject to an arbitrary distortion), and therefore, the basic building block in the coding scheme for any multi-terminal communication problem could not be filled by linear codes. For over thirty years, since the work of K\\\"orner and Marton came to light, neither did we know of a coding technique based on unstructured codes that performed as well, nor did we know of a framework for coding based on structured codes for which the above findings was a particular case.\n\nKrithivasan and Pradhan \\cite{201103TIT_KriPra} have proposed the ensemble of nested coset codes as the basic building block of algebraic codes for compressing sources subject to any arbitrary distortion. They employ this ensemble to propose a framework for communicating information from distributed encoders observing correlated sources to a centralized decoder. Firstly, this framework generalizes the technique proposed by K\\\"orner and Marton for the general problem of distributed function computation, joint quantization of distributed sources etc. Secondly, in conjunction with the Berger Tung \\cite{Berger-MSC} technique this strictly enlarges the largest known achievable rate region for the problem of distributed function computation. In the same spirit, we proposed the ensemble of nested coset codes \\cite{201108ISIT_PadPra} as an ensemble of codes possessing algebraic structure that achieves capacity of arbitrary point to point channels. We employed this ensemble to (i) elevate the technique proposed by Philosof and Zamir to derive a new achievable rate region \\cite{201301arXivMACDSTx_PadPra} for an arbitrary discrete multiple access channel with distributed states (ii) derive a new achievable rate region for the general discrete three user interference channel \\cite{201207ISIT_PadSahPra}.\n\n\\section{Broadcast channel: definitions and Marton's rate region}\n\\label{Sec:BroadcastChannelDefinitionsMartonRateRegion}\nWe begin with remarks on notation in section \\ref{SubSec:Notation}. In section \\ref{SubSec:DefinitionsBroadcastChannelCodeAchievabilityCapacity}, we state relevant definitions - code, achievability and capacity - with respect to a 3-DBC. In section \\ref{SubSec:MartonRateRegionFor2BC}, we briefly discuss Marton's coding technique and state Marton's rate region for a 2-DBC. In section \\ref{SubSec:NaturalExtensionOfMartonTo3BC}, we state a natural extension of Marton's coding technique for the three user case and provide a description of the corresponding achievable rate region.\n\\subsection{Notation}\n\\label{SubSec:Notation}\nWe employ notation that has now been widely accepted in the information theory literature\nsupplemented with the following. For a set $A \\subseteq \\reals^{k}$, $\\cocl \\left( A \\right)$ denotes closure of convex hull of $A$. Throughout this article, $\\log$ and $\\exp$ functions are taken with respect to the base $2$. Hence, units for information theoretic quantities such as entropy and mutual information are expressed in bits\/symbol. Let $h_{b}(x) \\define -x\\log_{2}x - (1-x)\\log_{2}(1-x)$ denote binary entropy function. Let $a * b \\define a(1-b)+(1-a)b$ denote binary convolution. For $K \\in \\naturals$, we let $[K]\\define \\left\\{ 1,2\\cdots,K \\right\\}$. We let $\\fieldq$ denote the finite field of cardinality $q$. While $+$ denotes addition in $\\reals$, we let $\\oplus$\ndenote addition in a finite field. The particular finite field,\nwhich is uniquely determined (up-to an isomorphism) by it's cardinality, is clear from\ncontext. When ambiguous, or to enhance clarity, we specify addition in $\\fieldq$ using\n$\\oplus_{q}$. For elements $a,b$, in a finite field, $a \\ominus b\n\\define a \\oplus (-b)$, where $(-b)$ is the additive inverse of $b$.\\begin{comment}{ In sections \\ref{Sec:3To1BCAndTheNeedForStructuredCodes} and \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, $\\oplus$ denotes addition in the binary field $\\BinaryField$.}\\end{comment} In this article, we will need to define multiple objects, mostly triples, of the same type. In order to reduce clutter, we use an \\underline{underline} to\ndenote aggregates of objects of similar type. For example, (i) if\n$\\OutputAlphabet_{1},\\OutputAlphabet_{2}, \\OutputAlphabet_{3}$ denote (finite) sets, we\nlet $\\underlineSetY$ either denote the Cartesian product\n$\\OutputAlphabet_{1} \\times \\OutputAlphabet_{2} \\times \\OutputAlphabet_{3}$ or\nabbreviate the collection $(\\OutputAlphabet_{1},\\OutputAlphabet_{2},\\OutputAlphabet_{3})$\nof sets, the particular reference being clear from context, (ii) if $y_{k} \\in \\OutputAlphabet_{k}:k=1,2,3$, we let $\\underliney \\in\n\\underlineSetY$ abbreviate $(y_{1},y_{2},y_{3}) \\in \\OutputAlphabet$ (iii) if\n$\\mathscr{M}_{1},\\mathscr{M}_{2}, \\mathscr{M}_{3}$ denote cardinalities of finite sets\n$\\MessageSetM_{1},\\MessageSetM_{2},\\MessageSetM_{3}$ respectively, then we let\n$\\underlineCardinalityMessageSet$ abbreviate the collection\n$\\mathscr{M}_{1},\\mathscr{M}_{2}, \\mathscr{M}_{3}$, (iv) if\n$d_{k}:\\OutputAlphabet_{k}^{n} \\rightarrow \\MessageSetM_{k}:k=1,2,3$ denote (decoding)\nmaps, then we let $\\underlined(\\underliney^{n})$ denote $(d_{1}(y_{1}^{n}),\nd_{2}(y_{2}^{n}), d_{3}(y_{3}^{n}))$.\n\\subsection{Definitions: Broadcast channel, code, achievability and capacity}\n\\label{SubSec:DefinitionsBroadcastChannelCodeAchievabilityCapacity}\nA 3-DBC consists of a finite input alphabet\nset $\\InputAlphabet$ and three finite output alphabet sets $\\OutputAlphabet_{1},\n\\OutputAlphabet_{2}, \\OutputAlphabet_{3}$. The discrete time channel is (i) time\ninvariant, i.e., the pmf of $\\underlineY_{t}=(Y_{1t},Y_{2t},Y_{3t})$, the output at time\n$t$, conditioned on $X_{t}$, the input at time $t$, is invariant with $t$, (ii)\nmemoryless, i.e., conditioned on present input $X_{t}$, the present output $\\underlineY_{t}$ is\nindependent of past inputs $X_{1},\\cdots, X_{t-1}$ and (iii) used without feedback, i.e.,\nthe encoder has no information of the symbols received by the decoder. Let\n$W_{\\underlineY|X}(\\underliney|x)=W_{Y_{1}Y_{2}Y_{3}|X}(y_{1},y_{2},y_{3}|x)$ denote\nprobability of observing $\\underliney \\in \\underlineSetY$ at the respective outputs\nconditioned on $x \\in \\InputAlphabet$ being input. Input is constrained with respect\nto a cost function $\\kappa : \\InputAlphabet \\rightarrow [0, \\infty)$. The cost function\nis assumed additive, i.e., cost of transmitting the vector $x^{n} \\in \\InputAlphabet^{n}$\nis $\\bar{\\kappa}^{n}(x^{n}) \\define \\sum_{i=1}^{n}\\kappa(x_{i})$. We refer to this 3-DBC as\n$(\\InputAlphabet,\\underlineSetY,W_{\\underlineY|X},\\kappa)$.\n\nIn this article, we restrict attention to communicating private messages to the three users. The focus of this article therefore is the (private message) capacity region of a 3-DBC, and in particular corresponding achievable rate regions. The following definitions make the relevant notions precise.\n\n\\begin{definition}\n \\label{Defn:3-BCCode}\n A 3-DBC code $(n,\\underlineCardinalityMessageSet,e,\\underlined)$ consist of (i) index\nsets $\\MessageSetM_{1}, \\MessageSetM_{2},\\MessageSetM_{3}$ of messages, of cardinality\n$\\mathscr{M}_{1},\\mathscr{M}_{2},\\mathscr{M}_{3}$ respectively, (ii) encoder map\n$e:\\underlineMessageSetM \\rightarrow\n\\InputAlphabet^{n}$, and (iii) three decoder maps $d_{k} : \\OutputAlphabet_{k}^{n}\n\\rightarrow \\MessageSetM_{k}:k=1,2,3$.\n\\end{definition}\n\n\\begin{definition}\n \\label{Defn:BCCodeErrorProbability}\n The error probability of a 3-DBC code\n$(n,\\underlineCardinalityMessageSet,e,\\underline{d})$ conditioned on message triple\n$(m_{1},m_{2},m_{3}) \\in \\underlineMessageSetM$ is\n\\begin{equation}\n \\label{Eqn:ErrorProbabilityOf3-BCCode}\n \\xi(e,\\underlined|\\underlinem) \\define 1-\\sum_{\\underliney^{n} : \\underlined\n(\\underliney^{n}) = \\underlinem}\nW_{\\underlineY|X}(\\underliney^{n}|e(\\underlinem)).\\nonumber\n\\end{equation}\nThe average error probability of\na 3-DBC code $(n,\\underlineCardinalityMessageSet,e,\\underlined)$ is\n$\\bar{\\xi}(e,\\underlined)\\define \\sum_{\\underlinem\n \\in \\underlineMessageSetM}\n\\frac{1}{\\mathscr{M}_{1}\\mathscr{M}_{2}\\mathscr{M}_{3}}\n\\xi(e,\\underlined|\\underlinem)$. Cost of transmitting message $\\underlinem \\in\n\\underlineMessageSetM$ per symbol is $\\tau(e|\\underlinem) \\define\n\\frac{1}{n}\\bar{\\kappa}^{n}(e(\\underlinem))$ and\naverage cost of 3-DBC code $(n,\\underlineCardinalityMessageSet,e,\\underlined)$ is $\\tau(e)\n\\define \\frac{1}{\\mathscr{M}_{1}\\mathscr{M}_{2}\\mathscr{M}_{3}}\\sum_{\\underlinem\n \\in \\underlineMessageSetM}\\tau(e|\\underlinem)$.\n\\end{definition}\n\n\\begin{definition}\n\\label{Defn:3-BCAchievabilityAndCapacity}\nA rate-cost quadruple $(R_{1},R_{2},R_{3},\\tau)\\in [0,\\infty)^{4}$ is\nachievable if for every $\\eta > 0$, there exists $N(\\eta)\\in \\naturals$ such that for all\n$n > N(\\eta)$, there exists a 3-DBC code $(n, \\underlineCardinalityMessageSet^{(n)},\ne^{(n)},\\underlined^{(n)})$ such that (i)\n$\\frac{\\log_{2}\\mathscr{M}_{k}^{(n)}}{n} \\geq R_{k}-\\eta:k=1,2,3$, (ii)\n$\\bar{\\xi}(e^{(n)},\\underlined^{(n)})\n\\leq \\eta$, and (iii) average cost $\\tau(e^{(n)}) \\leq \\tau+\\eta$. The capacity region is\n$\\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau) \\define \\cl{\\left\\{ \\underlineR \\in \\reals^{3}:\n(\\underlineR,\\tau)\\mbox{ is achievable} \\right\\}}$.\\begin{comment}{\nA sequence of 3-DBC codes $(n, \\underlineCardinalityMessageSet^{(n)},\ne^{(n)},\\underlined^{(n)}):n \\geq 1$ achieves $(\\underlineR,\\tau)$ if\n$\\lim_{n \\rightarrow \\infty } \\frac{\\log_{2}\\mathscr{M}_{k}^{(n)}}{n} \\geq\nR_{k}:k=1,2,3$ and $\\lim_{n \\rightarrow\n\\infty }\\bar{\\xi}(e^{(n)},\\underlined^{(n)}) = 0$ and $\\lim_{n \\rightarrow\n\\infty } \\tau(e^{(n)}) \\leq \\tau$.}\\end{comment}\n\\end{definition}\n\nThe currently known largest achievable rate region for 3-DBC is obtained by a natural extension of Marton's coding technique proposed for the 2-DBC. We begin with a brief review of Marton's coding technique for the 2-DBC in section \\ref{SubSec:MartonRateRegionFor2BC} and state it's natural extension to a general 3-DBC in section \\ref{SubSec:NaturalExtensionOfMartonTo3BC}.\n\n\\subsection{Marton's rate region}\n\\label{SubSec:MartonRateRegionFor2BC}\n\\begin{comment}{\nWe begin with a brief review of Marton's coding technique followed by a characterization of the corresponding achievable rate region.\n}\\end{comment}\nMarton's coding incorporates two fundamental coding techniques - superposition and precoding. Superposition involves each user decode a part of the signal carrying the other user's information and thereby enhance it's ability to decode the intended signals. The technique of jointly choosing each user's message bearing signal to contain mutual interference is precoding. Superposition coding is accomplished using a two layer coding scheme. First layer, which is public, contains a codebook over $\\PublicRVSet$. Second layer is private and contains two codebooks one each on $\\PrivateRVSet_{1}$ and $\\PrivateRVSet_{2}$. Precoding is accomplished by setting aside a \\textit{bin} of codewords for each private message, thus enabling the encoder to choose a compatible pair of codewords in the indexed bins. User $j$th message is split into two parts - public and private. The public parts together index a codeword in $\\PublicRVSet-$codebook and the private part of user $j$th message index a codeword in $\\PrivateRVSet_{j}-$codebook. Both users decode from the public codebook and their respective private codebooks.\n \nDefinition \\ref{Defn:MartonTestChannels} and theorem \\ref{Thm:MartonRateRegionFor2-BC}\nprovide a characterization of rate pairs achievable using Marton's coding technique.\nWe omit restating the definitions analogous to definitions \\ref{Defn:3-BCCode},\n\\ref{Defn:BCCodeErrorProbability}, \\ref{Defn:3-BCAchievabilityAndCapacity} for a 2-BC.\n\n\\begin{definition}\n \\label{Defn:MartonTestChannels}\nLet $\\SetOfDistributions_{M}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of distributions\n$p_{\\TimeSharingRV\\PublicRV \\PrivateRV_{1} \\PrivateRV_{2}XY_{1}Y_{2}}$ defined on $\\TimeSharingRVSet \\times \\PublicRVSet \\times\n\\PrivateRVSet_{1} \\times \\PrivateRVSet_{2} \\times \\InputAlphabet \\times\n\\OutputAlphabet_{1} \\times \\OutputAlphabet_{2}$, where (i) $\\TimeSharingRVSet$, $\\PublicRVSet$,\n$\\PrivateRVSet_{1}$ and $\\PrivateRVSet_{2}$ are finite sets of cardinality at most\n$|\\InputAlphabet|+4$, $|\\InputAlphabet|+4$, $|\\InputAlphabet|+1$ and $|\\InputAlphabet|+1$ respectively, (ii)\n$p_{\\underlineY|X \\underlinePrivateRV \\PublicRV \\TimeSharingRV}=p_{\\underlineY|X} = W_{\\underlineY|X}$,\n(iii) $\\Expectation\\left\\{\\kappa(X)\\right\\} \\leq \\tau$. For $p_{\\TimeSharingRV\\PublicRV\n\\underlinePrivateRV X \\underlineY} \\in \\SetOfDistributions_{M}(W_{\\underlineY|X},\\kappa,\\tau)$,\nlet $\\alpha_{M}(p_{\\TimeSharingRV\\PublicRV\n\\underlinePrivateRV X \\underlineY})$ denote the set of $(R_{1},R_{2}) \\in \\reals^{2}$ that satisfy\n\\begin{comment}{\n\\begin{flalign}\n \\label{Eqn:MartonAchievableRateRegionForParticularTestChannel}\n0 \\leq R_{k} &\\leq I(\\PublicRV\n\\PrivateRV_{k};Y_{k}|\\TimeSharingRV):k=1,2,\\nonumber\\\\ R_{1}+R_{2} &\\leq\\! I(W \\PrivateRV_{1};Y_{1}|\\TimeSharingRV)+\nI(\\PrivateRV_{2};Y_{2}|\\PublicRV,\\TimeSharingRV)\\!-\\!I(\\PrivateRV_{1};\\PrivateRV_{2}|\\PublicRV,\\TimeSharingRV) \\nonumber\\\\\nR_{1}+R_{2} &\\leq\\! I(\\PrivateRV_{1};Y_{1}|W,\\TimeSharingRV)+\nI(\\PublicRV\\PrivateRV_{2};Y_{2}|\\TimeSharingRV)\\!-\\!I(\\PrivateRV_{1};\\PrivateRV_{2}|\\PublicRV,\\TimeSharingRV)\n\\nonumber\n\\end{flalign}\n}\\end{comment}\n\\begin{flalign}\n \\label{Eqn:MartonAchievableRateRegionForParticularTestChannel}\n0 \\leq R_{k} &\\leq I(\\PublicRV\n\\PrivateRV_{k};Y_{k}|\\TimeSharingRV):k=1,2,\\nonumber\\\\ \nR_{1}+R_{2} &\\leq\\! \\min\\left\\{ I(W;Y_{1}|\\TimeSharingRV),I(W;Y_{2}|\\TimeSharingRV) \\right\\}+I(\\PrivateRV_{1};Y_{1}|\\TimeSharingRV W )+\nI(\\PrivateRV_{2};Y_{2}|\\PublicRV,\\TimeSharingRV)\\!-\\!I(\\PrivateRV_{1};\\PrivateRV_{2}|\\PublicRV,\\TimeSharingRV) \\nonumber\n\\end{flalign}\nand\n\\begin{equation}\n \\label{Eqn:MartonRateRegionAsAUnionOfTestChannelRateRegions}\n \\alpha_{M}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\TimeSharingRV\\PublicRV\n\\underlinePrivateRV X \\underlineY} \\\\\\in \\SetOfDistributions_{M}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\alpha_{M}(p_{\\TimeSharingRV\\PublicRV\n\\underlinePrivateRV X \\underlineY})\\right)\\nonumber\n\\end{equation}\n\\end{definition}\n\n\\begin{thm}\n \\label{Thm:MartonRateRegionFor2-BC}\nFor 2-DBC $(\\InputAlphabet,\\underlineSetY,W_{\\underlineY|X},\\kappa)$,\n$\\alpha(W_{\\underlineY|X},\\kappa,\\tau)$ is achievable, i.e., $\\alpha(W_{\\underlineY|X},\\kappa,\\tau) \\subseteq \\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau)$.\n\\end{thm}\n\\begin{remark}\n \\label{Rem:CardinalityBoundsDerivedByGohariAndAnantharam}\n The bounds on cardinality of $\\PublicRVSet,\\PrivateRVSet_{1}$ and $\\PrivateRVSet_{2}$\nwere derived by Gohari and Anantharam in \\cite{201202TIT_GohAna}.\n\\end{remark}\nWe refer the reader to \\cite{197905TIT_Mar} for a proof of achievability. El Gamal and Meulen\n\\cite{198101TIT_GamMeu} provide a simplified proof using the method of second moment.\\begin{comment}{ We make a\ncouple of observations that lead us to an alternate view of the coding technique. This\nwill motivate the use of structured codes in deriving a larger rate region.\n\nThe encoder splits each message into two parts - public and private. The public parts are\nused to index the codebook over $\\PublicRVSet$. Each user decodes both public parts and\nit's private part. We make a couple of observations. If we let $\\bar{V_{j}} \\define\n(W,V_{j})$ to be the pair of random variables decoded by user $j$, then $\\PublicRV$ is\nthe common part \\cite{1972MMPCT_GacKor} of $\\bar{V_{1}},\\bar{V_{2}}$. \\textit{When the\ndecoded random variables share a common part, this common information is best encoded\nand decoded in a separate layer.} Since the common part of the pair\n$\\bar{V}_{1},\\bar{V}_{2}$ is identified through univariate functions\n$f_{j}:\\PublicRVSet \\times \\PrivateRVSet_{j} \\rightarrow \\PublicRVSet:j=1,2$ such that\n$P(W=f_{1}(\\bar{V}_{1})=f_{2}(\\bar{V}_{2}))=1$, \\textit{each user decodes a function of\nthe random variable decoded by the other user.} The latter observation is central to\nMarton's coding technique. Indeed, Nair \\cite{} has proved the strategy of decoding a\nunivariate function of the other user's codeword subsumes that of Marton. It is natural\nto expect these observations to carry over for the natural extension of Marton to the\n3-BC.}\\end{comment}\n\\subsection{Natural extension of Marton's rate region for the 3-BC}\n\\label{SubSec:NaturalExtensionOfMartonTo3BC}\nThe natural extension to the case of 3 users will involve a 3 layer coding scheme. For simplicity, we describe the coding technique without referring to the time sharing random variable and employ the same in describing the achievable rate region. User $j$th message $M_{j}$ is split into four parts - two semi-private parts, and one, private and public parts each. We let message (i) $M^{W}_{j} \\in \\MessageSetM^{W}_{j}$ of rate $K_{j}$ denote it's public part (ii) $M^{U}_{i\\msout{j}} \\in \\MessageSetM^{U}_{i\\msout{j}},M^{U}_{\\msout{j}k} \\in \\MessageSetM^{U}_{\\msout{j}k}$ of rates $L_{ij},K_{jk}$ respectively, denote it's semi-private parts, where $(i,j,k)$ is an appropriate triple in $\\left\\{(1,2,3),(2,3,1),(3,1,2) \\right\\}$, and (iii) $M^{V}_{j} \\in \\MessageSetM^{V}_{j}$ of rate $T_{j}$ denote it's private part.\n\nThe first layer is public with a single codebook $(w^{n}(\\underline{m}^{W}):\\underline{m}^{W} \\in \\underlineMessageSetM^{W})$ of rate $K_{1}+K_{2}+K_{3}$ over $\\PublicRVSet$. $\\underline{M}^{W}$ indexes a codeword in $\\PublicRVSet-$codebook and each user decodes from\n$\\PublicRVSet-$codebook. \n\nEach codeword in $\\PublicRVSet-$codebook is linked to a triple of codebooks - one each on $\\SemiPrivateRVSet_{ij}:(i,j) \\in \\left\\{ (1,2),(2,3),(3,1) \\right\\}$- in the second layer. The second layer is semi-private. Each of the three semi-private codebooks is composed of \\textit{bins}, wherein each bin comprises a collection of codewords. For each pair $(i,j) \\in \\left\\{ (1,2),(2,3),(3,1) \\right\\}$ the following hold. $M^{U}_{i\\msout{j}}$ and $M^{U}_{\\msout{i}j}$ together index a bin in $\\SemiPrivateRVSet_{ij}-$codebook. Each bin in $\\SemiPrivateRVSet_{ij}-$codebook is of rate $S_{ij}$. Let $(u_{ij}^{n}(\\underlinem^{W},m_{\\msout{i}j}^{U},m_{i\\msout{j}}^{U},s_{ij}):s_{ij} \\in [\\exp\\{ nS_{ij} \\}])$ denote bin corresponding to semi-private messages $\\underline{m^{U}_{ij}}\\define (m_{\\msout{i}j}^{U},m_{i\\msout{j}}^{U}) \\comment{ \\in \\underline{\\MessageSetM_{ij}}^{U}}$ in the $\\SemiPrivateRVSet_{ij}-$codebook linked to public message $\\underlinem^{W}$. Users $i,j$ decode from $\\SemiPrivateRVSet_{ij}-$codebook and it maybe verified that $\\SemiPrivateRVSet_{ij}-$codebook is of rate $K_{ij}+L_{ij}+S_{ij}$.\n\nLet $(i,j)$ and $(j,k)$ be distinct pairs in $\\left\\{ (1,2),(2,3),(3,1) \\right\\}$. Every pair of codewords in $\\SemiPrivateRVSet_{ij}-$ and $\\SemiPrivateRVSet_{jk}-$codebooks is linked to a codebook on $\\PrivateRVSet_{j}$. The codebooks over $\\PrivateRVSet_{j}:j=1,2,3$ comprise the third layer which is private. $M_{j}^{V}$ indexes a bin in $\\PrivateRVSet_{j}-$codebook, each of which is of rate $S_{j}$, and thus $\\PrivateRVSet_{j}-$codebook is of rate $T_{j}+S_{j}$. Let $(v_{j}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij},\\underline{m^{U}_{jk}},s_{jk},m_{j}^{V},s_{j}):s_{j} \\in [\\exp\\{ nS_{j} \\}])$ denote bin corresponding to private message $m_{j}^{V}$ in the $\\PrivateRVSet_{j}-$codebook linked to codeword pair $(u_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}),u_{jk}^{n}(\\underlinem^{W},\\underline{m^{U}_{jk}},s_{jk}))$. User $j$ decodes from the private codebook over $\\PrivateRVSet_{j}$.\n\nHow does the encoder map messages to a codeword?\nLet $p_{W\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV}$ be a distribution on\n$\\PublicRVSet \\times \\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet \\times \\InputAlphabet$ such that $\\Expectation\\left\\{ \\kappa(\\InputRV) \\right\\} \\leq \\tau$. The encoder looks for $(s_{12},s_{23},s_{31},s_{1},s_{2},s_{3})$ such that the septuple\n\\begin{eqnarray}\n \\label{Eqn:JointlyTypicalSeptupleOfCodewords}\n\\left(\\substack{w^{n}(\\underlineM^{W}),u_{ij}^{n}(\\underlineM^{W},\\underline{M^{U}_{ij}},s_{ij}):(i,j)=(1,2),(2,3),(3,1),\\\\v_{j}^{n}(\\underlineM^{W},\\underline{M^{U}_{ij}},s_{ij},\\underline{M^{U}_{jk}},s_{jk},M_{j}^{V},s_{j}):(i,j,k)=(1,2,3),(2,3,1),(3,1,2)}\\right)\\nonumber\n\\end{eqnarray}\nof codewords is jointly typical with respect to $p_{W\\underlineSemiPrivateRV \\underlinePrivateRV}$. If such a septuple is found, this is mapped to a codeword on $\\InputAlphabet^{n}$ which is input to the channel. If it does not find any such septuple, an error is declared. \n\nDecoder $j$ looks for all quadruples $(\\underline{\\hatm}^{W},\\underline{\\hatm_{ij}}^{U},\\underline{\\hatm_{jk}}^{U},\\hatm_{j}^{V})$ such that\n\\begin{equation}\n \\label{Eqn:DecodingRuleOfDecoderj}\n\\left( w^{n}(\\underline{\\hatm}^{W}),u_{ij}^{n}(\\underline{\\hatm}^{W},\\underline{\\hatm^{U}_{ij}},s_{ij}),u_{jk}^{n}(\\underline{\\hatm}^{W},\\underline{\\hatm^{U}_{jk}},s_{jk}),v_{j}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij},\\underline{m^{U}_{jk}},s_{jk},m_{j}^{V},s_{j}),Y_{j}^{n} \\right) \\nonumber\n\\end{equation}\nis jointly typical with respect to $p_{W\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV\\underlineY}\\define p_{W\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV}W_{\\underline{Y}|X}$, where (i) $(i,j,k)$ is the appropriate triple in $\\left\\{ (1,2,3),(2,3,1),(3,1,2) \\right\\}$ and (ii) $Y_{j}^{n}$ is the received vector. If there is a unique such quadruple, it declares $\\hatm_{j} \\define (\\hatm_{j}^{W},\\hatm_{i\\msout{j}}^{U},\\hatm_{\\msout{j}k}^{U},\\hatm^{V}_{j})$ as user $j$th message. Otherwise, i.e., none or more than one such quadruple is found, it declares an error.\n\nAs is typical in information theory, we average error probability over the entire ensemble of codebooks and upper bound the same. Moreover, we incorporate the time sharing random variable in the above coding technique using the standard approach. Let $\\TimeSharingRV$, taking values over the finite alphabet $\\TimeSharingRVSet$, denote the time sharing random variable. Let $p_{\\TimeSharingRV}$ be a pmf on $\\TimeSharingRVSet$ and $q^{n} \\in \\TimeSharingRVSet^{n}$ denote a sequence picked according to $\\prod_{t=1}^{n}p_{Q}$. $q^{n}$ is revealed to the encoder and all decoders. The codewords in $\\PublicRVSet-$codebook are identically and independently distributed according to $\\prod_{t=1}^{n}p_{W|Q}(\\cdot|q_{t})$. Conditioned on entire public codebook $(W^{n}(\\underline{m}^{W})=w^{n}(\\underline{m}^{W}):\\underline{m}^{W} \\in \\underlineMessageSetM^{W})$ and the time sharing sequence $q^{n}$, each of the codewords $U_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}):(\\underline{m^{U}_{ij}},s_{ij}) \\in \\underline{\\MessageSetM_{ij}}^{U} \\times [\\exp\\{ nS_{ij} \\}]$ are independent and identically distributed according to $\\prod_{t=1}^{n}p_{U_{ij}|WQ}(\\cdot|w_{t}(\\underline{m}^{W}),q_{t})$. Conditioned on a realization of the entire collection of public and semi-private codebooks, the private codewords\n\\comment{\\begin{eqnarray}\n\\left(\\substack{W^{n}(\\underline{m}^{W}),U_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}),\\\\U_{jk}^{n}(\\underlinem^{W},\\underline{m^{U}_{jk}},s_{jk}),U_{ki}^{n}(\\underlinem^{W},\\underline{m^{U}_{ki}},s_{ki})}\\right)=\\nonumber\\\\\\left(\\substack{w^{n}(\\underline{m}^{W}),u_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}),\\\\u_{jk}^{n}(\\underlinem^{W},\\underline{m^{U}_{jk}},s_{jk}),u_{ki}^{n}(\\underlinem^{W},\\underline{m^{U}_{ki}},s_{ki})}\\right):\\left(\\substack{\\underlinem^{W},\\underlinem^{U}\\\\s_{ij},s_{jk},s_{ki}}\\right) \\in \\substack{\\underline{\\MessageSetM}^{W} \\times \\underline{\\MessageSetM}^{U}},\\nonumber \\end{eqnarray}\neach of the codewords }$(V_{j}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij},\\underline{m^{U}_{jk}},s_{jk},m_{j}^{V},s_{j}):s_{j} \\in [\\exp\\{ nS_{j} \\}])$ are independent and identically distributed according to\n\\begin{eqnarray}\n \\label{Eqn:DistributionOfVCodebookConditionedOnPublicAndSemiPrivate}\n\\prod_{t=1}^{n}p_{V_{j}|U_{ij}U_{jk}WQ}\\left(\\cdot|w_{t}(\\underline{m}^{W}),(u_{ij}^{n}(\\underlinem^{W},\\underline{m^{U}_{ij}},s_{ij}))_{t},(u_{jk}^{n}(\\underlinem^{W},\\underline{m^{U}_{jk}},s_{jk}))_{t},q_{t}\\right).\\nonumber\n\\end{eqnarray}\nWe now average error probability over the ensemble of codebooks. An upper bound on the error event at the encoder is derived using the method of second moment \\cite{198101TIT_GamMeu}. The probability of the error event at the encoder decays exponentially with $n$ if for each triple $(i,j,k) \\in \\left\\{ (1,2,3),(2,3,1),(3,1,2)\\right\\} $\n\\begin{eqnarray}\n \\label{Eqn:3BCSourceCodingBoundNonnegativity}\nS_{i}\\!\\!\\!\\!&>&\\!\\!\\!\\! 0\\\\\n\\label{Eqn:3BCSourceCodingPairwiseBound}\nS_{ij}+S_{jk}\\!\\!\\!\\! &>& \\!\\!\\!\\!I(U_{ij};U_{jk}|WQ) \\\\\n\\label{Eqn:3BCSourceCodingTripleBound}\nS_{ij}+S_{jk}+S_{ki} \\!\\!\\!\\!&>&\\!\\!\\!\\! I(U_{ij};U_{jk};U_{ki}|WQ)\\footnotemark\\\\\n\\label{Eqn:3BCSourceCodingQuadrapleBound}\nS_{i}+S_{ij}+S_{jk}+S_{ki}\\!\\!\\!\\!&>&\\!\\!\\!\\! I(U_{ij};U_{jk} ;U_{ki}|WQ)\n +I(V_{i};U_{jk}|U_{ij},U_{ki},WQ)\\\\\nS_{i}+S_{j}+S_{ij}+S_{jk}+S_{ki} \\!\\!\\!\\!&>& \\!\\!\\!\\!I(V_{i};U_{jk}|U_{ij},U_{ki},WQ)+I(V_{j};U_{ki}|U_{ij},U_{jk},WQ)\\nonumber\\\\\n\\label{Eqn:3BCSourceCodingPentaBound}\n&&\\!\\!\\!\\!+I(U_{ij};U_{jk};U_{ki}|WQ)+I(V_{i};V_{j}|U_{jk},U_{ij},U_{ki},WQ)\\\\\nS_{1}+S_{2}+S_{3}+S_{12}+S_{23}+S_{31} \\!\\!\\!\\!&>& \\!\\!\\!\\!I(V_{1};U_{23}|U_{12},U_{31},WQ)+I(V_{2};U_{31}|U_{12},U_{23},WQ)+I(V_{1};V_{2};V_{3}|\\TimeSharingRV \\PublicRV \\underlineSemiPrivateRV)\\nonumber\\\\\n\\label{Eqn:3BCSourceCodingSextupleBound}\n&&\\!\\!\\!\\!+I(U_{12};U_{23};U_{31}|WQ)+I(V_{3};U_{12}|U_{23},U_{31},WQ).\n\\end{eqnarray}\n\\footnotetext{For three random variables, $A,B,C, I(A;B;C)=I(A;B)+I(AB;C)$.}\nThe probability of error event at the decoders decays exponentially if for each triple $(i,j,k) \\in \\left\\{ (1,2,3),(2,3,1),(3,1,2)\\right\\} $\n\\begin{flalign}\n\\label{Eqn:3BCChannelCodingSingleBound}\nI(V_i;Y_i|QWU_{ij}U_{ki}) &\\geq T_i+S_i \\\\\n\\label{Eqn:3BCChannelCodingDoubleBound}\nI(U_{ij}V_i;Y_i|QWU_{ki})+I(U_{ij};U_{ki}|QW) &\\geq\nK_{ij}+L_{ij}+S_{ij}+T_i+S_i \\\\\n\\label{Eqn:3BCChannelCodingDoubleSecondBound}\nI(U_{ki}V_i;Y_i|QWU_{ij})+I(U_{ij};U_{ki}|QW) &\\geq\nK_{ki}+L_{ki}+S_{ki}+T_i+S_i \\\\\n\\label{Eqn:3BCChannelCodingTripleBound}\nI(U_{ij}U_{ki}V_i;Y_i|QW)+I(U_{ij};U_{ki}|QW) &\\geq\nK_{ij}+L_{ij}+S_{ij}+K_{ki}+L_{ki}+S_{ki}+T_i+S_i \\\\ \n\\label{Eqn:3BCChannelCodingSextupleBound}\nI(WU_{ij}U_{ki}V_i;Y_i|Q)+I(U_{ij};U_{ki}|QW) &\\geq\nK_i+K_j+K_k+K_{ij}+L_{ij}+S_{ij}+K_{ki}+L_{ki}+S_{ki}+T_i+S_i\n\\end{flalign}\n\\begin{comment}{\\begin{flalign}\n\\label{Eqn:3BCChannelCodingSingleBound}\nI(V_i;Y_i|QWU_{ij}U_{ki}) &\\geq \\bold{T_i} \\\\\n\\label{Eqn:3BCChannelCodingDoubleBound}\nI(U_{ij}V_i;Y_i|QWU_{ki})+I(U_{ij};U_{ki}|QW) &\\geq\n\\bold{K_{ij}}+\\bold{T_i} \\\\\n\\label{Eqn:3BCChannelCodingDoubleSecondBound}\nI(U_{ki}V_i;Y_i|QWU_{ij})+I(U_{ij};U_{ki}|QW) \\geq&\n\\bold{K_{ki}}+\\bold{T_i} \\\\\n\\label{Eqn:3BCChannelCodingTripleBound}\nI(U_{ij}U_{ki}V_i;Y_i|QW)+I(U_{ij};U_{ki}|QW) \\geq\n\\bold{K_{ij}}&+\\bold{K_{ki}}+\\bold{T_i} \\\\ \nI(WU_{ij}U_{ki}V_i;Y_i|Q)+I(U_{ij};U_{ki}|QW) &\\geq\nK_i+K_j+K_k\\nonumber\\\\\n\\label{Eqn:3BCChannelCodingSextupleBound}+\\bold{K_{ij}}+\\bold{K_{ki}}+&\\bold{T_i},\n\\end{flalign}\n}\\end{comment}\nFor each pmf $p_{\\TimeSharingRV W\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV}W_{\\underlineY|\\InputRV}$ defined on $\\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet \\times \\InputAlphabet \\times \\underline{\\OutputAlphabet}$, let $\\alpha_{NEM}(p_{QW\\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV\\underlineY})$ denote the set of all triples $(R_{1},R_{2},R_{3}) \\in [0,\\infty)^{4}$ such that (i) there exists non-negative real numbers $K_{ij},L_{ij},S_{ij},K_{j},T_{j},S_{j}$ that satisfies (\\ref{Eqn:3BCSourceCodingBoundNonnegativity})-(\\ref{Eqn:3BCChannelCodingSextupleBound}) for each pair $(i,j) \\in \\left\\{ (1,2),(2,3),(3,1) \\right\\}$ and (ii) $R_{j}=T_{j}+K_{jk}+L_{ij}+K_{j}$ for each triple $(i,j,k) \\in \\left\\{ (1,2,3),(2,3,1),(3,1,2)\\right\\} $. Natural extension of Marton's rate region for the 3-DBC with private messages is\n\\begin{equation}\n \\label{Eqn:3BCMartonRateRegionAsAUnionOfTestChannelRateRegions}\n \\alpha_{NEM}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\TimeSharingRV\\PublicRV \\underlineSemiPrivateRV\n\\underlinePrivateRV X \\underlineY} \\\\\\in \\SetOfDistributions_{NEM}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\alpha_{NEM}(p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\n\\underlinePrivateRV X \\underlineY})\\right),\\nonumber\n\\end{equation}\nwhere $\\SetOfDistributions_{NEM}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of distributions\n$p_{\\TimeSharingRV\\PublicRV \\underlineSemiPrivateRV \\underlinePrivateRV \\InputRV \\underlineY}$ defined on $\\TimeSharingRVSet \\times \\PublicRVSet \\times\n\\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet \\times \\InputAlphabet \\times\n\\underlineSetY$, where (i) $\\TimeSharingRVSet,\\PublicRVSet,\n\\underlineSemiPrivateRVSet, \\underlinePrivateRVSet$ are finite sets, (ii)\n$p_{\\underlineY|X \\underlinePrivateRV \\underlineSemiPrivateRV \\PublicRV\\TimeSharingRV}=p_{\\underlineY|X} = W_{\\underlineY|X}$,\n(iii) $\\Expectation\\left\\{\\kappa(X)\\right\\} \\leq \\tau$.\n\\section{\\3To1BC broadcast channels and the need for structured codes}\n\\label{Sec:3To1BCAndTheNeedForStructuredCodes}\n\nIn this section, we define \\3To1BC DBC, a class of BCs for which a natural extension of Marton's coding technique is strictly suboptimal. In particular, in section \\ref{SubSec:TheThreeUserBroadcastChannel} we present a novel 3-DBC for which the natural extension of Marton's coding technique is strictly suboptimal. The class of \\3To1BC DBCs, defined in section \\ref{SubSec:ThreeToOneBroadcastChannels}, is obtained by a natural generalization of this example. \\3To1BC DBCs also being the simplest class of DBCs for which the limitations of traditional unstructured independent codes and the need for structure can be easily demonstrated provides a pedagogical step for describing our findings. We begin by a discussion on the limitations of traditional unstructured independent codes.\n\n\\subsection{Limitations of unstructured codes}\n\\label{SubSec:LimitationsOfUnstructuredCodes}\nThe central aspect of any coding technique designed for a BC is \\textit{interference management}. Marton's coding incorporates two fundamental coding strategies for interference management - superposition and binning. In this discussion, we restrict attention to the strategy of superposition in Marton's coding. Transmission of information bearing signal of each user causes interference to the other user. As against to being completely oblivious to the other user's transmissions and therefore treating it as noise, superposition enables each user to decode a component of the other user's signal. Traditional unstructured independent codebooks facilitates this by building a two layer - cloud center and satellite - code. The cloud center code contains components of each user's signal that is decoded by the other user. Over a degraded BC, for which superposition is optimal, the more capable receiver decodes the entire signal carrying the less capable user's information.\n\nOver a general non-degraded channel what does it mean to say a \\textit{component} of the other users' signal? The coding technique as proposed by Marton does not provide an obvious answer to this question. However, a recent interpretation by Nair \\cite{2009MMICWCSP_Nai} makes this quite explicit. In particular, Nair proves superposition in Marton's coding is tantamount to each user decoding a (univariate) function of the signal carrying the other users' information. Due to the presence of just one interferer in a 2-BC this is a reasonable strategy.\n\nHow does one tackle interference in a 3-BC? The signal input to this channel is a function of three signals, one intended for each user. \\textit{The reception at each receiver is therefore plagued, in general, by a bivariate function of signals intended for the other users. It is therefore natural to enable each user decode the relevant bivariate interfering component, in addition to a univariate component of the other two user's signal.} Does the natural extension of Marton's coding `truly' decode a bivariate interference component of the other users' signals and if yes, how? Traditional unstructured independent coding, on which Marton's technique is based does not enable decoding a bivariate component of signals without decoding the arguments in their entirety. The latter strategy is in general inefficient. Before we provide a formal proof of this claim in section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, we lend credence by referring to the gamut of related problems.\n\nAptly describing this phenomenon, the problem of reconstructing modulo-2 sum of distributed binary sources \\cite{197903TIT_KorMar} brings to light the limitations of traditional independent unstructured coding. Even after three decades, we are unaware of a coding technique based on traditional unstructured independent coding that achieves rates promised by K\\\"orner and Marton. More recently, other problem instances \\cite{200906TIT_PhiZam}, \\cite{200710TIT_NazGas}, \\cite{201103TIT_KriPra}, \\cite{200809Allerton_SriJafVisJafSha},\\cite{201207ISIT_PadSahPra} have been identified, wherein the need to decode a bivariate function has been efficiently met by employing structured codebooks. While all of the above works propose ingenious coding techniques based on structured codes, the example studied in \\cite[Section V]{201207ISIT_PadSahPra} closely illustrates the role played by structured codes in decoding bivariate interference component for the problem in context.\\footnote{That all of the above works are centered around an additive problem - reconstruction of modulo-2 sum of binary sources or communicating over additive channels corroborates the phenomenon under question. Decoding a bivariate function without decoding it's arguments is more efficient than decoding the individual arguments if the bivariate function is compressive, i.e., the entropy of the function is significantly lower when compared to the joint entropy of the arguments. Indeed, the simplest such bivariate function of binary sources is an addition. Furthermore, a good understanding of linear codes that facilitated decoding the sum has constrained much of the attention thus far to additive problems. It is our belief that these examples, being only the tip of an iceberg, point to more efficient encoding and decoding techniques for multi-terminal communication problems based on structured codebooks.}\n\nHow do structured codes enable decode the bivariate interference component more efficiently? This is best described by the use of linear codes in decoding the sum interference component. Let us assume that codebooks of users $2$ and $3$ are built over the the binary field $\\BinaryField$ and the sum of user $2$ and $3$ codewords is the interference component at receiver $1$. If user $2$ and $3$ build independent codebooks of rate $R_{2}$ and $R_{3}$ respectively, the range of interference patterns has rate $R_{2}+R_{3}$. Enabling user $1$ decode the sum interference pattern constrains the sum $R_{2}+R_{3}$. Instead if codebooks of users $2$ and $3$ are sub-codes of a common linear code, then enabling user $1$ to decode the sum will only constrain $\\max\\left\\{ R_{2},R_{3}\\right\\}$.\n\n\\subsection{A three user broadcast channel}\n\\label{SubSec:TheThreeUserBroadcastChannel}\nIn this section, we present a novel 3-DBC that makes concrete our remarks in section \\ref{SubSec:LimitationsOfUnstructuredCodes}. In section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, we prove strict sub-optimality of Marton's coding technique for the 3-DBC presented herein. The novelty of this 3-DBC lies in our ability to prove strict containment of Marton's rate region even without a compact description of the latter. \n\\begin{example}\n \\label{Ex:3-BCExample}\nConsider the 3-DBC depicted in figure \\ref{Fig:TheThreeUserBroadcastChannel}. Let the input alphabet $\\InputAlphabet = \\InputAlphabet_{1} \\times \\InputAlphabet_{2} \\times \\InputAlphabet_{3}$ be a triple Cartesian product of the binary field $\\InputAlphabet_{1}=\\InputAlphabet_{2} = \\InputAlphabet_{3} = \\BinaryField$ and the output alphabets $\\OutputAlphabet_{1}=\\OutputAlphabet_{2}=\\OutputAlphabet_{3}=\\BinaryField$ be binary fields. If $X=X_{1}X_{2}X_{3}$ denote the three binary digits input to the channel, then the outputs are $Y_{1}=X_{1}\\oplus X_{2}\\oplus X_{3} \\oplus N_{1}$, $Y_{2} =X_{2} \\oplus N_{2}$ and $Y_{3} = X_{3} \\oplus N_{3}$, where (i) $N_{1}, N_{2}, N_{3}$ are independent binary random variables with $P(N_{j}=1)=\\delta_{j} \\in (0,\\frac{1}{2})$ and (ii) $(N_{1},N_{2},N_{3})$ is independent of the input $X$. The binary digit $X_{1}$ is constrained to an average Hamming weight of $\\tau \\in (0,\\frac{1}{2})$. In other words, $\\kappa (x_{1}x_{2}x_{3}) = 1_{\\left\\{x_{1}=1\\right\\}}$ and the average cost of input is constrained to $\\tau\\in (0,\\frac{1}{2})$. For the sake of clarity, we provide a formal description of this channel in terms of section \\ref{SubSec:DefinitionsBroadcastChannelCodeAchievabilityCapacity}. This 3-DBC maybe referred to as $(\\InputAlphabet,\\underlineSetY,W_{\\underlineY|X},\\kappa)$ where $\\InputAlphabet \\define \\left\\{ 0,1 \\right\\}\\times \\left\\{ 0,1 \\right\\} \\times \\left\\{ 0,1 \\right\\}, \\OutputAlphabet_{1}=\\OutputAlphabet_{2}=\\OutputAlphabet_{3}=\\left\\{ 0,1 \\right\\}, W_{\\underlineY|X}(y_{1},y_{2},y_{3}|x_{1}x_{2}x_{3})=BSC_{\\delta_{1}}(y_{1}|x_{1}\\oplus x_{2}\\oplus x_{3})BSC_{\\delta_{2}}(y_{2}|x_{2})BSC_{\\delta_{3}}(y_{3}|x_{3})$, where $\\delta_{j} \\in (0,\\frac{1}{2}):j=1,2,3$, $BSC_{\\eta}(1|0)=BSC_{\\eta}(0|1)=1-BSC_{\\eta}(0|0)=1-BSC_{\\eta}(1|1)=\\eta$ for any $\\eta \\in (0,\\frac{1}{2})$ and the cost function $\\kappa (x_{1}x_{2}x_{3}) = 1_{\\left\\{x_{1}=1\\right\\}}$.\n\\end{example}\n\\begin{figure}\n\\centering\n\\includegraphics[height=1.8in,width=1.9in]{binex}\n\\caption{A 3-DBC with octonary input and binary outputs described in example \\ref{Ex:3-BCExample}.}\n\\label{Fig:TheThreeUserBroadcastChannel}\n\\end{figure}\nWe begin with some observations for the above channel. Users $2$ and $3$ see \\textit{interference free point to point} links from the input. It is therefore possible to communicate to them simultaneously at their point to point capacities using any point to point channel codes achieving their respective capacities. For the purpose of this discussion, let us assume $\\delta \\define \\delta_{2}=\\delta_{3}$. This enables us employ the same capacity achieving code of rate $1-h_{b}(\\delta)$ for both users $2$ and $3$. What about user $1$? Three observations are in order. Firstly, if users $2$ and $3$ are being fed at their respective point to point capacities, then information can be pumped to user $1$ only through the first binary digit, henceforth referred to as $X_{1}$. In this case, we recognize that the sum of user $2$ and $3$'s transmissions interferes at receiver $1$. Thirdly, the first binary digit $X_{1}$ is costed, and therefore cannot cancel the interference caused by users $2$ and $3$.\n\nSince average Hamming weight of $X_{1}$ is restricted to $\\tau$, $X_{1}\\oplus N_{1}$ is restricted to an average Hamming weight of $\\tau * \\delta_{1}$. If the rates of users $2$ and $3$ are sufficiently small, receiver $1$ can attempt to decode codewords transmitted to users $2$ and $3$, cancel the interference and decode the desired codeword. This will require $2-2h_{b}(\\delta) \\leq 1- h_{b}(\\delta_{1} * \\tau)$ or equivalently $\\frac{1+h_{b}(\\delta_{1} * \\tau)}{2} \\leq h_{b}(\\delta)$. What if this were not the case?\n\nIn the case $\\frac{1+h_{b}(\\delta_{1} * \\tau)}{2} > h_{b}(\\delta)$, we are left with two choices. The first choice is to enable decoder $1$ decode as large a part of the interference as possible and precode for the rest of the uncertainty.\\footnote{Since $X_{1}$ is costed, precoding results in a rate loss, i.e., in terms of rate achieved, the technique of precoding is in general inferior to the technique of decoding interference. This motivates a preference for decoding the interference as against to precoding. However, for the Gaussian case, precoding suffers \\textit{no} rate loss. This is the precise reason for dirty paper coding being optimal for vector Gaussian BCs \\cite{200609TIT_WeiSteSha}.} The second choice is to attempt decoding the sum of user $2$ and $3$'s codewords, instead of the pair. In the sequel, we pursue the second choice using linear codes. In section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, we prove Marton's coding technique is forced to take the first choice which results in it's sub-optimality.\n\nSince linear codes achieve capacity of binary symmetric channels, there exists a single linear code, or a coset thereof, of rate $1-h_{b}(\\delta)$ that achieves capacity of both user $2$ and $3$ channels. Let us employ this linear code for communicating to users $2$ and $3$. The code being linear or affine, the collection of sums of all possible pairs of codewords is restricted to a coset of rate $1-h_{b}(\\delta)$. This suggests that decoder $1$ decode the sum of user $2$ and $3$ codewords. Indeed, if $1-h_{b}(\\delta) \\leq 1-h_{b}(\\tau * \\delta_{1})$, or equivalently $\\tau*\\delta_{1} \\leq \\delta$, then user $1$ can first decode the interference, peel it off, and then go on to decode the desired signal. Under this case, a rate $h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$ is achievable for user $1$ even while communicating independent information at rate $1-h_{b}(\\delta)$ for both users $2$ and $3$. We have therefore proposed a coding technique based on linear codes that achieves the rate triple $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta),1-h_{b}(\\delta))$ if $\\tau*\\delta_{1} \\leq \\delta=\\delta_{2}=\\delta_{3}$. \n\nLet us now consider the general case with respect to $\\delta_{2}, \\delta_{3}$. Without loss of generality we may assume $\\delta_{2} \\leq \\delta_{3}$. We employ a capacity achieving linear code to communicate to user $2$. This code is sub sampled (uniformly and randomly) to yield a capacity achieving code for user $3$. This construction ensures the sum of all pairs of user $2$ and $3$ codewords to lie within user $2$'s linear code, or a coset thereof, of rate $1-h_{b}(\\delta_{2})$. If $1-h_{b}(\\delta_{2}) \\leq 1-h_{b}(\\tau*\\delta_{1})$, or equivalently $\\tau*\\delta_{1} \\leq \\delta_{2}$, then decoder $1$ can decode the sum of user $2$ and $3$'s codewords, i.e., the interfering signal, peel it off and decode the desired message at rate $h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$. If $\\delta_{3} \\leq \\delta_{2}$, then user $2$'s code is obtained by sub-sampling a capacity achieving linear code provided to user $3$. In this case, user $1$ can be fed at rate of $h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$ if $1-h_{b}(\\delta_{3}) \\leq 1-h_{b}(\\tau*\\delta_{1})$, or equivalently $\\tau*\\delta_{1} \\leq \\delta_{3}$\nThe above arguments are summarized in the following lemma.\n\n\\begin{lemma}\n \\label{Lem:RateTripleAchievableUsingLinearCodes}\nConsider the 3-DBC in example \\ref{Ex:3-BCExample}. If $\\tau*\\delta_{1} \\leq \\min\\left\\{\\delta_{2},\\delta_{3}\\right\\}$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\mathbb{C}(\\tau)$.\n\\end{lemma}\n\nIn section \\ref{Sec:StrictSubOptimalityOfMartonCodingTechnique}, we prove that if $1+h_{b}(\\delta_{1} * \\tau) > h_{b}(\\delta_{2})+h_{b}(\\delta_{3})$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\notin \\alpha_{NEM}(\\tau)$. We therefore conclude in corollary \\ref{Cor:ConditionsUnderWhichMartonCodingTechniqueIsStrictlySubOptimal} that if $\\tau,\\delta_{1},\\delta_{2},\\delta_{3}$ are such that $1+h_{b}(\\delta_{1} * \\tau) > h_{b}(\\delta_{2})+h_{b}(\\delta_{3})$ and $\\min\\left\\{ \\delta_{2},\\delta_{3}\\right\\}\\geq \\delta_{1} * \\tau$, then Marton's coding technique is strictly suboptimal for the 3-DBC presented in example \\ref{Ex:3-BCExample}. In particular, if $\\tau, \\delta_{1}, \\delta=\\delta_{2}=\\delta_{3}$ are such that $\\frac{1+h_{b}(\\delta_{1} * \\tau)}{2} > h_{b}(\\delta) \\geq h_{b}(\\delta_{1} * \\tau)$, then Marton's coding technique is strictly suboptimal for the 3-DBC presented in example \\ref{Ex:3-BCExample}. While the proof of this statement is long, the curious reader may sample our conclusion in theorem \\ref{Thm:ConditionsUnderWhichMartonCodingTechniqueDoesNotAchieveRateTriple} and corollary \\ref{Cor:ConditionsUnderWhichMartonCodingTechniqueIsStrictlySubOptimal}.\n\\subsection{\\3To1BC broadcast channels}\n\\label{SubSec:ThreeToOneBroadcastChannels}\n\nIn this section, we characterize \\3To1BC DBCs - a class of 3-DBCs for which the 3-DBC presented in section \\ref{SubSec:TheThreeUserBroadcastChannel} is an example. This charcterization resembles the notion of \\3To1BC interference channel studied in \\cite{200912arXiv_CadJaf,201009TIT_BreParTse}, \\cite{201207ISIT_PadSahPra}. Being the first known class of BCs for which Marton's coding technique is suboptimal, it's importance cannot be overemphasized. In addition, it provides a pedagogical step in deriving an achievable rate region for the 3-DBC using structured codes. Section \\ref{Sec:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes} is dedicated to deriving achievable rate regions for \\3To1BC DBC using nested coset codes \\cite{201108ISIT_PadPra}.\n\nA 3-DBC\n$(\\mathcal{X},\\underlineSetY,W_{\\underlineY|X})$\nis a 3-to-1 DBC if $\\mathcal{X}=\\mathcal{X}_{1}\\times \\mathcal{X}_{2} \\times\n\\mathcal{X}_{3}$ is a Cartesian product of three alphabet sets such that\n$W_{Y_{2}|X}(y_{2} |\n(x_{1},x_{2},x_{3}))=W_{Y_{2}|X_{2}}(y_{2}|x_{2})$ and $W_{Y_{3}|X}(y_{3} |\n(x_{1},x_{2},x_{3}))=W_{Y_{3}|X_{3}}(y_{3}|x_{3})$. Note that transition probabilities\n$W_{Y_1,Y_2,Y_3|X}$ of a 3-to-1 DBC can be denoted as\n$W_{Y_1,Y_2,Y_3|X_{1}X_{2}X_{3}}$. Since users $2$ and $3$ enjoy interference free point to point links, the corresponding receivers need to decode only signals intended for them. Therefore, there is no need for a (i) public codebook that each of the users decode into and (ii) a codebook that both users $2$ and $3$ decode into.\n\n\\section{Achievable rate regions for \\3To1BC broadcast channels using nested coset codes}\n\\label{Sec:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes}\n\nIn this section, we present the first step in deriving an achievable rate region for the general 3-DBC using nested coset codes \\cite{201108ISIT_PadPra}. In particular, we restrict attention to \\3To1BC DBCs and derive an achievable rate region for this class using the ensemble of nested coset codes. The coding technique we propose is a generalization of the simple linear coding strategy proposed for example \\ref{Ex:3-BCExample}. The reader may wish to revisit the same.\n\n\\subsection{Decoding sum of codewords using nested coset codes}\n\\label{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}\nThe essential aspect of the linear coding strategy proposed for example \\ref{Ex:3-BCExample} is that users $2$ and $3$ employ a code that is closed under addition, the linear code being the simplest such example. Since linear codes only achieve symmetric capacity, we are forced to bin codewords from a larger linear code in order to find codewords that are typical with respect to a nonuniform distribution. This is akin to binning for channels with state information, wherein $\\exp\\left\\{ nI(U;S) \\right\\}$ codewords, each picked according to $\\prod_{t=1}^{n}p_{T}$, are chosen for each message in order to find a codeword in $T_{\\delta}(U|s^{n})$ jointly typical with state sequence $s^{n}$.\n\nWe now generalize the coding technique proposed for example \\ref{Ex:3-BCExample}. Consider auxiliary alphabet sets $\\PrivateRVSet_{1},\\SemiPrivateRVSet_{21},\\SemiPrivateRVSet_{31}$ where $\\SemiPrivateRVSet_{21}=\\SemiPrivateRVSet_{31} =\\fieldq$ be the finite field of cardinality $q$ and let $p_{\\PrivateRV_{1}\\SemiPrivateRV_{21}\\SemiPrivateRV_{31} X \\underlineY}$ be a pmf on $\\PrivateRVSet_{1} \\times \\SemiPrivateRVSet_{21} \\times \\SemiPrivateRVSet_{31} \\times \\InputAlphabet \\times \\underlineSetY$.\\footnote{The seemingly queer notation employed herein is in anticipation of a generalization in section \\ref{SubSec:AnEnlargedAchievableRateRegion}. As earlier $\\PrivateRV$ stands for a private codebook and $\\SemiPrivateRV$ stands for a semi-private codebook. This will be evident from the coding technique described further.} For $j=2,3$, let $\\lambda_{j} \\subseteq \\SemiPrivateRVSet_{j1}^{n}$ be coset of a linear code $\\overline{\\lambda_{j}}\\subseteq\\fieldq^{n}$ of rate $S_{j1}+T_{j1}$. The linear codes are contained in one another, i.e., if $S_{j_{1}1}+T_{j_{1}1} \\leq S_{j_{2}2}+T_{j_{2}2}$, then $\\overline{\\lambda_{j_{1}}} \\subseteq \\overline{\\lambda_{j_{2}}}$. Codewords of $\\lambda_{j}$ are partitioned independently and uniformly into $\\exp \\left\\{ nT_{j} \\right\\}$ bins. A codebook $\\mathcal{C}_{1}$ of rate $K_{1}+L_{1}$ is built over $\\PrivateRVSet_{1}$. The codewords of $\\mathcal{C}_{1}$ are independently and uniformly partitioned into $\\exp \\left\\{ nL_{1} \\right\\}$ bins. Messages of users $1,2,3$ at rates $L_{1},T_{21},T_{31}$ is used to index a bins in $\\mathcal{C}_{1},\\lambda_{2},\\lambda_{3}$ respectively. The encoder looks for a jointly typical triple, with respect to $p_{\\PrivateRV_{1}\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}}$, of codewords in the indexed triple of bins. Following a second moment method similar to that employed in \\cite{201108ISIT_PadPra}, it can be proved that the encoder finds at least one jointly typical triple if\n\\begin{eqnarray}\n \\label{Eqn:SingleLayerSourceCodingBounds}\nS_{j1} \\geq \\log_{2} q - H(U_{j1}):j=2,3,~~~ K_{1} \\geq 0,~~~\nS_{21}+S_{31} \\geq 2\\log_{2} q - H(U_{21})-H(U_{31})+I(U_{21};U_{31})\\\\\nS_{j1}+K_{1} \\geq \\log_{2} q - H(U_{j1})+I(U_{j1};V_{1}):j=2,3,\n\\sum_{j=2}^{3}S_{j1}+K_{1} \\geq 2\\log_{2} q - \\sum_{j=2}^{3}H(U_{j1})+I(U_{21};U_{31};V_{1}).\n\\end{eqnarray}\nHaving chosen one such jointly typical triple, say $V_{1}^{n},U_{21}^{n},U_{31}^{n}$, it generates a vector $X^{n}$ according to $\\prod_{t=1}^{n}p_{X|\\PrivateRV_{1}\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}}(\\cdot | \\PrivateRV_{1}^{n},\\SemiPrivateRV_{21}^{n},\\SemiPrivateRV_{31}^{n})$ and inputs the same on the channel.\n\nDecoders $2$ and $3$ perform a standard point to point channel decoding. For example, decoder $2$ receives $Y_{2}^{n}$ and looks for all codewords in $\\lambda_{2}$ that are jointly typical with $Y_{2}^{n}$. If it finds all such codewords in a unique bin it declares the corresponding bin index as the decoded message. It can be proved by following the technique similar to \\cite[Proof of Theorem 1]{201108ISIT_PadPra} that if\n\\begin{equation}\n \\label{Eqn:BoundsOnUser2And3Rates}\nS_{j1}+T_{j1} \\leq \\log_{2} q - H(\\SemiPrivateRV_{j1}|Y_{j}):j=2,3 \n\\end{equation}\nthen probability of decoding error at decoders $2$ and $3$ can be made arbitrarily small for sufficiently large $n$.\n\nHaving received $Y_{1}^{n}$, decoder $1$ looks for all codewords $v_{1}^{n} \\in \\mathcal{C}_{1}$ for which there exists a codeword $u_{2 \\oplus 3}^{n} \\in (\\lambda_{2} \\oplus \\lambda_{3})$ such that $(v_{1}^{n},u_{2 \\oplus 3}^{n},Y_{1}^{n})$ are jointly typical with respect to $p_{\\SemiPrivateRV_{21}\\oplus \\SemiPrivateRV_{3},\\PrivateRV_{1},Y_{1}}$. Here $(\\lambda_{2}\\oplus \\lambda_{3}) \\define \\left\\{ v_{2}^{n}\\oplus v_{3}^{n}: v_{j}^{n} \\in \\lambda_{j}^{n}:j=2,3 \\right\\}$.\\footnote{Recall that structure of $\\lambda_{2},\\lambda_{3}$ contains cardinality of $(\\lambda_{2}\\oplus \\lambda_{3})$. In particular $|(\\lambda_{2}\\oplus \\lambda_{3})|\\leq \\exp \\left\\{ \\min \\left\\{ S_{21}+T_{21},S_{31}+T_{31} \\right\\} \\right\\}$} If all such codewords in $\\mathcal{C}_{1}$ belong to a unique bin, the corresponding bin index is declared as the decoded message. Again following the technique similar to \\cite[Proof of Theorem 1]{201108ISIT_PadPra}, it can be proved, that if\n\\begin{flalign}\n \\label{Eqn:BoundsOnUser1Rates}\nK_{1}\\!+\\!L_{1} \\!\\leq \\!H(V_{1})\\! - \\!H(\\PrivateRV_{1}|\\SemiPrivateRV_{21}\\oplus \\SemiPrivateRV_{31},Y_{1}) ,~~~\nK_{1}\\!+\\!L_{1}\\!+\\!S_{j1}\\!+\\!T_{j1} \\leq \\log_{2} q +H(V_{1})\\!-\\! H(\\PrivateRV_{1},\\SemiPrivateRV_{21}\\oplus \\SemiPrivateRV_{31}|Y_{1})\n\\end{flalign}\nfor $j=2,3$, then probability of decoding error at decoder $1$ falls exponentially with $n$.\n\nSince $L_{1},T_{21},T_{31}$ denotes rate achievable by users $1,2,3$ respectively, eliminating $K_{1},S_{21},S_{31}$ from the set of equations (\\ref{Eqn:SingleLayerSourceCodingBounds})-(\\ref{Eqn:BoundsOnUser1Rates}) yields an achievable rate region. The following definition and theorem provide a precise mathematical characterization of this achievable rate region.\n\\begin{definition}\n \\label{Defn:TestChannelsForSingleLayerWithOneUserDecodingInterference}\nLet $\\SetOfDistributions_{1}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of pmf's $p_{\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}\\PrivateRV_{1} X \\underlineY}$ defined on $\\SemiPrivateRVSet_{21}\\times \\SemiPrivateRVSet_{31}\\times \\PrivateRVSet_{1} \\times \\InputAlphabet \\times \\underlineSetY$, where (i) $\\SemiPrivateRVSet_{21}=\\SemiPrivateRVSet_{31}=\\fieldq$ is the finite field of cardinality $q$, $\\PrivateRVSet_{1}$ is a finite set, (ii) $p_{\\underlineY|X\\PrivateRV_{1}\\underlineSemiPrivateRV}=p_{\\underlineY|X}=W_{\\underlineY|X}$, and (iii) $\\Expectation\\left\\{ \\kappa(X) \\right\\} \\leq \\tau$. For $p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY} \\in \\SetOfDistributions_{1}(W_{\\underlineY|X},\\kappa,\\tau)$, let $\\beta_{1}(p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY})$ be defined as the set of triples $(R_{1},R_{2},R_{3})$ that satisfy\n\\begin{flalign}\n\\label{Eqn:OneLayerRateRegionForSpecificTestChannel}\n0 \\leq R_{1} &\\leq I(V_{1};U_{21} \\oplus U_{31},Y_{1}),~~~~R_{1}+R_{j} \\leq H(V_{1}U_{j1})-H(U_{j1}|Y_{j})-H(V_{1}|U_{21}\\oplus U_{31},Y_{1}):j=2,3\\nonumber\\\\\n0 &\\leq R_{j} \\leq I(U_{j1};Y_{j}),~~~~R_{1}+R_{j} \\leq H(V_{1}U_{j1})-H(V_{1},U_{21}\\oplus U_{31}|Y_{1}):j=2,3\\nonumber\\\\\nR_{2}+R_{3} &\\leq I(U_{21};Y_{2})+I(U_{31};Y_{3})-I(U_{21};U_{31}),\\nonumber\\\\ \n\\sum_{k=1}^{3}R_{k} &\\leq H(U_{21}U_{31}V_{1})-H(V_{1},U_{21}\\oplus U_{31}|Y_{1})-\\max \\left\\{H(U_{21}|Y_{2}),H(U_{31}|Y_{3}) \\right\\}\\nonumber\\\\\n\\sum_{k=1}^{3}R_{k}&\\leq H( U_{21}U_{31}V_{1})\\!-\\!H(V_{1}|U_{21}\\!\\oplus\\! U_{31},Y_{1})-\\sum_{k=2}^{3}H(U_{k1}|Y_{k})\\nonumber\\\\\nR_{1}+\\sum_{k=1}^{3}R_{k} &\\leq H(V_{1})+H(U_{21}U_{31}V_{1})\\!-\\!2H(V_{1},U_{21}\\!\\oplus\\! U_{31}|Y_{1}) \\nonumber\\\\\nR_{j}+\\sum_{k=1}^{3}R_{k} &\\leq H(V_{1}U_{j1})+H(U_{21}U_{31})-2H(U_{j1}|Y_{j})-H(V_{1},U_{21}\\oplus U_{31}|Y_{1}):j=2,3\\nonumber\n\\end{flalign}\nand\n\\begin{eqnarray}\n \\label{Eqn:OneLayerAchievableRateRegionWithOneUserDecodingInterference}\n \\beta_{1}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY} \\\\\\in \\SetOfDistributions_{1}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\beta_{1}(p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY})\\right).\\nonumber\n\\end{eqnarray}\n\\end{definition}\n\\begin{thm}\n \\label{Thm:OneLayerAchievableRateRegionWithOneUserDecodingInterference}\nFor a 3-DBC $(\\mathcal{X},\\underlineSetY,W_{\\underlineY|X},\\kappa)$, $\\beta_{1}(W_{\\underlineY|X},\\kappa,\\tau)$ is achievable, i.e., $\\beta_{1}(W_{\\underlineY|X},\\kappa,\\tau) \\subseteq \\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau)$.\n\\end{thm}\nThe proof is based on upper bounding the average probability of error for the coding scheme described earlier. The codewords in user $1$'s codebook, the $\\mathcal{V}_{1}-$codebook, are picked independently according to $\\prod_{t=1}^{n}p_{V_{1}}$. Assuming $S_{j_{1}1}+T_{j_{1}1} \\leq S_{j_{2}1}+T_{j_{2}1}$, we first pick generator matrix of $\\overline{\\lambda_{j_{1}}}$ by picking each entry independently and uniformly over $\\mathcal{U}_{j_{1}1}=\\mathcal{F}_{q}$. We append this with $n\\left[S_{j_{2}1}+T_{j_{2}1}-(S_{j_{1}1}+T_{j_{1}1})\\right]$ rows, again picking each new entry independently and uniformly over $\\mathcal{U}_{j_{2}1}=\\mathcal{F}_{q}$ to yield generator matrix for $\\overline{\\lambda_{j_{2}}}$. The row space of these generator matrices are shifted by uniformly distributed independent bias vectors $B_{j_{1}1}^{n}:j=2,3$ and the resulting collection of codewords in uniformly and independently partitioned into $\\exp\\left\\{nT_{j_{1}1}\\right\\}$ bins. This induces a distribution over the collection of all triple codebooks and as is typical in information theory, we upper bound the average probability of error. The key element is the use of joint typical encoding and decoding. This enables analyze decoding sum of codewords over arbitrary channels. While we do not provide details in this version of the article, we refer the diligent reader to \\cite[Proof of theorem 2]{201301arXivMACDSTx_PadPra} where a similar decoding rule is analyzed. In the interest of brevity, we omit a proof. The new elements in the proof is the interplay of joint typical encoding and decoding with correlated codebooks.\\footnote{While particular decoding rules such as syndrome decoding of linear codes can achieve capacity of particular channels such as binary symmetric, we will need to employ typical decoding to achieve capacity of arbitrary channels.} Indeed, codebooks being cosets of a common linear code are correlated and moreover, it's codewords are correlated. The analysis of error events contain several new elements.\n\nFor example \\ref{Ex:3-BCExample}, if $\\tau * \\delta_{1} \\leq \\min \\left\\{ \\delta_{2},\\delta_{3} \\right\\}$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\beta_{1}(W_{\\underlineY|X},\\kappa,\\tau)$. Indeed, it can be verified that if $\\tau * \\delta_{1} \\leq \\min \\left\\{ \\delta_{2},\\delta_{3} \\right\\}$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\beta_{1}(p_{\\underlineSemiPrivateRV\\PrivateRV_{1} X \\underlineY})$, where $p_{\\underlineSemiPrivateRV \\PrivateRV_{1}X}=p_{V_{1}}p_{U_{21}}p_{U_{31}}1_{\\left\\{X_{1}=V_{1}\\right\\}}1_{\\left\\{X_{2}=U_{21}\\right\\}}1_{\\left\\{X_{3}=U_{31}\\right\\}}$, $p_{U_{21}}(1)=p_{U_{31}}(1)=\\frac{1}{2}$ and $p_{V_{1}}(1)=\\tau$.\n\\subsection{An enlarged achievable rate region}\n \\label{SubSec:AnEnlargedAchievableRateRegion}\nWe revisit the coding technique proposed in section \\ref{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}. Observe that (i) user $1$ decodes a sum of the entire codewords\/signals transmitted to users $2$ and $3$ and (ii) users $2$ and $3$ decode only their respective codewords. This technique may be enhanced in the following way. User $1$ can decode the sum of \\textit{one component} of user $2$ and $3$ signals. In other words, we may include private codebooks for users $2$ and $3$.\n\nWe begin with a description of the coding technique. In addition to the codebooks $\\mathcal{C}_{1},\\lambda_{2},\\lambda_{3}$ described in section \\ref{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}, we incorporate private layer codebooks for users $2$ and $3$. Specifically, in addition to auxiliary alphabet sets $\\PrivateRVSet_{1},\\SemiPrivateRVSet_{21},\\SemiPrivateRVSet_{31}$ introduced in section \\ref{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}, let $\\PrivateRVSet_{2},\\PrivateRVSet_{3}$ denote arbitrary finite sets and $p_{U_{21}U_{31}\\underlinePrivateRV}$ denote a pmf on $\\SemiPrivateRVSet_{21} \\times \\SemiPrivateRVSet_{31} \\times \\underlinePrivateRVSet$. Two codebooks $\\mathcal{C}_{2} \\subseteq \\PrivateRVSet_{2}^{n},\\mathcal{C}_{3} \\subseteq \\PrivateRVSet_{3}^{n}$ of rates $K_{2}+L_{2}$, $K_{3}+L_{3}$ are partitioned independently and uniformly into $\\exp \\left\\{ nL_{2} \\right\\}, \\exp \\left\\{ nL_{3} \\right\\}$ bins respectively. Messages of users' $2$ and $3$ are split into two parts each. One part of user $2$'s ($3$'s) message, of rate $T_{21}$ ($T_{31}$), index a bin in $\\lambda_{2}$ ($\\lambda_{3}$), and the other part, of rate $L_{2}$ ($L_{3}$), index a bin in $\\mathcal{C}_{2}$ ($\\mathcal{C}_{3}$). The sole part of user $1$'s message indexes a bin in $\\mathcal{C}_{1}$. The encoder looks for a quintuple of jointly typical codewords with respect to $p_{U_{21}U_{31}\\underlinePrivateRV}$, in the quintuple of indexed bins. Following a second moment method similar to that employed in \\cite{201108ISIT_PadPra}, it can be proved that the encoder finds at least one jointly typical triple if\n\\begin{equation}\n \\label{Eqn:ManyToOneSourceCodingBounds}\nS_{A}+K_{B} \\geq |A|\\log_{2} q + \\sum_{b \\in B} H(V_{b}) - H(U_{A},V_{B})\n\\end{equation}\nfor all $A \\subseteq \\left\\{21,31\\right\\}, B \\subseteq \\left\\{ 1,2,3 \\right\\}$, where $S_{A} = \\sum _{j1 \\in A}S_{j1}$, $K_{B} = \\sum_{b \\in B}K_{b}$, $U_{A} = (U_{j1}:j1 \\in A)$ and $V_{B}=(V_{b}:b \\in B)$. Having chosen one such jointly typical quintuple, say $(\\SemiPrivateRV_{21}^{n},\\SemiPrivateRV_{31}^{n},\\underlinePrivateRV^{n})$, the encoder generates a vector $X^{n}$ according to $\\prod_{t=1}^{n}p_{X|\\underlinePrivateRV\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}}(\\cdot | \\underlinePrivateRV^{n},\\SemiPrivateRV_{21}^{n},\\SemiPrivateRV_{31}^{n})$ and inputs the same on the channel.\n\nThe operations of decoders $2$ and $3$ are identical and we describe one of them. Decoder $3$ receives $Y_{3}^{n}$ and looks for all pairs of codewords in the Cartesian product $\\lambda_{3} \\times \\mathcal{C}_{3}$ that are jointly typical with $Y_{3}^{n}$ with respect to $p_{U_{31}V_{3}Y_{3}}$. If all such pairs belong to a unique pair of bins, the corresponding pair of bin indices is declared as the decoded message of user $3$. Else an error is declared. It can be proved that if\n\\begin{eqnarray}\n \\label{Eqn:UpperboundOnPublicCodebook}\nS_{j1}+T_{j1} \\leq \\log_{2} q-H(U_{j1}|V_{j},Y_{j}),&&\nK_{j}+L_{j} \\leq H(V_{j})-H(V_{j}|Y_{j},U_{j1})\\\\\n\\label{Eqn:UpperboundOnPublicAndPrivateCodebook}\nS_{j1}+T_{j1}+K_{j}+L_{j} &\\leq& \\log_{2} q+H(V_{j})-H(V_{j},U_{j1}|Y_{j})\n\\end{eqnarray}\nfor $j=2,3$, then probability of users $2$ or $3$ decoding into an incorrect message falls exponentially with $n$.\n\nOperation of decoder $1$ is identical to that described in \\ref{SubSec:DecodingSumOfCodewordsUsingNestedCosetCodes}. If (\\ref{Eqn:BoundsOnUser1Rates}) holds, then probability of error at decoder 1 falls exponentially with $n$. Substituting $R_{1}=K_{1}, R_{2}=T_{21}+L_{2}, R_{3}=T_{31}+L_{3}$ and eliminating $S_{21},S_{31},K_{1},K_{2},K_{3}$ in (\\ref{Eqn:BoundsOnUser1Rates})-(\\ref{Eqn:UpperboundOnPublicAndPrivateCodebook}) yields an achievable rate region. We provide a mathematical characterization of this achievable rate region.\n\\begin{definition}\n \\label{Defn:TestChannelsForTwoLayersWithOneUserDecodingInterference}\nLet $\\SetOfDistributions_{2}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of pmf's $p_{\\SemiPrivateRV_{21}\\SemiPrivateRV_{31}\\PrivateRV_{1}\\PrivateRV_{2}\\PrivateRV_{3} X \\underlineY}$ defined on $\\SemiPrivateRVSet_{21}\\times \\SemiPrivateRVSet_{31}\\times \\PrivateRVSet_{1} \\times \\PrivateRVSet_{2} \\times\\PrivateRVSet_{3} \\times \\InputAlphabet \\times \\underlineSetY$, where (i) $\\SemiPrivateRVSet_{21}=\\SemiPrivateRVSet_{31}=\\fieldq$ is the finite field of cardinality $q$, $\\PrivateRVSet_{1},\\PrivateRVSet_{2}, \\PrivateRVSet_{3}$ are finite sets, (ii) $p_{\\underlineY|X\\underlinePrivateRV\\underlineSemiPrivateRV}=p_{\\underlineY|X}=W_{\\underlineY|X}$, and (iii) $\\Expectation\\left\\{ \\kappa(X) \\right\\} \\leq \\tau$. For $p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY} \\in \\SetOfDistributions_{2}(W_{\\underlineY|X},\\kappa,\\tau)$, let $\\beta_{2}(p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY})$ be defined as the set of triples $(R_{1},R_{2},R_{3}) \\in [0,\\infty)^{3}$ for which there exists nonnegative numbers $S_{21},T_{21},S_{31},T_{31},K_{j},L_{j}:j=1,2,3$ such that $R_{1}=K_{1}, R_{2}=T_{21}+L_{2}, R_{3}=T_{31}+L_{3}$ and (\\ref{Eqn:BoundsOnUser1Rates}),(\\ref{Eqn:ManyToOneSourceCodingBounds}),(\\ref{Eqn:UpperboundOnPublicCodebook}),(\\ref{Eqn:UpperboundOnPublicAndPrivateCodebook}) hold. Let\n\\begin{eqnarray}\n \\label{Eqn:TwoLayerAchievableRateRegionWithOneUserDecodingInterference}\n \\beta_{2}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY} \\\\\\in \\SetOfDistributions_{2}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\beta_{2}(p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY})\\right).\\nonumber\n\\end{eqnarray}\n\\end{definition}\n\\begin{thm}\n \\label{Thm:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes}\nFor a 3-DBC $(\\mathcal{X},\\underlineSetY,W_{\\underlineY|X},\\kappa)$, $\\beta_{2}(W_{\\underlineY|X},\\kappa,\\tau)$ is achievable, i.e., $\\beta_{2}(W_{\\underlineY|X},\\kappa,\\tau) \\subseteq \\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau)$.\n\\end{thm}\nThe proof is similar to that of theorem \\ref{Thm:OneLayerAchievableRateRegionWithOneUserDecodingInterference}. The only differences being (i) the encoder looks for a quintuple of codewords instead of a triple, and (ii) decoders $2$ and $3$ decode from a pair of codebooks. From \\cite[Proof of theorem 1]{201108ISIT_PadPra}, the informed reader can see why the second difference can be easily handled. Indeed, in \\cite[Theorem 1]{201108ISIT_PadPra}, we prove nested coset codes achieve capacity of arbitrary point to point channels. This indicates that for $j=2,3$, $\\SemiPrivateRVSet_{j1}-$codebook can be used to communicate at rate $I(\\SemiPrivateRV_{j1};Y_{j})$ and the private layer $\\PrivateRVSet_{j}-$codebook can be used to communicate at rate $I(\\SemiPrivateRV_{j}:Y_{j}|\\SemiPrivateRV_{j1})$, thereby satisfying user $j$'s rate. This leaves us to argue only the first difference pointed above. Employ a second moment method similar to that employed in \\cite{198101TIT_GamMeu}, \\cite{201108ISIT_PadPra}, it can be shown that probability of encoder not finding a jointly typical quintuple decays exponentially if (\\ref{Eqn:ManyToOneSourceCodingBounds}) holds. We defer a detailed proof of theorem \\ref{Thm:AchievableRateRegionsFor3To1BCUsingNestedCosetCodes} to a subsequent enlarged version of this article.\n\\section{Achievable rate region for $3-$user broadcast channel using nested coset codes}\n\\label{Sec:AchievableRateRegionFor3BCUsingNestedCosetCodes}\nIn this section, we derive an achievable rate region for the general 3-DBC. This is obtained by a simple extension of the coding technique proposed for the \\3To1BC DBC. In section \\ref{SubSec:AnEnlargedAchievableRateRegion}, only user $1$ decoded a bivariate interference component. Users $2$ and $3$ only decoded from their respective codebooks. This suffices for a \\3To1BC DBC since receivers $2$ and $3$ enjoyed interference free point to point links. In general, each user would attempt to decode a bivariate interference component of the other users' signals. In the sequel, we propose a simple extension of the technique presented in section \\ref{SubSec:AnEnlargedAchievableRateRegion} to enable each user decode a bivariate interference component.\n\nUser $j$ splits it's message $M_{j}$ into three parts $(M_{ji}^{U},M_{jk}^{U},M_{j}^{V})$, where $i,j,k$ are distinct indices in $\\left\\{ 1,2,3 \\right\\}$. Let $\\SemiPrivateRVSet_{ji}=\\fieldqi, \\SemiPrivateRVSet_{jk}=\\fieldqk$ be finite fields and $\\PrivateRVSet_{j}$ be an arbitrary finite set. Let $\\lambda_{ji} \\subseteq \\SemiPrivateRVSet_{ji}^{n}$, $\\lambda_{jk} \\subseteq \\SemiPrivateRVSet_{jk}^{n}$ denote cosets of linear codes $\\overline{\\lambda_{ji}}, \\overline{\\lambda_{jk}}$ of rates $S_{ji}+T_{ji}, S_{jk}+T_{jk}$ respectively. Observe that cosets $\\lambda_{ji}$ and $\\lambda_{ki}$ are built over the same finite field $\\fieldqi$. To enable contain the range the sum of these cosets, the larger of $\\lambda_{ji}$, $\\lambda_{ki}$ contains the other. Codewords of $\\lambda_{ji}$ and $\\lambda_{jk}$ are independently and uniformly partitioned into $\\exp\\left\\{nT_{ji}\\right\\}$ and $\\exp\\left\\{nT_{jk}\\right\\}$ bins respectively. A codebook $\\mathcal{C}_{j}$ of rate $K_{j}+L_{j}$ is built over $\\PrivateRVSet_{j}$. Codewords of $\\mathcal{C}_{j}$ are partitioned into $\\exp\\left\\{ nL_{j}\\right\\}$ bins uniformly and independently at random. \n\n\n$M_{ji}^{U}$,$M_{jk}^{U}$ and $M_{j}^{V}$ index bins in $\\lambda_{ji}$, $\\lambda_{jk}$ and $\\mathcal{C}_{j}$ respectively. The encoder looks for a collection of $9$ codewords from the indexed bins that are jointly typical with respect to a pmf $p_{\\underlineSemiPrivateRV \\underlinePrivateRV}$ defined on $\\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet$.\\footnote{$\\underlineSemiPrivateRV$ abbreviates $\\SemiPrivateRV_{12}\\SemiPrivateRV_{13}\\SemiPrivateRV_{21}\\SemiPrivateRV_{23}\\SemiPrivateRV_{31}\\SemiPrivateRV_{32}$.} Following a second moment method similar to that employed in \\cite{201108ISIT_PadPra}, it can be proved that the encoder finds at least one jointly typical collection if\n\\begin{equation}\n \\label{Eqn:ManyToManySourceCodingBounds}\nS_{A}+K_{B} \\geq \\sum_{a \\in A}\\log |\\mathcal{U}_{a}| + \\sum_{b \\in B} H(V_{b}) - H(U_{A},V_{B})\n\\end{equation}\nfor all $A \\subseteq \\left\\{12,13,21,23,31,32\\right\\}, B \\subseteq \\left\\{ 1,2,3 \\right\\}$, where $S_{A} = \\sum _{jk \\in A}S_{jk}$, $K_{B} = \\sum_{b \\in B}K_{b}$, $U_{A} = (U_{jk}:jk \\in A)$ and $V_{B}=(V_{b}:b \\in B)$. Having chosen one such jointly typical collection, say $(\\underlineSemiPrivateRV^{n},\\underlinePrivateRV^{n})$, the encoder generates a vector $X^{n}$ according to $\\prod_{t=1}^{n}p_{X|\\underlineSemiPrivateRV\\underlinePrivateRV}(\\cdot | \\underlineSemiPrivateRV^{n},\\underlinePrivateRV^{n})$ and inputs the same on the channel.\n\nDecoder $j$ receives $Y_{j}^{n}$ and looks for all triples $(u_{ji}^{n},u_{jk}^{n},v_{j}^{n})$ of codewords in $\\lambda_{ji} \\times \\lambda_{jk} \\times \\mathcal{C}_{j}$ such that there exists a $u^{n}_{ij\\oplus kj} \\in (\\lambda_{ij} \\oplus \\lambda_{kj})$ such that $(u_{ij\\oplus kj}^{n},u_{ji}^{n},u_{jk}^{n},v_{j}^{n},Y_{j}^{n})$ are jointly typical with respect to $p_{U_{ij}\\oplus U_{kj},U_{ji},U_{jk},V_{j},Y_{j}}$. If it finds all such triples in a unique triple of bins, the corresponding triple of bin indices is declared as decoded message of user $j$. Else an error is declared. The probability of error at decoder $j$ can be made arbitrarily small for sufficiently large block length if\n\\begin{equation}\n\\begin{aligned}\n\\label{Eqn:ManyToManyBCChannelCodingBounds}\n\\lefteqn{S_{A_{j}}+T_{A_{j}} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| - H(U_{A_{j}}|U_{A_{j}^{c}},U_{ij}\\oplus U_{kj},V_{j},Y_{j})}\n\\\\\n\\lefteqn{S_{A_{j}}+T_{A_{j}}+S_{ij}+T_{ij} \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| + \\log q_{j} - H(U_{A_{j}},U_{ij}\\oplus\nU_{kj}|U_{A_{j}^{c}},V_{j},Y_{j})} \\\\\n\\lefteqn{S_{A_{j}}+T_{A_{j}}+S_{kj}+T_{kj} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| + \\log q_{j} - H(U_{A_{j}},U_{ij}\\oplus\nU_{kj}|U_{A_{j}^{c}},V_{j},Y_{j})} \\\\\n\\lefteqn{S_{A_{j}}+T_{A_{j}}+K_{j}+L_{j} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}|+H(V_{j}) - H(U_{A_{j}},V_{j}|U_{A_{j}^{c}},U_{ij}\\oplus\nU_{kj},Y_{j})} \\\\\n\\lefteqn{S_{A_{j}}+T_{A_{j}}+K_{j}+L_{j}+S_{ij}+T_{ij} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| + \\log q_{j}+H(V_{j}) -\nH(U_{A_{j}},V_{j},U_{ij}\\oplus U_{kj}|U_{A_{j}^{c}},Y_{j})} \\\\\n&S_{A_{j}}+T_{A_{j}}+K_{j}+L_{j}+S_{kj}+T_{kj} \\leq \\sum_{a \\in A_{j}}\\log |\\mathcal{U}_{a}| + \\log q_{j}+H(V_{j}) - H(U_{A_{j}},V_{j},U_{ij}\\oplus\nU_{kj}|U_{A_{j}^{c}},Y_{j}), \n\\end{aligned}\n\\end{equation}\nwhere for every $A_{j} \\subseteq \\left\\{ ji,jk\\right\\}$ with distinct indices $i,j,k$ in $\\left\\{ 1,2,3 \\right\\}$, \n\\begin{equation}\n\\label{Eqn:DefinitionOfS_A_jAndT_A_j}\nS_{A_{j}} \\define \\left\\{ \\begin{array}{ll} S_{ji}&\\mbox{ if }S_{A_{j}}=\\left\\{ ji \\right\\} \\\\ S_{jk}&\\mbox{ if }S_{A_{j}}=\\left\\{ jk \\right\\} \\\\ S_{ji}+S_{jk}&\\mbox{ if }S_{A_{j}}=\\left\\{ ji,jk \\right\\} \\end{array} \\right. \nT_{A_{j}} \\define \\left\\{ \\begin{array}{ll} T_{ji}&\\mbox{ if }T_{A_{j}}=\\left\\{ ji \\right\\} \\\\ T_{jk}&\\mbox{ if }T_{A_{j}}=\\left\\{ jk \\right\\} \\\\ T_{ji}+T_{jk}&\\mbox{ if }T_{A_{j}}=\\left\\{ ji,jk \\right\\} \\end{array} \\right. .\n\\end{equation}\nRecognize that user $j$'s rate $R_{j}=T_{ji}+T_{jk}+L_{j}$. We are now equipped to state an achievable rate region for the general 3-DBC using nested coset codes.\n\n\\begin{definition}\n \\label{Defn:CollectionOfTestChannelsForCommunicatingOverGeneral3BCUsingNCC}\nLet $\\mathbb{D}_{f}(W_{\\underlineY|X},\\kappa,\\tau)$ denote the collection of probability mass functions $p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY}$ defined on $\\underlineSemiPrivateRVSet \\times \\underlinePrivateRVSet \\times \\mathcal{X} \\times \\underlineSetY$, where $\\underlineSemiPrivateRV\\define (\\SemiPrivateRV_{12},\\SemiPrivateRV_{13},\\SemiPrivateRV_{21},\\SemiPrivateRV_{23},\\SemiPrivateRV_{31},\\SemiPrivateRV_{32})$, $\\underlinePrivateRV\\define (\\PrivateRV_{1},\\PrivateRV_{2},\\PrivateRV_{3})$, $\\underlineSemiPrivateRVSet \\define \\SemiPrivateRVSet_{12} \\times \\SemiPrivateRVSet_{13} \\times \\SemiPrivateRVSet_{21} \\times \\SemiPrivateRVSet_{23} \\times \\SemiPrivateRVSet_{31} \\times \\SemiPrivateRVSet_{32},\\underlinePrivateRVSet \\define \\PrivateRVSet_{1} \\times \\PrivateRVSet_{2} \\times \\PrivateRVSet_{3}, \\SemiPrivateRVSet_{ij}=\\fieldqj$, the finite field of cardinality $q_{j}$ for each $1\\leq i,j\\leq3$, $\\SemiPrivateRVSet_{i}$ is an arbitrary finite set such that (i) $p_{\\underlineY| X \\underlinePrivateRV \\underlineSemiPrivateRV}=p_{\\underlineY |X}=W_{\\underlineY |X}$, (ii) $\\Expectation\\left\\{ \\kappa(X) \\right\\} \\leq \\tau$.\n\nFor $p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY} \\in \\mathbb{D}_{f}(W_{\\underlineY|X},\\kappa,\\tau)$, let $\\beta_{f}(p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY})$ is defined as the set of rate triples $(R_{1},R_{2},R_{3}) \\in [0,\\infty]^{3}$ for which there exists nonnegative numbers $S_{ij}:ij \\in \\left\\{12,13,21,23,31,32 \\right\\}, T_{jk}:jk \\in \\left\\{12,13,21,23,31,32 \\right\\}, K_{j}:j \\left\\{ 1,2,3\\right\\}, L_{j}:\\left\\{1,2,3 \\right\\}$ that satisfy (\\ref{Eqn:ManyToManySourceCodingBounds})-(\\ref{Eqn:DefinitionOfS_A_jAndT_A_j}) and $R_{1}=T_{12}+T_{13}+L_{1}, R_{2}=T_{21}+T_{23}+L_{2}, R_{3}=T_{31}+T_{32}+L_{3}$. Let\n\\begin{eqnarray}\n \\label{Eqn:AchievableRateRegionFor3BCUsingNestedCosetCodes}\n \\beta_{f}(W_{\\underlineY|X},\\kappa,\\tau) = \\cocl\\left(\\underset{\\substack{p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY} \\in\\\\ \\mathbb{D}_{f}(W_{\\underlineY|X},\\kappa,\\tau)}\n}{\\bigcup}\\beta_{f}(p_{\\underlineSemiPrivateRV\\underlinePrivateRV X \\underlineY})\\right).\\nonumber\n\\end{eqnarray}\n\\end{definition}\n\\begin{thm}\n \\label{Thm:AchievableRateRegionFor3BCUsingNestedCosetCodes}\nFor 3-DBC $(\\mathcal{X},\\underlineSetY,W_{\\underlineY|X},\\kappa)$, $\\beta_{f}(W_{\\underlineY|X},\\kappa,\\tau)$ is achievable, i.e., $\\beta_{f}(W_{\\underlineY|X},\\kappa,\\tau) \\subseteq \\mathbb{C}(W_{\\underlineY|X},\\kappa,\\tau)$.\n\\end{thm}\nWe defer the proof of this theorem to a subsequent enlarged version of this article.\n\\begin{comment}{\n\\begin{eqnarray}\n \\label{Eqn:ManyToManyBCChannelCodingBounds}\n\\lefteqn{ST_{A} \\leq |A|\\log_{2} q - H(U_{A}|U_{A^{c}},U_{ij}\\oplus U_{kj},V_{j},Y_{j})}\n\\nonumber\\\\\n\\lefteqn{ST_{A}+ST_{ij} \\leq (|A|+1)\\log_{2} q - H(U_{A},U_{ij}\\oplus\nU_{kj}|U_{A^{c}},V_{j},Y_{j})} \\nonumber\\\\\n\\lefteqn{ST_{A}+ST_{kj} \\leq (|A|+1)\\log_{2} q - H(U_{A},U_{ij}\\oplus\nU_{kj}|U_{A^{c}},V_{j},Y_{j})} \\nonumber\\\\\n\\lefteqn{ST_{A}+KL_{j} \\leq |A|\\log_{2} q+H(V_{j}) - H(U_{A},V_{j}|U_{A^{c}},U_{ij}\\oplus\nU_{kj},Y_{j})} \\nonumber\\\\\n\\lefteqn{ST_{A}+KL_{j}+ST_{ij} \\leq (|A|+1)\\log_{2} q+H(V_{j}) -\nH(U_{A},V_{j},U_{ij}\\oplus U_{kj}|U_{A^{c}},Y_{j})} \\nonumber\\\\\n&\\!\\!\\!\\!\\!\\!ST_{A}+KL_{j}+ST_{kj} (\\leq |A|+1)\\log_{2} q+H(V_{j}) -\nH(U_{A},V_{j},U_{ij}\\oplus\nU_{kj}|U_{A^{c}},Y_{j}) \\nonumber\n\\end{eqnarray}\nwhere $ST_{A} \\define S_{A}+T_{A}$, $ST_{ij}\\define S_{ij}+T_{ij}$, $ST_{kj}\n=S_{kj}+T_{kj}$, $KL_{j}=K_{j}+L_{j}$\n}\\end{comment}\n\n\\section{Enlarging Marton's rate region using nested coset codes}\n\\label{Sec:EnlargingMarton'sRateRegionUsingNestedCosetCodes}\nThe natural question that arises is whether the achievable rate region using nested coset codes $\\beta(W_{\\underlineY|X},\\kappa,\\tau)$ contains Marton's rate region $\\alpha_{NEM}(W_{\\underlineY|X},\\kappa,\\tau)$. It is our belief that coding techniques based on structured codes do not substitute their counterparts based on traditional unstructured independent codes, but enhance the same. Indeed, the technique proposed by K\\\"orner and Marton \\cite{197903TIT_KorMar} is strictly suboptimal to that studied by Berger and Tung \\cite{Berger-MSC} if the function is not sufficiently compressive, i.e., entropy of the sum is larger than one half of the joint entropy of the sources.\\footnote{If $X$ and $Y$ are the distributed binary sources whose modulo$-2$ sum is to be reconstructed at the decoder, then K\\\"orner and Marton technique is strictly suboptimal if $H(X \\oplus Y) > \\frac{H(X,Y)}{2}$.} The penalty paid in terms of the binning rate for endowing structure is not sufficiently compensated for by the function. This was (recognized)\/(hinted at) by Ahlswede and Han \\cite[Section VI]{198305TIT_AhlHan} for the problem studied by K\\\"orner and Marton.\n\nWe follow the approach of Ahlswede and Han \\cite[Section VI]{198305TIT_AhlHan} to build upon Marton's rate region by gluing to it the coding technique proposed herein. In essence the coding techniques studied in section \\ref{SubSec:NaturalExtensionOfMartonTo3BC} and \\ref{Sec:AchievableRateRegionFor3BCUsingNestedCosetCodes} are glued together.\\footnote{This is akin to the use of superposition and binning in Marton's coding.} Indeed, a description of the resulting rate region is quite involved and we spare the reader of these details. The resulting coding technique will involve each user split it's message into six parts - one public and private part each, two semi-private and \\textit{bivariate} parts each. This can be understood by splitting the message as proposed in sections \\ref{SubSec:NaturalExtensionOfMartonTo3BC} and \\ref{Sec:AchievableRateRegionFor3BCUsingNestedCosetCodes} and identifying the private parts. In essence each user decodes a univariate component of every other user's transmission particularly set apart for it, and furthermore decodes a bivariate component of the other two user's transmissions.\\footnote{An informed and inquisitive reader may begin to see a relationship emerge between the several layers of coding and common parts of a collection of random variables. Please refer to section \\ref{Sec:ConcludingRemarks} for a discussion.} A mathematical characterization of the resulting achievable rate region will be provided in a subsequent enlarged version of this article. For now, we conclude by stating that the resulting achievable rate region 1) contains and strictly enlarges Marton's rate region for the general 3-DBC and 2) is therefore the currently known largest achievable rate region for the same.\n\n\\section{Concluding remarks : Common parts of random variables and the need for structure}\n\\label{Sec:ConcludingRemarks}\nLet us revisit Marton's coding technique for 2-BC. Define the pair $\\overline{\\PrivateRV_{j}}\\define (\\PublicRV, \\PrivateRV_{j}):j=1,2$ of random variables decoded by the two users and let $\\overline{\\PrivateRVSet_{j}}\\define \\PublicRVSet \\times \\PrivateRVSet_{j}:j=1,2$. Let us stack the collection of compatible codewords over $\\overline{\\PrivateRVSet_{1}}^{n} \\times \\overline{\\PrivateRVSet_{2}}^{n}$. The encoder can work with this stack, being oblivious to the distinction between $\\PublicRVSet$ and $\\PrivateRVSet_{j}:j=1,2$. In other words, it does not recognize that a symbol over $\\overline{V_{j}}$ is indeed a pair of symbols. A few key observations of this stack of codewords is in order. Recognize that many pairs of compatible codewords agree in their `$\\PublicRVSet-$coordinate'. In other words, they share the same codeword on the $\\PublicRVSet-$codebook. $\\PublicRV$ is a common part \\cite{1972MMPCT_GacKor} of the pair $(\\overline{\\PrivateRV_{1}},\\overline{\\PrivateRV_{2}})$. Being a common part, it can be realized through univariate functions. Let us say $W=f_{1}(V_{1})=f_{2}(V_{2})$. This indicates, \\textit{$\\PublicRVSet-$codebook is built such that, the range of these univariate functions when applied on the collection of codewords in this stack, is contained.}\n\nHow did Marton accomplish this containment? Marton proposed building the $\\PublicRV-$codebook first, followed by conditional codebooks over $\\PrivateRV_{1},\\PrivateRV_{2}$. Conditional coding with a careful choice of order therefore contained the range under the action of univariate function. How is all of this related to the need for containing bivariate functions of a pair of random variables. The fundamental underlying thread is the notion of common part \\cite{1972MMPCT_GacKor}. What are the common parts of a triple of random variables? Clearly, one can simply extend the notion of common part defined for a pair of random variables. This yields four common parts - one part that is simultaneously to common to all three random variables and one common part each, corresponding to each pair in the triple.\nIndeed, if $\\overline{V_{1}}=(W,U_{12},U_{31},V_{1}),\\overline{V_{2}}=(W,U_{12},U_{23},V_{2}),\\overline{V_{3}}=(W,U_{23},U_{31},V_{3})$, then $W$ is the part simultaneously to common to $\\overline{V_{1}},\\overline{V_{2}},\\overline{V_{3}}$ and $U_{ij}:ij\\in\\left\\{12,23,31\\right\\}$ are the pairwise common parts. A simple extension of Marton's coding suggests a way to handle these common parts.\n\nThis does not yet answer the need for containment under bivariate function. We envision a fundamentally richer notion of common part for a triple of random variables. Indeed, three nontrivial binary random variables $X,Y, Z=X\\oplus Y$ have no common parts as defined earlier, since each pair has no common part and the triple does not admit a simultaneous common part. Yet, the degeneracy in the joint probability matrix hints at a common part. Indeed, they possess a \\textit{conferencing} common part. For example, the pair $(X,Y),Z$ have a common part. In other words, there exists a \\textit{bivariate} function of $X,Y$ and a univariate function of $Z$ that agree with probability $1$. Containment of this bivariate function brings in the need for structured codes. Indeed, the resemblance to the problem studied by K\\\"orner and Marton \\cite{197903TIT_KorMar} is striking. We therefore believe the need for structured codes for three (multi) user communication problems is closely linked to the notion of common parts of a triple (collection) of random variables. Analogous to conditional coding that contained univariate functions, endowing codebooks with structure is an inherent need to carefully handle additional degrees of freedom prevalent in larger dimensions.\n\n\\section{Strict sub-optimality of Marton's coding technique}\n\\label{Sec:StrictSubOptimalityOfMartonCodingTechnique}\n\nIn this section, we prove strict sub-optimality of Marton's coding technique for the 3-DBC presented in example \\ref{Ex:3-BCExample}. In particular, we prove that if parameters $\\tau,\\delta_{1},\\delta_{2},\\delta_{3}$ are such that $1+h_{b}(\\delta_{1} * \\tau) > h_{b}(\\delta_{2})+h_{b}(\\delta_{3})$ and $(R_{1},1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\alpha_{NEM}(\\tau)$, then $R_{1} < h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$.\n\nWhy is Marton's coding technique suboptimal for the case described above. As mentioned in section \\ref{SubSec:TheThreeUserBroadcastChannel}, in this case, receiver $1$ is unable to decode the pair of codewords transmitted to users $2$ and $3$. Furthermore, based on unstructured independent coding, it does not attempt to decode a function of transmitted codewords - in this case the modulo$-2$ sum. This forces decoder $1$ to be content by decoding only individual components of user $2$ and $3$'s transmissions, leaving residual uncertainty in the interference. The encoder helps out by precoding for this residual uncertainty. However, as a consequence of the cost constraint on $X_{1}$, it is forced to live with a rate loss.\n\nSince our proof traces through the above arguments in three stages, it is instructive. In the first stage, we characterize all test channels $p_{\\TimeSharingRV\\PublicRV \\underlineSemiPrivateRV \\underlinePrivateRV X \\underlineY}$ for which $(R_{1},1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\alpha_{NEM}(p_{\\TimeSharingRV\\PublicRV \\underlineSemiPrivateRV \\underlinePrivateRV X \\underlineY})$. This stage enables us identify `active' codebooks, their corresponding rates and characterize two upper bounds on $R_{1}$. One of these contains the rate loss due to precoding. In the second stage, we therefore characterize the condition under which there is no rate loss. As expected, it turns out that there is no rate loss only if decoder $1$ has decoded codewords of users $2$ and $3$. This gets us to the third stage, where we conclude that $1+h_{b}(\\delta_{1} * \\tau) > h_{b}(\\delta_{2})+h_{b}(\\delta_{3})$ precludes this possibility. The first stage is presented in lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple}, second stage is stated in lemma \\ref{Lem:CharacterizationOfConditionThatEnsuresNoRateLoss} and proved in appendices \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx}, \\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss}. Third stage can be found in arguments following lemma \\ref{Lem:CharacterizationOfConditionThatEnsuresNoRateLoss}.\n\nWe begin with a characterization of a test channel $p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\\underlinePrivateRV X\\underlineY}$ for which $(R_{1},1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\alpha_{NEM}(p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\\underlinePrivateRV X\\underlineY})$. Since independent information needs to be communicated to users $2$ and $3$ at their respective point to point capacities, it is expected that their codebooks are not precoded for each other's signal, and moreover none of users $2$ and $3$ decode a part of the other users' signal. The following lemma establishes this. We remind the reader that $X_{1}X_{2}X_{3}=X$ denote the three binary digits at the input, where $Y_{2}$, the output at receiver $2$ is obtained by passing $X_{2}$ through a BSC with cross over probability $\\delta_{2}$, $Y_{3}$, the output at receiver $3$ is obtained by passing $X_{3}$ through a BSC with cross over probability $\\delta_{3}$ and $Y_{1}$ is obtained by passing $X_{1}\\oplus X_{2}\\oplus X_{3}$ through a BSC with cross over probability $\\delta_{1}$. Moreover, the binary symmetric channels (BSC's) are independent. Input symbol $X_{1}$ is constrained with respect to a Hamming cost function and the constraint on the average cost per symbol is $\\tau$. Formally, $\\kappa(x_{1}x_{2}x_{3})=1_{\\left\\{ x_{1}=1 \\right\\}}$ is the cost function and the average cost per symbol is not to exceed $\\tau$.\n\n\\begin{lemma}\n \\label{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple}\nIf there exists a test channel $p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\\underlinePrivateRV X\\underlineY} \\in \\SetOfDistributions_{NEM}(\\tau)$ and nonnegative numbers $K_{i},S_{ij},K_{ij},L_{ij},S_{i},T_{i}$ that satisfy (\\ref{Eqn:3BCSourceCodingBoundNonnegativity})-(\\ref{Eqn:3BCChannelCodingSextupleBound}) for each triple $(i,j,k) \\in \\left\\{ (1,2,3),(2,3,1),(3,1,2) \\right\\}$ such that $R_{2}=K_{2}+K_{23}+L_{12}+T_{2}=1-h_{b}(\\delta_{2}),R_{3}=K_{3}+K_{31}+L_{23}+T_{3}=1-h_{b}(\\delta_{3})$, then\n\\begin{enumerate}\n \\item \\label{Item:RatesOfPublicAndSemiPrivateCodebooks}$K_1=K_2=K_3=K_{23}=L_{23}=K_{12}=L_{31}=S_2=S_3=0$ and $I(\\SemiPrivateRV_{31} V_{1}V_{3};Y_{2}|\\TimeSharingRV WU_{23}U_{12}V_{2})=0$,\n\\item \\label{Item:BinningRatesOfSemiPrivateCodebooks}$S_{31}=I(U_{31};U_{23}|\\TimeSharingRV W),S_{12}=I(U_{12};U_{23}|\\TimeSharingRV W)$, $S_{23}=I(U_{12};U_{31}|\\TimeSharingRV WU_{23})=0$,\n\\item \\label{Item:WU23IsConditionallyIndependentOfY2Y3GivenQ}$I(V_{2}U_{12};V_{3}U_{31}|\\TimeSharingRV WU_{23})=0$, $I(WU_{23};Y_{j}|\\TimeSharingRV )=0:j=2,3,I(V_{2}U_{12};Y_{2}|\\TimeSharingRV W U_{23})=1-h_{b}(\\delta_{2})$ and $I(V_{3}U_{31};Y_{3}|\\TimeSharingRV W U_{23})=1-h_{b}(\\delta_{3})$, \n\\item \\label{ItemNumber:MarkovChains} $(V_3,X_3,V_1,U_{31}) - (\\TimeSharingRV WU_{23}U_{12}V_2) - (X_2,Y_2) $ and $(V_2,X_2,V_1,U_{12}) - (\\TimeSharingRV WU_{23}U_{31}V_3) - (X_3,Y_3)$ are Markov chains,\n\\item \\label{Item:X2AndX3AreConditionallyIndependentGivenQWU12U23U31}$X_{2} - \\TimeSharingRV \\PublicRV \\SemiPrivateRV_{12} \\SemiPrivateRV_{23} \\SemiPrivateRV_{31} - X_{3}$ is a Markov chain,\n\\item \\label{Item:U31AndX2AreConditionallyIndependentGivenQWU12U23}$\\SemiPrivateRV_{12}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23} \\SemiPrivateRV_{31}-X_{3}$ and $\\SemiPrivateRV_{31}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23} \\SemiPrivateRV_{12}-X_{2}$ are Markov chains.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Substituting (i) $(2,3,1)$ for $(i,j,k)$ in (\\ref{Eqn:3BCChannelCodingSextupleBound}), (ii) $(1,2,3)$ for $(i,j,k)$ in (\\ref{Eqn:3BCSourceCodingPairwiseBound}) and combining the resulting bounds yields \n\\begin{flalign}\n\\label{Eqn:UpperBoundOnR_2PlusABunchOfTerms}\nI(WU_{23}U_{12}V_{2};Y_{2}|\\TimeSharingRV) \\geq R_{2}+K_{3}+K_{1}+L_{23}+K_{12}+S_{2}\\geq R_{2}=1-h_{b}(\\delta_{2}),\n\\end{flalign}\nwhere the second inequality follows from non-negativity of $K_{3},K_{1},L_{23},K_{12},S_{2}$. Moreover, \n\\begin{eqnarray}\n\\label{Eqn:UsingDataProcessingInequalityInDerivingUpperBoundOnR2}\n1-h_{b}(\\delta_{2}) &\\geq& I(X_{2};Y_{2}) = I(\\TimeSharingRV \\PublicRV \\underlineSemiPrivateRV\\underlinePrivateRV X_{1}Y_{1}X_{3}Y_{3}X_{2};Y_{2}) \\geq I(W \\SemiPrivateRV_{23} \\SemiPrivateRV_{12} \\PrivateRV_{2} ; Y_{2}|\\TimeSharingRV) \\\\\n\\label{Eqn:SubstitutingUpperBoundOnR_2}\n&\\geq& R_{2}\\!+\\! K_{3}\\!+\\! K_{1}\\!+\\! L_{23}\\!+\\! K_{12}\\!+\\! S_{2} \\geq R_{2}=1-h_{b}(\\delta_{2}),\n\\end{eqnarray}\nwhere (i) equality in (\\ref{Eqn:UsingDataProcessingInequalityInDerivingUpperBoundOnR2}) follows from Markov chain $\\TimeSharingRV \\PublicRV \\underlineSemiPrivateRV \\underlinePrivateRV X_{1}Y_{1}X_{3}Y_{3}- X_{2} -Y_{2}$, and (ii) (\\ref{Eqn:SubstitutingUpperBoundOnR_2}) follows from substituting (\\ref{Eqn:UpperBoundOnR_2PlusABunchOfTerms}). Since all the terms involved are non-negative, equality holds through the above chain of inequalities to yield\n\\begin{eqnarray}\n\\label{Eqn:BinningRatesOfUser2SemiPrivateCodebooks}\n&S_{12}+S_{23}=I(U_{12};U_{23}|\\TimeSharingRV W),K_{1}\\!=\\!K_{3}\\!=\\!L_{23}\\!=\\!K_{12}\\!=\\!S_{2}\\!=\\!\nI(\\TimeSharingRV;Y_{2})\\!=\\!0\\\\\n\\label{Eqn:K3K1L23K12S2AreZero}\n&I(U_{31}V_{1}X_{1}Y_{1}V_{3}X_{3}Y_{3}X_{2};Y_{2}|QWU_{12}U_{23}V_{2})\\!=\\!0\\!\\\\\n\\label{Eqn:PartOfmarkovChainUsedInGettingToRateLoss}\n&\\mbox{ and therefore }\n(V_1,V_3,X_3,U_{31}) - (\\TimeSharingRV WU_{12}U_{23}V_2) - Y_2\\mbox{ is a Markov chain}\n\\end{eqnarray}\nwhere the first equality in (\\ref{Eqn:BinningRatesOfUser2SemiPrivateCodebooks}) follows from condition for equality in the first inequality of (\\ref{Eqn:UpperBoundOnR_2PlusABunchOfTerms}). The above sequence of steps are repeated by substituting (i) $(3,1,2)$ for $(i,j,k)$ in (\\ref{Eqn:3BCChannelCodingSextupleBound}), (ii) $(2,3,1)$ for $(i,j,k)$ in (\\ref{Eqn:3BCSourceCodingPairwiseBound}). It can be verified that\n\\begin{eqnarray} \n\\label{Eqn:BinningRatesOfUser3SemiPrivateCodebooks}\n&S_{31}+S_{23}=I(U_{31};U_{23}|\\TimeSharingRV W), K_{1}\\!=\\!K_{2}\\!=\\!L_{31}\\!=\\!K_{23}\\!=\\!S_{3}\\!=\\!I(\\TimeSharingRV; Y_{3})\\!=\\!0,\\\\\n\\label{Eqn:K2K1L31K23S3AreZero}\n&I(\\SemiPrivateRV_{12} V_{1}X_{1}Y_{1}V_{2}X_{2}Y_{2}X_{3};Y_{3}|\\TimeSharingRV WU_{23}U_{31}V_{3})\\!=\\!0\\!\\\\\n\\label{Eqn:MarkovChainFoundInTestChannelThatIncludesY2ForUser2}\n&\\mbox{ and therefore }\n(V_1,V_2,X_2,U_{12}) - (\\TimeSharingRV WU_{23}U_{31}V_3) - Y_3\\mbox{ is a Markov chain}.\n\\end{eqnarray}\nThe second set of equalities in (\\ref{Eqn:BinningRatesOfUser2SemiPrivateCodebooks}), (\\ref{Eqn:BinningRatesOfUser3SemiPrivateCodebooks}) lets us conclude\n\\begin{equation}\n \\label{Eqn:RatesOfUsers12And3InTermsOfSimplifiedCodebooks}\nR_{1}=T_{1},R_{2}=L_{12}+T_{2} \\mbox{ and }R_{3}=K_{31}+T_{3}.\n\\end{equation}\nFrom $I(U_{12};U_{23}|\\TimeSharingRV W) + I(U_{31};U_{23}|\\TimeSharingRV W) = S_{12}+S_{23}+S_{31}+S_{23}$, and (\\ref{Eqn:3BCSourceCodingTripleBound}), we have $I(U_{12};U_{23}|\\TimeSharingRV W) + I(U_{31};U_{23}|\\TimeSharingRV W) \\geq I(U_{12};U_{23};U_{31}|\\TimeSharingRV W)+S_{23}$. The non-negativity of $S_{23}$ (\\ref{Eqn:3BCSourceCodingBoundNonnegativity}) implies $S_{23}=0$ and $I(U_{31};U_{12}|\\TimeSharingRV WU_{23})=0$. We therefore conclude\n\\begin{eqnarray}\n\\label{Eqn:SemiPrivateLayerBinningRates}\nS_{12}=I(U_{12};U_{23}|\\TimeSharingRV W),S_{31}=I(U_{31};U_{23}|\\TimeSharingRV W),S_{23}=0,\nI(U_{31};U_{12}|\\TimeSharingRV W U_{23})=0\n\\end{eqnarray}\nSubstituting (\\ref{Eqn:BinningRatesOfUser2SemiPrivateCodebooks}), (\\ref{Eqn:BinningRatesOfUser3SemiPrivateCodebooks}), (\\ref{Eqn:SemiPrivateLayerBinningRates}) in (\\ref{Eqn:3BCSourceCodingQuadrapleBound}) for $(i,j,k)=(2,3,1)$ and $(i,j,k)=(3,1,2)$ and (\\ref{Eqn:3BCSourceCodingPentaBound}) for $(i,j,k)=(2,3,1)$, we obtain\n\\begin{eqnarray}\n\\label{Eqn:IndependenceOfV2WithU31AndV3WithU12ConditionedOnWU23}\nI(V_{2};U_{31}|\\TimeSharingRV WU_{12}U_{23})=I(V_{3};U_{12}|\\TimeSharingRV WU_{23}U_{31})\n=I(V_{2};V_{3}|\\TimeSharingRV WU_{12}U_{23}U_{31})=0.\n\\end{eqnarray}\n(\\ref{Eqn:IndependenceOfV2WithU31AndV3WithU12ConditionedOnWU23}) and last equality in (\\ref{Eqn:SemiPrivateLayerBinningRates}) yield \n\\begin{equation}\n\\label{Eqn:V2U12AndV3U31AreIndependentConditionedOnWU23}\nI(V_{2}U_{12};V_{3}U_{31}|\\TimeSharingRV WU_{23})=0.\n\\end{equation}\nSubstituting (\\ref{Eqn:RatesOfUsers12And3InTermsOfSimplifiedCodebooks}), (\\ref{Eqn:SemiPrivateLayerBinningRates}) in (\\ref{Eqn:3BCChannelCodingDoubleBound})\\begin{comment}{ and (\\ref{Eqn:3BCChannelCodingSextupleBound})}\\end{comment} with $(i,j,k)=(2,3,1)$ yields the upper bound $R_2 \\leq I(U_{12}V_2;Y_2|\\TimeSharingRV WU_{23})$. Since\n\\begin{eqnarray}\n&\\!\\!\\!\\!1-h_{b}(\\delta_{2}) \\!=\\! R_{2} \\!\\leq\\! I(\\SemiPrivateRV_{12} \\PrivateRV_{2};Y_{2}|\\TimeSharingRV\\PublicRV \\SemiPrivateRV_{23}) \\!\\leq\\! I(\\PublicRV \\SemiPrivateRV_{12}\\SemiPrivateRV_{23} \\PrivateRV_{2};Y_{2}|\\TimeSharingRV) \\leq 1-h_{b}(\\delta_{2}),\\nonumber\n\\end{eqnarray}\nwhere the last inequality follows from (\\ref{Eqn:UsingDataProcessingInequalityInDerivingUpperBoundOnR2}), equality holds in all of the above inequalities to yield $I(W\\SemiPrivateRV_{23};Y_{2}|\\TimeSharingRV )=0$ and $I(\\SemiPrivateRV_{12} \\PrivateRV_{2};Y_{2}|\\TimeSharingRV\\PublicRV \\SemiPrivateRV_{23})=1-h_{b}(\\delta_{2})$. A similar argument proves $I(W\\SemiPrivateRV_{23};Y_{3}|\\TimeSharingRV )=0$ and $I(\\SemiPrivateRV_{31} \\PrivateRV_{3};Y_{3}|\\TimeSharingRV\\PublicRV \\SemiPrivateRV_{23})=1-h_{b}(\\delta_{3})$.\n\nWe have proved the Markov chains in (\\ref{Eqn:PartOfmarkovChainUsedInGettingToRateLoss}), (\\ref{Eqn:MarkovChainFoundInTestChannelThatIncludesY2ForUser2}). In order to prove Markov chains in item \\ref{ItemNumber:MarkovChains}, we prove the following lemma.\n\\begin{lemma}\n\\label{Lem:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput}\nIf $A,B,X,Y$ are discrete random variables such that (i) $X,Y$ take values in $\\left\\{0,1\\right\\}$ with $P(Y=0|X=1)=P(Y=1|X=0)=\\eta \\in (0,\\frac{1}{2})$, (ii) $A-B-Y$ and $AB-X-Y$ are Markov chains, then $A-B-XY$ is also a Markov chain.\n\\end{lemma}\nPlease refer to appendix \\ref{AppSec:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput} for a proof. Markov chains in (\\ref{Eqn:PartOfmarkovChainUsedInGettingToRateLoss}), (\\ref{Eqn:MarkovChainFoundInTestChannelThatIncludesY2ForUser2}) in conjunction with lemma \\ref{Lem:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput} establishes Markov chains in item \\ref{ItemNumber:MarkovChains}.\n\n(\\ref{Eqn:V2U12AndV3U31AreIndependentConditionedOnWU23}) and (\\ref{Eqn:K3K1L23K12S2AreZero}) imply $I(\\SemiPrivateRV_{31}\\PrivateRV_{3};\\SemiPrivateRV_{12}\\PrivateRV_{2}Y_{2}|\\TimeSharingRV\\PublicRV \\SemiPrivateRV_{23})=0$. This in conjunction with (\\ref{Eqn:K2K1L31K23S3AreZero}) implies\n\\begin{equation}\n \\label{Eqn:U31V3Y3AndU12V2Y2AreIndependentConditionedOnWU23}\nI(\\SemiPrivateRV_{31}\\PrivateRV_{3}Y_{3};\\SemiPrivateRV_{12}\\PrivateRV_{2}Y_{2}|\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23})=0\\mbox{ and thus }\\SemiPrivateRV_{31}\\PrivateRV_{3}Y_{3}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23}-\\SemiPrivateRV_{12}\\PrivateRV_{2}Y_{2}\\mbox{ is a Markov chain.}\n\\end{equation}\n(\\ref{Eqn:U31V3Y3AndU12V2Y2AreIndependentConditionedOnWU23}) implies $\\SemiPrivateRV_{31}Y_{3}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23}-\\SemiPrivateRV_{12}Y_{2}$ is a Markov chain, and therefore $Y_{3}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{12}\\SemiPrivateRV_{23}\\SemiPrivateRV_{31}-Y_{2}$ is a Markov chain. Employing lemma \\ref{Lem:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput} twice we observe $Y_{3}X_{3}-\\PublicRV \\SemiPrivateRV_{12}\\SemiPrivateRV_{23}\\SemiPrivateRV_{31}-X_{2}Y_{2}$ is a Markov chain and furthermore $X_{3}-\\TimeSharingRV \\PublicRV \\SemiPrivateRV_{12}\\SemiPrivateRV_{23}\\SemiPrivateRV_{31}-X_{2}$ is a Markov chain, thus proving item \\ref{Item:X2AndX3AreConditionallyIndependentGivenQWU12U23U31}.\n\nFinally, we prove Markov chains in item \\ref{Item:U31AndX2AreConditionallyIndependentGivenQWU12U23}. From Markov chain $(V_3,X_3,V_1,U_{31}) - (\\TimeSharingRV WU_{23}U_{12}V_2) - (X_2,Y_2)$ proved in item \\ref{ItemNumber:MarkovChains}, we have $I(X_{2};U_{31}|\\TimeSharingRV WU_{23}U_{12}V_{2})=0$. From (\\ref{Eqn:V2U12AndV3U31AreIndependentConditionedOnWU23}), we have $I(V_{2};U_{31}|\\TimeSharingRV WU_{23}U_{12})=0$. Summing these two, we have $I(X_{2}V_{2};U_{31}|\\TimeSharingRV WU_{23}U_{12})=0$ and therefore $I(X_{2};U_{31}|\\TimeSharingRV WU_{23}U_{12})=0$ implying the Markov chain $X_{2}-\\TimeSharingRV WU_{23}U_{12}-U_{31}$. Similarly, Markov chain $(V_2,X_2,V_1,U_{12}) - (\\TimeSharingRV WU_{23}U_{31}V_3) - (X_3,Y_3)$ proved in item \\ref{ItemNumber:MarkovChains} implies $I(X_{3};U_{12}|WU_{23}U_{31}V_{3}\\TimeSharingRV )=0$. From (\\ref{Eqn:V2U12AndV3U31AreIndependentConditionedOnWU23}), we have $I(V_{3};U_{12}|\\TimeSharingRV WU_{23}U_{31})=0$. Summing these two, we have $I(X_{3}V_{3};U_{12}|\\TimeSharingRV WU_{23}U_{31})=0$ and therefore $I(X_{3};U_{12}|\\TimeSharingRV WU_{23}U_{31})=0$ implying the Markov chain $X_{3}-\\TimeSharingRV WU_{23}U_{31}-U_{12}$.\n\\end{proof}\nLemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple} enables us simplify the bounds (\\ref{Eqn:3BCSourceCodingBoundNonnegativity})-(\\ref{Eqn:3BCChannelCodingSextupleBound}) for the particular test channel under consideration. Substituting (\\ref{Eqn:BinningRatesOfUser2SemiPrivateCodebooks})-(\\ref{Eqn:IndependenceOfV2WithU31AndV3WithU12ConditionedOnWU23}) in (\\ref{Eqn:3BCSourceCodingBoundNonnegativity})-(\\ref{Eqn:3BCChannelCodingSextupleBound}) and employing statements of lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple}, we conclude that if $(R_{1},1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\in \\alpha_{NEM}(p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV\\underlinePrivateRV X\\underlineY})$, then there exists nonnegative numbers $S_{1},T_{1},L_{12},K_{31}$ that satisfy $R_{1}=T_{1},R_{2}=L_{12}+T_{2}=1-h_{b}(\\delta_{2}),R_{3}=K_{31}+T_{3}=1-h_{b}(\\delta_{3})$,\n\\begin{eqnarray}\n \\label{Eqn:BoundsOnUser1CodebookAsAResultOfUser1Decoding}\n&S_1 \\geq I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31})\\begin{comment}{\\define a_{1}}\\end{comment},~~~T_1+S_1\\leq I(V_1;Y_1|\\TimeSharingRV WU_{12}U_{31})\\begin{comment}{\\define a_{2}}\\end{comment} \\\\\n\\label{Eqn:BoundsOnRatesOfUser123CodebooksAsAResultOfUser1Decoding}\n&L_{12}+K_{31}+T_1+S_1 \\leq I(U_{12};U_{31}|\\TimeSharingRV W)-I(U_{23};U_{12}|\\TimeSharingRV W)+I(V_1U_{12}U_{31};Y_1|\\TimeSharingRV W)-I(U_{23};U_{31}|\\TimeSharingRV W)\\begin{comment}{\\define a_{5}}\\end{comment}\\\\\n\\label{Eqn:BoundsOnRatesOfUser2CodebookAsAResultOfUser2Decoding}\n&0 \\leq T_{2} \\leq I(V_2;Y_2|\\TimeSharingRV WU_{12}U_{23})\\begin{comment}{\\define a_{6}}\\end{comment},~~~1-h_{b}(\\delta_{2})=T_{2}+L_{12} = I(U_{12}V_2;Y_2|\\TimeSharingRV WU_{23})\\begin{comment}{\\define a_{7}}\\end{comment}\\\\\n\\label{Eqn:BoundsOnRatesOfUser3CodebookAsAResultOfUser3Decoding}\n&0 \\leq T_{3}\\leq I(V_3;Y_3|\\TimeSharingRV WU_{31}U_{23})\\begin{comment}{\\define a_{8}}\\end{comment},~~~1-h_{b}(\\delta_{3})=T_{3}+K_{31}= I(U_{31}V_3;Y_3|\\TimeSharingRV WU_{23})\\begin{comment}{\\define a_{9}}\\end{comment}.\n\\end{eqnarray}\n(\\ref{Eqn:BoundsOnRatesOfUser2CodebookAsAResultOfUser2Decoding}), (\\ref{Eqn:BoundsOnRatesOfUser3CodebookAsAResultOfUser3Decoding}) imply\n\\begin{equation}\n \\label{Eqn:LowerBoundsOnRateOfCodebooksSharedWithUser1}\nL_{12}\\geq I(U_{12};Y_{2}|\\TimeSharingRV \\PublicRV U_{23}),~~~~~~~~~~~ K_{31} \\geq I(U_{31};Y_3|\\TimeSharingRV WU_{23}),\n\\end{equation}\n(\\ref{Eqn:BoundsOnUser1CodebookAsAResultOfUser1Decoding}) implies\n\\begin{eqnarray}\nT_{1}\\!\\!\\!\\!&=&\\!\\!\\!\\!R_1 \\leq I(V_1;Y_1|\\TimeSharingRV W\\SemiPrivateRV_{12}\\SemiPrivateRV_{31})\n -I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31}), \\nonumber\\\\\n\\label{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}\n&\\leq&\\!\\!\\!\\! I(V_1;Y_1U_{23}|\\TimeSharingRV W\\SemiPrivateRV_{12}\\SemiPrivateRV_{31})\n -I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31})=I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV)\n -I(V_1;V_2V_3|\\TimeSharingRV W\\underlineSemiPrivateRV),\n\\end{eqnarray}\nand (\\ref{Eqn:BoundsOnRatesOfUser123CodebooksAsAResultOfUser1Decoding}) in conjunction with (\\ref{Eqn:LowerBoundsOnRateOfCodebooksSharedWithUser1}), and lower bound on $S_{1}$ in (\\ref{Eqn:BoundsOnUser1CodebookAsAResultOfUser1Decoding}) imply\n\\begin{eqnarray}\nR_1 \\!\\!\\!\\!&\\leq&\\!\\!\\!\\! I(U_{12}U_{31}V_1;Y_1|\\TimeSharingRV W)\n-I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31})-I(U_{12};Y_2|\\TimeSharingRV WU_{23})-I(U_{31};Y_3|\\TimeSharingRV WU_{23}) \\nonumber\\\\\n&&\\!\\!\\!\\!+I(U_{12};U_{31}|\\TimeSharingRV W)-I(U_{23};U_{12}|\\TimeSharingRV W)-I(U_{23};U_{31}|\\TimeSharingRV W)\\nonumber\\\\\n\\!\\!\\!\\!&\\leq&\\!\\!\\!\\! I(U_{12}U_{31}V_{1};Y_1 U_{23}|\\TimeSharingRV W)\n-I(V_1;U_{23}V_2V_3|\\TimeSharingRV WU_{12}U_{31})-I(U_{12};Y_2|\\TimeSharingRV WU_{23})-I(U_{31};Y_3|\\TimeSharingRV WU_{23})\\nonumber\\\\\n&&\\!\\!\\!\\!+I(U_{12};U_{31}|\\TimeSharingRV W)-I(U_{23};U_{12}|\\TimeSharingRV W)-I(U_{23};U_{31}|\\TimeSharingRV W)\\nonumber\\\\\\label{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}\n\\!\\!\\!\\!&=&\\!\\!\\!\\!I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) \\!-\\!I(V_1;V_2V_3|\\TimeSharingRV W\\underlineSemiPrivateRV)\\! +\\!I(U_{12}U_{31};Y_1|\\TimeSharingRV W U_{23}) \\!-\\!I(U_{12};Y_2|\\TimeSharingRV WU_{23})\\!-\\!I(U_{31};Y_3|\\TimeSharingRV WU_{23}),\n\\end{eqnarray}\nwhere (\\ref{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}) follows from the last equality in (\\ref{Eqn:SemiPrivateLayerBinningRates}). We have thus obtained (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}) and (\\ref{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}), two upper bounds on $R_{1}$ we were seeking, and this concludes the first stage of our proof. In the sequel, we prove the minimum of the above upper bounds on $R_{1}$ is strictly lesser than $h_{b}(\\tau * \\delta_{1})-h_{b}(\\delta_{1})$. Towards, that end, note that upper bound (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}) contains the rate loss due to precoding. In the second stage, we work on (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}) and derive conditions under which there is \\textit{no} rate loss.\n\nMarkov chains of lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple} item \\ref{ItemNumber:MarkovChains} imply $V_{1}-QW\\underlineSemiPrivateRV V_{2}V_{3}-X_{2}$ and $V_{1}-QW\\underlineSemiPrivateRV V_{2}V_{3}X_{2}-X_{3}$ are Markov chains. Therefore, $I(V_{1};X_{2}|QW\\underlineSemiPrivateRV V_{2}V_{3})=0$ and $I(V_{1};X_{3}|QW\\underlineSemiPrivateRV V_{2}V_{3}X_{2})=0$. Summing these, we have $I(V_{1};X_{2}X_{3}|QW\\underlineSemiPrivateRV V_{2}V_{3})=0$. Employing this in (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}),\\begin{comment}{ abbreviating $\\tildeQRV \\define \\TimeSharingRV \\PublicRV \\SemiPrivateRV_{23}$,\\footnote{This notation is employed for the rest of this section.} $\\tildeQRVSet \\define \\TimeSharingRVSet \\times \\PublicRVSet \\times \\SemiPrivateRVSet_{23} $,\\addtocounter{footnote}{-1}\\footnotemark and letting $\\tildeq \\define (q,w,u_{23})$\\addtocounter{footnote}{-1}\\footnotemark denote a generic element in $\\tildeQRVSet$,}\\end{comment} we note\n\\begin{eqnarray}\n\\label{eq:inequality1}\nR_1 &\\leq& I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) -I(V_1;V_2V_3|\\TimeSharingRV W\\underlineSemiPrivateRV) =I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) -I(V_1;V_2V_3X_2X_3|\\TimeSharingRV W\\underlineSemiPrivateRV) \\\\\n&\\leq& I(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) - I(V_1;X_2,X_3|\\TimeSharingRV W\\underlineSemiPrivateRV) \\leq I(V_1;Y_1 |\\TimeSharingRV W\\underlineSemiPrivateRV) -I(V_1;X_2\\oplus X_3|\\TimeSharingRV W\\underlineSemiPrivateRV) \\label{eq:inequality3}\\\\\n&=&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}}\\!\\!\\!\\! p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) \\left[ I(V_1;Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u})) \\right]\\label{eq:inequality4}\\\\\n&\\leq&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}}\\!\\!\\!\\! p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) I(X_{1}X_{2}X_{3}V_1;Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\\nonumber\\\\\n&\\leq&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}}\\!\\!\\!\\! p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) \\left[H(Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))-H(Y_1|X_{1}X_{2}X_{3}V_{1},(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\\right]\\nonumber\\\\\n&=&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}}\\!\\!\\!\\! p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) \\left[H(X_{1}\\oplus N_{1}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))-h_{b}(\\delta_{1})\\right]\\nonumber\\\\\n\\label{Eqn:TheStepJustBeforeApplicationOfJensenInequality}\n&=&\\sum_{\\substack{(q,w,\\underline{u})\\in\\\\\\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV}} p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u}) h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1}), \\mbox{ where }\\tau_{q,w,\\underline{u}}=p_{X_{1}|QW\\underlineSemiPrivateRV}(1|q,w,\\underline{u})\\\\\n\\begin{comment}{\n&=&\\sum_{\\tildeq \\in \\tildeQRVSet}p_{\\tildeQRV}(\\tildeq)\\sum_{\\substack{(u_{12},u_{31}) \\in\\\\ \\SemiPrivateRVSet_{12} \\times \\SemiPrivateRVSet_{31}}}p_{ U_{12} U_{31}|\\tildeQRV}(u_{12},u_{31}|\\tildeq) h_{b}(\\tau_{\\tildeq,u_{12},u_{31}}*\\delta_{1})-h_{b}(\\delta_{1}), \\nonumber\\\\\n&=&\\sum_{\\tildeq \\in \\tildeQRVSet}p_{\\tildeQRV}(\\tildeq)\\Expectation_{U_{12}U_{31}|\\tildeQRV=\\tildeq} \\left\\{h_{b}(\\tau_{\\tildeq,u_{12},u_{31}}*\\delta_{1})\\right\\}-h_{b}(\\delta_{1}), \\nonumber\\\\\n\\label{Eqn:ApplicationOfJensensInequalityFirstInstance}\n&\\leq&\\sum_{\\tildeq \\in \\tildeQRVSet}p_{\\tildeQRV}(\\tildeq)h_{b}(\\Expectation_{U_{12}U_{31}|\\tildeQRV=\\tildeq} \\left\\{\\tau_{\\tildeq,u_{12},u_{31}}*\\delta_{1}\\right\\})-h_{b}(\\delta_{1})=\\sum_{\\tildeq \\in \\tildeQRVSet}p_{\\tildeQRV}(\\tildeq)\\left\\{h_{b}(\\tau_{\\tildeq}*\\delta_{1})-h_{b}(\\delta_{1})\\right\\}, \\\\\n}\\end{comment}\n\\label{Eqn:ApplicationOfJensensInequality}\n&=&\\Expectation_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV} \\left\\{h_{b}(\\tau_{q,w,\\underline{u}} * \\delta_{1})\\right\\}-h_{b}(\\delta_{1})\\leq h_{b}(\\Expectation_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV} \\left\\{ \\tau_{q,w,\\underline{u}} * \\delta_{1}\\right\\})-h_{b}(\\delta_{1})\\leq h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})\n\\end{eqnarray}\nwhere (\\ref{Eqn:ApplicationOfJensensInequality}) follows from application of Jensen's inequality to the strictly concave function $h_{b}(\\cdot)$, and second inequality in (\\ref{Eqn:ApplicationOfJensensInequality}) follows from $\\delta \\in (0,\\frac{1}{2})$. We conclude that $R_{1}=h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})$ if and only if equality holds in the above chain of inequalities, and in particular, equality holds in (\\ref{Eqn:ApplicationOfJensensInequality}), which by the condition for equality in Jensen's inequality implies $\\tau_{q,w,\\underline{u}}=\\tau $ for every $(q,w,\\underline{u}) \\in \\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV$ that satisfies $p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u})>0$. This in conjunction with\n\\begin{equation}\nI(V_1;Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u})) \\leq h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1}) \\nonumber\n\\end{equation}\nwhich follows from the chain of inequalities from (\\ref{eq:inequality4}) through (\\ref{Eqn:TheStepJustBeforeApplicationOfJensenInequality}) implies\n\\begin{equation}\n \\label{Eqn:AfterProcessingTheRateLossUpperBoundAndFitForApplicationOfRateLossLemma}\nI(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV) -I(V_1;V_2V_3|\\TimeSharingRV W\\underlineSemiPrivateRV) \\leq h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1})\n\\end{equation}\nwith equality if and only if\n\\begin{eqnarray}\n\\label{Eqn:EqualityInJensenImpliesChannelMustOperateAtUniformCost}\n&\\mbox{for every }(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRV \\mbox{ that satisfies }p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u})>0, p_{X_{1}|QW\\underlineSemiPrivateRV}(1|q,w,\\underline{u})=\\tau_{q,w,\\underline{u}}=\\tau,~~~~\\\\\n\\label{Eqn:NoRateLossForEveryConditionalTestChannel}\n&\\mbox{ and }I(V_1;Y_1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u})) = h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1}).\n\\end{eqnarray}\n\nAn informed reader, by now must have made the connection to capacity of the point to point channel with non-causal state\n\\cite{1980MMPCT_GelPin}. We develop this connection in appendix \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx}. For now, we provide a characterization for (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) to hold. This will require us to define a few mathematical objects that may initially seem unrelated to a reader unaware of findings in \\cite{1980MMPCT_GelPin}. Very soon, we argue the relevance. An informed reader will find the following development natural.\n\nLet $\\mathbb{D}_{T}(\\tau,\\delta,\\epsilon)$ denote the collection of all probability mass functions $p_{\\GPV\\GPS\\GPX\\GPY}$ defined on $\\GPSetV \\times \\left\\{ 0,1 \\right\\}\\times \\left\\{ 0,1 \\right\\}\\times \\left\\{ 0,1 \\right\\}$, where $\\GPSetV$ is an arbitrary finite set such that (i) $p_{\\GPY|\\GPX\\GPS\\GPV}(x\\oplus s|x,s,v)=p_{\\GPY|\\GPX\\GPS}(x\\oplus s|x,s)=1-\\delta$, where $\\delta \\in (0,\\frac{1}{2})$, (ii) $p_{\\GPS}(1)=\\epsilon \\in [0,1]$, and (iii) $p_{\\GPX}(1)\\leq \\tau \\in (0,\\frac{1}{2})$. For $p_{\\GPV\\GPS\\GPX\\GPY} \\in \\mathbb{D}_{T}(\\tau,\\delta,\\epsilon)$, let $\\alpha_{T}(p_{\\GPV\\GPS\\GPX\\GPY})=I(\\GPV;\\GPY)-I(\\GPV;\\GPS)$ and $\\alpha_{T}(\\tau,\\delta,\\epsilon)=\\sup_{p_{\\GPV\\GPS\\GPX\\GPY} \\in \\mathbb{D}_{T}(\\tau,\\delta,\\epsilon)}\\alpha_{T}(p_{\\GPV\\GPS\\GPX\\GPY})$.\n\n\nFor every $(q,w,\\underline{u}) \\in \\mathcal{Q} \\times \\mathcal{W} \\times \\underlineSemiPrivateRV$ that satisfies $p_{\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(q,w,\\underline{u})>0$, we note $p_{Y_{1}|X_{1},X_{2}\\oplus X_{3}\\PrivateRV_{1}\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(x_{1}\\oplus x_{2}\\oplus x_{3}|x_{1},x_{2}\\oplus x_{3},v_{1},q,w,\\underline{u})=p_{Y_{1}|X_{1},X_{2}\\oplus X_{3}\\TimeSharingRV\\PublicRV\\underlineSemiPrivateRV}(x_{1}\\oplus x_{2}\\oplus x_{3}|x_{1},x_{2}\\oplus x_{3},q,w,\\underline{u})=1-\\delta_{1}$. In other words, conditioned on the event $\\left\\{ (\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)= (q,w,\\underline{u})\\right\\}$, $V_{1}-X_{1},X_{2}\\oplus X_{3}-Y_{1}$ is a Markov chain. We conclude $p_{V_{1}X_{2}\\oplus X_{3}X_{1}Y_{1}|QW\\underlineSemiPrivateRV}(\\cdots|q,w,\\underline{u}) \\in \\mathbb{D}_{T}(\\tau_{q,w,\\underline{u}},\\delta_{1},\\epsilon_{q,w,\\underline{u}})$, where $\\epsilon_{q,w,\\underline{u}}=p_{X_{2}\\oplus X_{3}|QW\\underlineSemiPrivateRV}(1|q,w,\\underline{u})$, and hence\n\\begin{equation}\n \\label{Eqn:CharacterizationOfAchievableRateForUser1InTermsOfGelfandPinskerCapacity}\nI(V_1;Y_1|(\\TimeSharingRV ,\\PublicRV, \\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV ,\\PublicRV ,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))\\leq \\alpha_{T}(\\tau_{q,w,\\underline{u}},\\delta_{1},\\epsilon_{q,w,\\underline{u}}). \\nonumber\n\\end{equation}\nTherefore, (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) holds only if $\\alpha_{T}(\\tau_{q,w,\\underline{u}},\\delta_{1},\\epsilon_{q,w,\\underline{u}}) = h_{b}(\\tau_{q,w,\\underline{u}}*\\delta_{1})-h_{b}(\\delta_{1})$, where $\\tau_{q,w,\\underline{u}}=\\tau \\in (0,\\frac{1}{2})$. The following lemma characterizes conditions under which this is the case. Please refer to appendices \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx},\\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss} for a proof.\n\\begin{lemma} \n\\label{lem:GP}\nIf $\\tau ,\\delta \\in (0,\\frac{1}{2})$ and $\\epsilon \\in (0,1)$, then $\\alpha_{T}(\\tau,\\delta,\\epsilon) < h_b(\\tau * \\delta)-h_b(\\delta)$. Alternatively, if $\\tau ,\\delta \\in (0,\\frac{1}{2})$ and $\\epsilon \\in [0,1]$, then either $\\alpha_{T}(\\tau,\\delta,\\epsilon) < h_b(\\tau * \\delta)-h_b(\\delta)$ or $\\epsilon \\in \\left\\{ 0,1\\right\\}$.\\end{lemma}\nRecall that arguments in relation to (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) imply that if for any $(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRVSet$ that satisfies $P((\\TimeSharingRV, \\PublicRV, \\underlineSemiPrivateRV)=(q,w,\\underline{u})) >0$, \n\\begin{equation}\n\\label{Eqn:NoRateLossForEveryConditionalTestChannelRestated}\n I(V_1;Y_1|(\\TimeSharingRV ,\\PublicRV, \\underlineSemiPrivateRV)=(q,w,\\underline{u}))\n\\!-\\!I(V_1;X_{2}\\oplus X_{3}|(\\TimeSharingRV ,\\PublicRV ,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))0$. From lemma \\ref{lem:GP}, we conclude for every such $(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underlineSemiPrivateRVSet$, $\\epsilon_{q,w,\\underline{u}}=p_{X_{2}\\oplus X_{3}|QW\\underlineSemiPrivateRV}(1|q,w,\\underline{u}) \\in \\left\\{ 0,1 \\right\\}$. We therefore assume \n\\begin{equation}\n \\label{Eqn:CaseOfNoRateLoss}\nI(V_1;Y_1|\\TimeSharingRV W\\underlineSemiPrivateRV)\n-I(V_1;X_2\\oplus X_3|\\TimeSharingRV W\\underlineSemiPrivateRV) = h_b(\\tau * \\delta_{1}) -h_b(\\delta_{1})\\mbox{ and }H(X_2 \\oplus X_3|\\TimeSharingRV W\\underlineSemiPrivateRV)=0.\n\\end{equation}\nThis has got us to the third and final stage. Here we argue (\\ref{Eqn:CaseOfNoRateLoss}) implies RHS of (\\ref{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}) is strictly smaller than $h_b(\\tau * \\delta_{1}) -h_b(\\delta_{1})$. Towards that end, note that Markov chain $X_{2}-\\TimeSharingRV WU_{23}U_{12}U_{31}-X_{3}$ proved in lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple} item \\ref{Item:X2AndX3AreConditionallyIndependentGivenQWU12U23U31}\nand (\\ref{Eqn:CaseOfNoRateLoss}) imply $H(X_{2}|\\TimeSharingRV W\\underlineSemiPrivateRV)=H(X_{3}|\\TimeSharingRV W\\underlineSemiPrivateRV)=0$.\\footnote{Indeed, for any $(q,w,\\underline{u}) \\in \\TimeSharingRVSet \\times \\PublicRVSet \\times \\underline{\\SemiPrivateRVSet}$ that satisfies $P((\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))>0$, if $P(X_{j}=1|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))=\\alpha_{j}:j=2,3$, then $0=H(X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u}))=h_{b}(\\alpha_{2}*\\alpha_{3}) \\geq \\alpha_{2}h_{b}(\\alpha_{3})+(1-\\alpha_{2})h_{b}(1-\\alpha_{3})=\\alpha_{2}h_{b}(\\alpha_{3})+(1-\\alpha_{2})h_{b}(\\alpha_{3})=h_{b}(\\alpha_{3})\\geq 0$, where the first inequality follows from concavity of binary entropy function, and similarly, interchanging the roles of $\\alpha_{2}, \\alpha_{3}$, we obtain $0=H(X_{2}\\oplus X_{3}|(\\TimeSharingRV,\\PublicRV,\\underlineSemiPrivateRV)=(q,w,\\underline{u})) \\geq h_{b}(\\alpha_{2})\\geq 0$.} Furthermore, Markov chains $\\SemiPrivateRV_{12}-\\PublicRV \\SemiPrivateRV_{23} \\SemiPrivateRV_{31}-X_{3}$ and $\\SemiPrivateRV_{31}-\\PublicRV \\SemiPrivateRV_{23} \\SemiPrivateRV_{12}-X_{2}$ proved in lemma \\ref{Lem:CharacterizationOfTestChannelThatContainsTheRateTriple} item \\ref{Item:U31AndX2AreConditionallyIndependentGivenQWU12U23} imply\n\\begin{equation}\n\\label{Eqn:InTheCaseOfNoRateLossReeciever1KnowsX2AndX3}\nH(X_2|WU_{23}U_{12})=H(X_3|WU_{23}U_{31})=0.\n\\end{equation}\nObserve that\n\\begin{eqnarray}\n\\label{Eqn:SubstitutingForNoRateLossCaseInUpperBound1}\nh_b(\\tau * \\delta_{1}) -h_b(\\delta_{1})\\!\\!&=&\\!\\!I(V_1;Y_1|W\\underlineSemiPrivateRV) -I(V_1;X_2\\oplus X_3 W\\underlineSemiPrivateRV) = I(V_1;Y_1| W \\underlineSemiPrivateRV ) =I(V_1;Y_1| W \\underlineSemiPrivateRV ,X_2,X_3) \\\\\n\\!\\!&=&\\!\\! H(Y_{1}|W\\underlineSemiPrivateRV X_{2}X_{3})-H(Y_{1}|W\\underlineSemiPrivateRV V_{1}X_{2}X_{3})\n\\leq H(Y_{1}|W\\underlineSemiPrivateRV X_{2}X_{3})-H(Y_{1}|W\\underlineSemiPrivateRV V_{1}X_{1}X_{2}X_{3})\\nonumber\\\\\n\\label{Eqn:EntropyOfY1ConditionedOnX2AndX3}\n\\!\\!&=&\\!\\!H(Y_{1}| W \\underlineSemiPrivateRV ,X_2,X_3)-h_b(\\delta_{1})\n\\end{eqnarray}\nwhere the first two equalities in (\\ref{Eqn:SubstitutingForNoRateLossCaseInUpperBound1}) follows from (\\ref{Eqn:CaseOfNoRateLoss}) and the last equality follows from (\\ref{Eqn:InTheCaseOfNoRateLossReeciever1KnowsX2AndX3}). (\\ref{Eqn:EntropyOfY1ConditionedOnX2AndX3}) and first equality in (\\ref{Eqn:SubstitutingForNoRateLossCaseInUpperBound1}) enables us conclude\n\\begin{equation}\n \\label{Eqn:LowerBoundOnEntropyOfY1ConditionedOnX2AndX3}\nH(Y_{1}| W \\underlineSemiPrivateRV ,X_2,X_3) \\geq h_b(\\tau * \\delta_{1})\n\\end{equation}\nWe now upper bound RHS of (\\ref{Eqn:BoundOnUser1RateAsAConsequenceOfDecoderDecoding3Codebooks}). Note that it suffices to prove $I(U_{12}U_{31};Y_1|WU_{23})-I(U_{12};Y_2|WU_{23}) -I(U_{31};Y_3|WU_{23})$ is negative. Observe that\n\\begin{eqnarray}\n\\lefteqn{I(U_{12}U_{31};Y_1|WU_{23})-I(U_{12};Y_2|WU_{23})\n-I(U_{31};Y_3|WU_{23})}\\nonumber\\\\ \n&&=H(Y_{1}|WU_{23})-H(Y_{1}|W\\underlineSemiPrivateRV)-H(Y_{2}|WU_{23})+H(Y_{2}|WU_{23}U_{12})-H(Y_{3}|WU_{23})+H(Y_{3}|WU_{23}U_{31})\n\\nonumber\\\\\n&&=H(Y_{1}|WU_{23})-H(Y_{1}|W\\underlineSemiPrivateRV)-H(Y_{2})+H(Y_{2}|WU_{23}U_{12})-H(Y_{3})+H(Y_{3}|WU_{23}U_{31})\n\\nonumber\\\\\n\\label{Eqn:X2IsAFunctionOfWU23U12AndX3IsAFunctionOfWU23U31}\n&&=H(Y_{1}|WU_{23})-H(Y_{1}|WX_{2}X_{3}\\underlineSemiPrivateRV)-H(Y_{2})+H(Y_{2}|WU_{23}U_{12}X_{2})-H(Y_{3})+H(Y_{3}|WU_{23}U_{31}X_{3})\n\\\\\n&&= H(Y_{1}|WU_{23})-H(Y_{1}|WX_{2}X_{3}\\underlineSemiPrivateRV)-2+h_{b}(\\delta_{2})+h_{b}(\\delta_{3})\n\\nonumber\\\\\n\\label{Eqn:SubstitutingLowerBoundOnEntropyOfY1}\n&&\\leq 1-H(Y_{1}|WX_{2}X_{3}\\underlineSemiPrivateRV)-2+h_{b}(\\delta_{2})+h_{b}(\\delta_{3})\n\\leq h_{b}(\\delta_{2})+h_{b}(\\delta_{3})-h_b(\\delta_{1}*\\tau)-1\n\\end{eqnarray}\nwhere (\\ref{Eqn:X2IsAFunctionOfWU23U12AndX3IsAFunctionOfWU23U31}) follows from (\\ref{Eqn:CaseOfNoRateLoss}) and (\\ref{Eqn:InTheCaseOfNoRateLossReeciever1KnowsX2AndX3}), second inequality in (\\ref{Eqn:SubstitutingLowerBoundOnEntropyOfY1}) follows from (\\ref{Eqn:LowerBoundOnEntropyOfY1ConditionedOnX2AndX3}).\nIf $\\tau,\\delta_{1},\\delta_{2},\\delta_{3}$ are such that $h_b(\\delta_{2})+h_{b}(\\delta_{3}) < 1+h_b(\\delta_{1}*\\tau)\n$, then $R_1 < h_b(\\tau *\\delta_{1})-h_b(\\delta_{1})$ and RHS of (\\ref{Eqn:SubstitutingLowerBoundOnEntropyOfY1}) is negative. We summarize our findings in the following theorem and corollary.\n\n\\begin{thm}\n \\label{Thm:ConditionsUnderWhichMartonCodingTechniqueDoesNotAchieveRateTriple}\nConsider the 3-DBC in example \\ref{Ex:3-BCExample}. If $h_b(\\delta_{2})+h_{b}(\\delta_{3}) < 1+h_b(\\delta_{1}*\\tau)\n$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta_{2}),1-h_{b}(\\delta_{3})) \\notin \\alpha_{NEM}(\\tau)$.\n\\end{thm}\n\n\\begin{corollary}\n\\label{Cor:ConditionsUnderWhichMartonCodingTechniqueIsStrictlySubOptimal}\nConsider the 3-DBC in example \\ref{Ex:3-BCExample} with $\\delta=\\delta_{2}=\\delta_{3}$. If $h_{b}(\\tau*\\delta_{1}) \\leq h_b(\\delta) < \\frac{1+h_b(\\delta_{1}*\\tau)}{2}$, then $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta),1-h_{b}(\\delta)) \\notin \\alpha_{NEM}(\\tau)$ but $(h_{b}(\\tau*\\delta_{1})-h_{b}(\\delta_{1}),1-h_{b}(\\delta),1-h_{b}(\\delta)) \\in \\mathbb{C}(\\tau)$ and thus $\\alpha_{NEM}(\\tau) \\neq \\mathbb{C}(\\tau)$. In particular, if $\\delta_{1}=0.01$ and $\\delta_{2} \\in (0.1325,0.21)$, then $\\alpha_{NEM}(\\frac{1}{8}) \\neq \\mathbb{C}(\\frac{1}{8})$.\n\\end{corollary}\n\n\\appendices\n\\section{Characterization for no rate loss in point to point channels with channel state information}\n\\label{AppSec:CharacterizationForNoRateLossInPTP-STx}\n\nWe now develop the connection between upper bound (\\ref{Eqn:UpperBoundOnRateOfUser1ThatContainsRateLoss}) and the capacity of a point to point channel with non-causal state \\cite{1980MMPCT_GelPin}. We only describe the relevant additive channel herein and refer the interested reader to either to \\cite{1980MMPCT_GelPin} or \\cite[Chapter 7]{201201NIT_ElgKim} for a detailed study. The notation employed in this section and appendix \\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss} is specific to these sections.\n\nConsider a point to point channel with binary input and output alphabets $\\mathcal{X}=\\mathcal{Y}=\\left\\{0,1\\right\\}$. The channel transition probabilities depend on a random parameter, called state that takes values in the binary alphabet $\\mathcal{S}=\\left\\{ 0,1 \\right\\}$. The discrete time channel is time-invariant, memoryless and used without feedback. The channel is additive, i.e., if $S,X$ and $Y$ denote channel state, input and output respectively, then $P(Y=x \\oplus s | X=x, S=s)=1-\\delta$, where $\\oplus$ denotes addition in binary field and $\\delta \\in (0,\\frac{1}{2})$. The state is independent and identically distributed across time with $P(S=1)=\\epsilon \\in (0,1)$.\\footnote{Through appendices \\ref{AppSec:CharacterizationForNoRateLossInPTP-STx},\\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss} we prove if $\\delta,\\tau \\in (0,\\frac{1}{2})$ and $\\epsilon \\in (0,1)$, then $\\alpha_{T}(\\tau,\\eta,\\epsilon) < h_b(\\tau * \\eta)-h_b(\\eta)$. This implies statement of lemma \\ref{Lem:CharacterizationOfConditionThatEnsuresNoRateLoss}.} The input is constrained by an additive Hamming cost, i.e., the cost of transmitting $x^{n} \\in \\InputAlphabet^{n}$ is $\\sum_{t=1}^{n}1_{\\left\\{x_{t}=1\\right\\}}$ and average cost of input per symbol is constrained to be $\\tau \\in (0,\\frac{1}{2})$. \n\nThe quantities of interest - left and right hand sides of (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) - are related to two scenarios with regard to knowledge of state for the above channel. In the first scenario we assume the state sequence is available to the encoder non-causally and the decoder has no knowledge of the same. In the second scenario, we assume knowledge of state is available to both the encoder and decoder non-causally. Let $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon),\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$ denote the capacity of the channel in the first and second scenarios respectively. It turns out, the left hand side of (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) is upper bounded by $\\mathcal{C}(\\tau,\\delta,\\epsilon)$ and the right hand side of (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) is $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$. A necessary condition for (\\ref{Eqn:NoRateLossForEveryConditionalTestChannel}) to hold, is therefore $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$. For the point to point channel with non-causal state, this equality is popularly referred to as \\textit{no rate loss}. We therefore seek the condition for no rate loss.\n\nThe objective of this section and appendix \\ref{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss} is to study the condition under which $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$. In this section, we characterize each of these quantities, in the standard information theoretic way, in terms of a maximization of an objective function over a particular collection of probability mass functions.\n\nWe begin with a characterization of $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ and $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\n\\begin{definition}\n \\label{Defn:TestChannelsForPTP-STx}\nLet $\\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ denote the set of all probability mass functions $p_{USXY}$ defined on $\\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ that satisfy (i) $p_{S}(1) = \\epsilon$, (ii) $p_{Y|XSU}(x\\oplus s|x,s,u)=p_{Y|XS}(x\\oplus s | x,s)=1-\\delta$, (iii) $P(X=1)\\leq \\tau$. For $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, let $\\alpha_{T}(p_{USXY}) = I(U;Y)-I(U;S)$ and $\\alpha_{T}(\\tau,\\delta,\\epsilon) = \\underset{p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)}{\\sup} \\alpha_{T}(p_{USXY})$.\n\\end{definition}\n\n\\begin{thm}\n \\label{Thm:CapacityOfBinaryAdditivePTP-STx}\n$\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon) = \\alpha_{T}(\\tau,\\delta,\\epsilon)$\n\\end{thm}\nThis is a well known result in information theory and we refer the reader to \\cite{1980MMPCT_GelPin} or \\cite[Section 7.6, Theorem 7.3]{201201NIT_ElgKim} for a proof.\n\\begin{definition}\n \\label{Defn:TestChannelsForPTP-STxRx}\nLet $\\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ denote the set of all probability mass functions $p_{SXY}$ defined on $\\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ that satisfy (i) $p_{S}(1) = \\epsilon$, (ii) $p_{Y|XS}(x\\oplus s|x,s)=1-\\delta$, (iii) $P(X=1)\\leq \\tau$. For $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$, let $\\alpha_{TR}(p_{SXY}) = I(X;Y|S)$ and $\\alpha_{TR}(\\tau,\\delta,\\epsilon) = \\underset{p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)}{\\sup} \\alpha_{TR}(p_{SXY})$.\n\\end{definition}\n\n\\begin{thm}\n \\label{Thm:CapacityOfBinaryAdditivePTP-STxRx}\n$\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon) = \\alpha_{TR}(\\tau,\\delta,\\epsilon)$\n\\end{thm}\nThis can be argued using Shannon's characterization of point to point channel capacity \\cite{194807BSTJ_Sha} and we refer the reader to \\cite[Section 7.4.1]{201201NIT_ElgKim} for a proof.\n\\begin{remark}\n\\label{Rem:SideInformationAtBothEncoderAndDecoderResultsInLargerCapacity}\n From the definition of $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ and $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, it is obvious that $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon) \\leq \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, we provide an alternative argument based on theorems \\ref{Thm:CapacityOfBinaryAdditivePTP-STx}, \\ref{Thm:CapacityOfBinaryAdditivePTP-STxRx}. For any $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, it is easy to verify the corresponding marginal $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ and moreover $\\alpha_{T}(p_{USXY})=I(U;Y)-I(U;S) \\leq I(U;YS)-I(U;S) = I(U;Y|S)=H(Y|S)-H(Y|US) \\leq H(Y|S)-H(Y|USX) \\overset{(a)}{=} H(Y|S)-H(Y|SX)=I(X;Y|S)=\\alpha_{TR}(p_{SXY})\\leq \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, where (a) follows from Markov chain $U-(S,X)-Y$ ((ii) of definition \\ref{Defn:TestChannelsForPTP-STx}). Since this this true for every $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, we have $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon) \\leq \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\end{remark}\n\nWe provide an alternate characterization for $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\begin{lemma}\n \\label{Lem:AlternateCharacterizationForCapacityOfBinaryAdditivePTP-STxRx}\nFor $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, let $\\beta_{TR}(p_{USXY}) = I(U;Y|S)$ and $\\beta_{TR}(\\tau,\\delta,\\epsilon) = \\underset{p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)}{\\sup} \\beta_{TR}(p_{USXY})$. Then $\\beta_{TR}(\\tau,\\delta,\\epsilon) = \\alpha_{TR}(\\tau,\\delta,\\epsilon) = \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\end{lemma}\n\\begin{proof}\n We first prove $\\beta_{TR}(\\tau,\\delta,\\epsilon) \\leq \\alpha_{TR}(\\tau,\\delta,\\epsilon)$. Note that for any $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$, the corresponding marginal $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$. Moreover, $\\beta_{TR}(p_{USXY})=I(U;Y|S) = H(Y|S)-H(Y|US) \\leq H(Y|S)-H(Y|USX) \\overset{(a)}{=} H(Y|S)-H(Y|SX)=I(X;Y|S)=\\alpha_{TR}(p_{SXY})$, where (a) follows from Markov chain $U-(S,X)-Y$ ((ii) of definition \\ref{Defn:TestChannelsForPTP-STx}). Therefore, $\\beta_{TR}(\\tau,\\delta,\\epsilon) \\leq \\alpha_{TR}(\\tau,\\delta,\\epsilon)$. Conversely, given $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$, define $\\AuxiliaryAlphabet=\\left\\{ 0,1\\right\\}$ and a probability mass function $q_{USXY}$ defined on $\\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ as $q_{USXY}(u,s,x,y)=p_{SXY}(s,x,y)1_{\\left\\{ u=x \\right\\}}$. Clearly $q_{SXY}=p_{SXY}$ and hence (i) and (iii) of definition \\ref{Defn:TestChannelsForPTP-STx} are satisfied. Note that $q_{USX}(x,s,x)=p_{SX}(s,x)$, and hence $q_{Y|XSU}(y|x,s,x)=p_{Y|XS}(y|x,s)=W_{Y|XS}(y|x,s)$. Hence $q_{USXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$. It is easy to verify $\\beta_{TR}(q_{USXY}) = \\alpha_{TR}(p_{SXY})$ and therefore $\\beta_{TR}(\\tau,\\delta,\\epsilon) \\geq \\alpha_{TR}(\\tau,\\delta,\\epsilon)$.\n\\end{proof}\nWe now derive a characterization of the condition under which $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$. Towards that end, we first prove uniqueness of the pmf that achieves $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\begin{lemma}\n \\label{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTP-STxRx}\nSuppose $p_{SXY},q_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ are such that $\\alpha_{TR}(p_{SXY})=\\alpha_{TR}(q_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, then $p_{SXY}=q_{SXY}$. Moreover, if $\\alpha_{TR}(p_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, then $p_{SX}=p_{S}p_{X}$, i.e., $S$ and $X$ are independent.\n\\end{lemma}\n\\begin{proof}\nClearly, if $q_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ satisfies $q_{SX}=q_{S}q_{X}$ with $q_{X}(1)=\\tau$, then $\\alpha_{TR}(q_{SXY})=h_{b}(\\tau * \\delta)-h_{b}(\\delta)$ and since $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon) \\leq h_{b}(\\tau * \\delta)-h_{b}(\\delta)$,\\footnote{This can be easily verified using standard information theoretic arguments.} we have $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon) = h_{b}(\\tau * \\delta)-h_{b}(\\delta)$. Let $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$ be another pmf for which $\\alpha_{TR}(p_{SXY})=h_{b}(\\tau * \\delta)-h_{b}(\\delta)$. Let $\\chi_{0}\\define p_{X|S}(1|0)$ and $\\chi_{1}\\define p_{X|S}(1|1)$. $\\alpha_{TR}(p_{SXY})=I(X;Y|S)=H(Y|S)-H(Y|X,S)=H(X\\oplus S \\oplus N|S)-h_{b}(\\delta)$. We focus on the first term\n\\begin{eqnarray}\n\\lefteqn{H(X\\oplus S \\oplus N|S) = (1-\\epsilon)H(X\\oplus 0 \\oplus N|S=0)+\\epsilon H(X\\oplus 1 \\oplus N|S=1)}\\nonumber\\\\\n&=&(1-\\epsilon)h_{b}(\\chi_{0}(1-\\delta)+(1-\\chi_{0})\\delta)+\\epsilon h_{b}(\\chi_{1} (1-\\delta)+(1-\\chi_{1})\\delta)\n \\nonumber\\\\\n\\label{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromConcavity}\n&\\leq& h_{b}((1-\\epsilon)\\chi_{0}(1-\\delta)+(1-\\epsilon)(1-\\chi_{0})\\delta+\\epsilon\\chi_{1} (1-\\delta)+\\epsilon(1-\\chi_{1})\\delta)\\\\\n\\label{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromRangeOfEta}\n&=&h_{b}(p_{X}(1)(1-\\delta)+(1-p_{X}(1))\\delta)=h_{b}(\\delta+p_{X}(1)(1-2\\delta)) \\leq h_{b}(\\delta+\\tau(1-2\\delta))=h_{b}(\\tau * \\delta)\n\\end{eqnarray}\nwhere (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromConcavity}) follows from concavity of binary entropy function $h_{b}(\\cdot)$ and inequality in (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromRangeOfEta}) follows from $\\delta \\in (0,\\frac{1}{2})$. We therefore have $\\alpha_{TR}(p_{SXY})=h_{b}(\\tau * \\delta)-h_{b}(\\delta)$ if and only if equality holds in (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromConcavity}), (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromRangeOfEta}). $h_{b}(\\cdot)$ being strictly concave, equality holds in (\\ref{Eqn:UniquenessOfPMFThatAchievesCapacityOfPTPWithStateFollowsFromConcavity}) if and only if $\\epsilon \\in \\left\\{ 0,1\\right\\}$ or $\\chi_{0}=\\chi_{1}$. The range of $\\epsilon$ precludes the former and therefore $\\chi_{0}=\\chi_{1}$. This proves $p_{SX}=p_{S}p_{X}$ and $p_{X}(1)=\\tau$. Given $p_{SXY} \\in \\SetOfDistributions_{TR}(\\tau,\\delta,\\epsilon)$, these constrains completely determine $p_{SXY}$ and we have $p_{SXY}=q_{SXY}$.\n\\end{proof}\nFollowing is the main result of this section.\n\\begin{lemma}\n \\label{Lem:CharacterizationOfConditionThatEnsuresNoRateLoss}\n$\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ if and only if there exists a pmf $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ such that\n\\begin{enumerate}\n \\item the corresponding marginal achieves $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, i.e., $\\alpha_{TR}(p_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$,\n\\item $S-Y-U$ is a Markov chain.\n\\item $X-(U,S)-Y$ is a Markov chain.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nWe first prove the reverse implication, i.e., the if statement. Note that $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)=\\alpha_{TR}(p_{SXY})=I(X;Y|S)=H(Y|S)-H(Y|XS)\\overset{(a)}{=}H(Y|S)-H(Y|XSU)\\overset{(b)}{=}H(Y|S)-H(Y|US)=I(U;Y|S)=I(U;YS)-I(U;S)\\overset{(c)}{=}I(U;Y)-I(U;S) \\leq \\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$, where (a) follows from (ii) of definition \\ref{Defn:TestChannelsForPTP-STx}, (b) follows from hypothesis 3) and (c) follows from hypothesis 2). We therefore have $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)\\leq \\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$, and the reverse inequality follows from remark \\ref{Rem:SideInformationAtBothEncoderAndDecoderResultsInLargerCapacity}.\n\nConversely, let $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ achieve $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$, i.e., $\\alpha_{T}(p_{USXY})=\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$. We have $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)=\\alpha_{T}(p_{USXY}) = I(U;Y)-I(U;S) \\overset{(b)}{\\leq} I(U;YS)-I(U;S) = I(U;Y|S)=H(Y|S)-H(Y|US) \\overset{(c)}{\\leq} H(Y|S)-H(Y|USX) \\overset{(a)}{=} H(Y|S)-H(Y|SX)=I(X;Y|S)=\\alpha_{TR}(p_{SXY})\\leq \\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, where (a) follows from Markov chain $U-(S,X)-Y$ ((ii) of definition \\ref{Defn:TestChannelsForPTP-STx}). Equality of $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon),\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ implies equality in (b), (c) and thus $I(U;S|Y)=0$ and $H(Y|US)=H(Y|USX)$ and moreover $\\alpha_{TR}(p_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$.\n\\end{proof}\n\nFor the particular binary additive point to point channel with state, we strengthen the condition for no rate loss in the following lemma.\n\\begin{lemma}\n \\label{Lem:XShouldBeAFunctionOfUAndS}\nIf $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ satisfies\n\\begin{enumerate}\n\\item $S-Y-U$ is a Markov chain.\n\\item $X-(U,S)-Y$ is a Markov chain.\n\\end{enumerate}\nthen $H(X|U,S)=0$, or in other words, there exists a function $f : \\AuxiliaryAlphabet \\times \\StateAlphabet \\rightarrow \\InputAlphabet$ such that $P(X=f(U,S))=1$.\n\\end{lemma}\n\\begin{proof}\nWe prove this by contradiction. In particular, we prove $H(X|U,S) > 0$ violates Markov chain $X-(U,S)-Y$.\n\nIf $H(X|U,S) > 0$, then $H(X \\oplus S |U,S) >0$. Indeed, $0 < H(X|U,S)\\leq H(X,S|U,S)=H(X\\oplus S, S |U,S) = H(S|U,S)+H(X\\oplus S | U,S)=H(X \\oplus S | U, S)$. Since $(U,S,X)$ is independent of $X\\oplus S \\oplus Y$ and in particular, $(U,S,S\\oplus X)$ is independent of $X\\oplus S \\oplus Y$, we have $H((X\\oplus S)\\oplus(X\\oplus S \\oplus Y)|U,S)> H(X\\oplus S \\oplus Y|U,S)=h_{b}(\\delta)=H(Y|U,S,X)$, where the first inequality follows from concavity of binary entropy function. But $(X\\oplus S)\\oplus(X\\oplus S \\oplus Y)=Y$ and we have therefore proved $H(Y|U,S)>H(Y|U,S,X)$ contradicting Markov chain $X-(U,S)-Y$.\n\\end{proof}\nWe summarize the conditions for no rate loss below.\n\\begin{thm}\n \\label{Thm:CharacterizationOfConditionThatEnsuresNoRateLossForBinaryAdditive}\n$\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)=\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$ if and only if there exists a pmf $p_{USXY} \\in \\SetOfDistributions_{T}(\\tau,\\delta,\\epsilon)$ such that\n\\begin{enumerate}\n \\item the corresponding marginal achieves $\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, i.e., $\\alpha_{TR}(p_{SXY})=\\mathcal{C}_{TR}(\\tau,\\delta,\\epsilon)$, and in particular $S$ and $X$ are independent,\n\\item $S-Y-U$ is a Markov chain.\n\\item $X-(U,S)-Y$ is a Markov chain,\n\\item $H(X|U,S)=0$, or in other words, there exists a function $f : \\AuxiliaryAlphabet \\times \\StateAlphabet \\rightarrow \\InputAlphabet$ such that $P(X=f(U,S))=1$.\n\\end{enumerate}\n\\end{thm}\n\\section{The binary additive dirty point to point channel suffers a rate loss}\n\\label{AppSec:TheBinaryAdditiveDirtyPointToPointChannelSuffersARateLoss}\nThis section is dedicated to proving proposition \\ref{Prop:PropositionRegardindRateLoss}. We begin with an upper bound on cardinality of auxiliary set involved in characterization of $\\mathcal{C}_{T}(\\tau,\\delta,\\epsilon)$.\n\\begin{lemma}\n \\label{Lem:CardinalityBoundOnCostConstrainedGelfandPinskerAuxiliaryRV}\nConsider a point to point channel with state information available at transmitter. Let $\\mathcal{S}, \\mathcal{X}$ and $\\mathcal{Y}$ denote state, input and output alphabets respectively. Let $W_{S},W_{Y|XS}$ denote pmf of state, channel transition probabilities respectively. The input is constrained with respect to a cost function $\\kappa : \\mathcal{X} \\times \\mathcal{S} \\rightarrow [0,\\infty)$. Let $\\mathbb{D}_{T}(\\tau)$ denote the collection of all probability mass functions $p_{UXSY}$ defined on $\\mathcal{U} \\times \\mathcal{X} \\times \\mathcal{S} \\times \\mathcal{Y}$, where $\\mathcal{U}$ is an arbitrary set, such that (i) $p_{S}=W_{S}$, (ii) $p_{Y|XSU}=p_{Y|XS}=W_{Y|XS}$ and (iii) $\\Expectation \\left\\{ \\kappa(X,S) \\right\\} \\leq \\tau$. Moreover, let \\[\\overline{\\mathbb{D}_{T}}(\\tau) = \\left\\{ p_{UXSY} \\in \\mathbb{D}_{T}(\\tau): |\\mathcal{U}| \\leq \\min \\left\\{ \\substack{|\\mathcal{X}|\\cdot |\\mathcal{S}|, \\\\|\\mathcal{X}|+|\\mathcal{S}|+|\\mathcal{Y}|-2 }\\right\\} \\right\\}.\\] For $p_{UXSY} \\in \\mathbb{D}_{T}(\\tau)$, let $\\alpha(p_{UXSY})=I(U;Y)-I(U;S)$. Let \n\\begin{eqnarray} \n \\alpha_{T}(\\tau)=\\sup_{p_{UXSY} \\in \\mathbb{D}_{T}(\\tau)}\\alpha(p_{UXSY}) ,~~~~ \\overline{\\alpha_{T}}(\\tau)=\\sup_{p_{UXSY} \\in \\overline{\\mathbb{D}_{T}}(\\tau)}\\alpha(p_{UXSY}). \\nonumber\n\\end{eqnarray}\nThen $\\alpha_{T}(\\tau)=\\overline{\\alpha_{T}}(\\tau)$.\n\\end{lemma}\n\\begin{proof}\n The proof is based on Fenchel-Eggelston-Carath\\'eodory \\cite{Egg-Convexity1958}, \\cite[Appendix C]{201201NIT_ElgKim} theorem which is stated here for ease of reference.\n\\begin{lemma}\n\\label{Lem:FenchelEgglestonCaratheodory}\n let $\\mathcal{A}$ be a finite set and $\\mathcal{Q}$ be an arbitrary set. Let $\\mathcal{P}$ be a connected compact subset of pmf's on $\\mathcal{A}$ and $p_{A|Q}(\\cdot | q) \\in \\mathcal{P}$ for each $q \\in \\mathcal{Q}$. For $j=1,2,\\cdots,d$ let $g_{j}: \\mathcal{P} \\rightarrow \\reals$ be continuous functions. Then for every $Q \\sim F_{Q}$ defined on $\\mathcal{Q}$, there exist a random variable $\\overline{Q} \\sim p_{\\overline{Q}}$ with $|\\overline{\\mathcal{Q}}| \\leq d$ and a collection of pmf's $p_{A|\\overline{Q}}(\\cdot|\\overline{q}) \\in \\mathcal{P}$, one for each $\\overline{q} \\in \\overline{\\mathcal{Q}}$, such that\n\\begin{eqnarray}\n \\int_{\\mathcal{Q}}g_{j}(p_{A|Q}(a|q))dF_{Q}(q) = \\sum_{\\overline{q} \\in \\overline{\\mathcal{Q}}}g_{j}(p_{A|\\overline{Q}}(a|\\overline{q}))p_{\\overline{Q}}(\\overline{q}).\\nonumber\n\\end{eqnarray}\n\\end{lemma}\nThe proof involves identifying $g_{j}:j=1,2\\cdots,d$ such that rate achievable and cost expended are preserved. We first prove the bound $|\\mathcal{U}| \\leq |\\mathcal{X}|\\cdot|\\mathcal{S}|$.\n\nSet $\\mathcal{Q}=\\mathcal{U}$ and $\\mathcal{A}=\\mathcal{X} \\times \\mathcal{S}$ and $\\mathcal{P}$ denote the connected compact subset of pmf's on $\\mathcal{X} \\times \\mathcal{S}$. Without loss of generality, let $\\mathcal{X} = \\left\\{1,2,\\cdots, |\\mathcal{X}| \\right\\}$ and $\\mathcal{S} = \\left\\{1,2,\\cdots, |\\mathcal{S}| \\right\\}$. For $i=1,2, \\cdots, |\\mathcal{X}|$ and $k=1,2,\\cdots,|\\mathcal{S}|-1$, let $g_{i,k}(\\pi_{X,S})=\\pi_{X,S}(i,k)$ and $g_{l,|\\mathcal{S}|}(\\pi_{X,S})=\\pi_{X,S}(l,|\\mathcal{S}|)$ for $l=1,2,\\cdots, |\\mathcal{X}|-1$. Let $g_{|\\mathcal{X}|\\cdot |\\mathcal{S}|}(\\pi_{X,S})=H(S)-H(Y)$. It can be verified that\n\\begin{flalign}\n g_{|\\mathcal{X}|\\cdot |\\mathcal{S}|}(\\pi_{X,S}) =& -\\sum_{s \\in \\mathcal{S}} (\\sum_{x \\in \\mathcal{X}}\\pi_{X,S}(x,s))\\log_{2}(\\sum_{x \\in \\mathcal{X}}\\pi_{X,S}(x,s))+ \\sum_{y \\in \\mathcal{Y}} \\theta(y)\\log_{2}(\\theta(y)),\\mbox{ where }\\nonumber\\\\\n\\label{Eqn:ThetaOfy}\n\\!\\!\\!\\!\\!\\!\\!\\theta(y)=&\\sum_{(x,s) \\in \\mathcal{X}\\times \\mathcal{S}}\\pi_{X,S}(x,s)W_{Y|XS}(y|x,s)\n\\end{flalign}\nwhere, is continuous. An application of lemma \\ref{Lem:FenchelEgglestonCaratheodory} using the above set of functions, the upper bound $|\\mathcal{X}|\\cdot|\\mathcal{S}|$ on $|\\mathcal{U}|$ can be verified.\n\nWe now outline proof of upper bound $|\\mathcal{X}|+|\\mathcal{S}|+|\\mathcal{Y}|-2$ on $|\\mathcal{U}|$. Without loss of generality, we assume $\\mathcal{X}=\\left\\{ 1,\\cdots, |\\mathcal{X}| \\right\\}$, $\\mathcal{S}=\\left\\{ 1,\\cdots, |\\mathcal{S}| \\right\\}$ and $\\mathcal{Y}=\\left\\{ 1,\\cdots, |\\mathcal{Y}| \\right\\}$. As earlier, set $\\mathcal{Q}=\\mathcal{U}$ and $\\mathcal{A}=\\mathcal{X} \\times \\mathcal{S}$ and $\\mathcal{P}$ denote the connected compact subset of pmf's on $\\mathcal{X} \\times \\mathcal{S}$. For $j=1,\\cdots,|\\mathcal{S}|-1$, let $g_{j}(\\pi_{X,S})=\\sum_{x \\in \\mathcal{X}} \\pi_{X,S}(x,j)$. For $j=|\\mathcal{S}|, \\cdots, |\\mathcal{S}|+|\\mathcal{Y}|-2$, let $g_{j}(\\pi_{X,S})=\\sum_{(x,s) \\in \\mathcal{X} \\times \\mathcal{S}}\\pi_{X,S}(x,s)W_{Y|X,S}(j-|\\mathcal{S}|+1|x,s)$. For $j=|\\mathcal{S}|+|\\mathcal{Y}|-1, \\cdots, |\\mathcal{S}|+|\\mathcal{Y}|+|\\mathcal{X}|-3$, let $g_{j}(\\pi_{X,S})=\\sum_{s \\in \\mathcal{S}}\\pi_{X,S}(j-|\\mathcal{S}|-|\\mathcal{Y}|+2,s)$. Let $g_{t}(\\pi_{X,S})=H(S)-H(Y)$, i.e.,\n\\begin{eqnarray}\ng_{t}(\\pi_{X,S}) = -\\sum_{s \\in \\mathcal{S}} (\\sum_{x \\in \\mathcal{X}}\\pi_{X,S}(x,s))\\log_{2}(\\sum_{x \\in \\mathcal{X}}\\pi_{X,S}(x,s))+ \\sum_{y \\in \\mathcal{Y}} \\theta(y)\\log_{2}(\\theta(y)),\\nonumber\n\\end{eqnarray}\nwhere $t=|\\mathcal{S}|+|\\mathcal{Y}|+|\\mathcal{X}|-2$, and $\\theta(y)$ as is in (\\ref{Eqn:ThetaOfy}). The rest of the proof follows by simple verification.\n\\end{proof}\n\n\\begin{prop}\n \\label{Prop:PropositionRegardindRateLoss}\nThere exists no probability mass function $p_{UXSY}$ defined on $\\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ where $\\AuxiliaryAlphabet = \\left\\{ 0,1,2,3 \\right\\}, \\InputAlphabet = \\StateAlphabet = \\OutputAlphabet = \\left\\{ 0,1 \\right\\}$, such that\n\\begin{enumerate}\n \\item $X$ and $S$ are independent with $P(S=1)=\\epsilon$, $P(X=1)=\\tau$, where $\\epsilon \\in (0,1)$, $\\tau \\in (0,\\frac{1}{2})$,\n\\item $p_{Y|X,S,U}(x \\oplus s|x,s,u)=p_{Y|X,S}(x\\oplus s |x,s)=1-\\delta$ for every $(u,x,s,y) \\in \\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$, where $\\delta \\in (0,\\frac{1}{2})$,\n\\item $U-Y-S$ and $X-(U,S)-Y$ are Markov chains, and\n\\item $p_{X|US}(x|u,s) \\in \\left\\{ 0,1 \\right\\}$ for each $(u,s,x) \\in \\AuxiliaryAlphabet \\times \\StateAlphabet \\times \\InputAlphabet$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n The proof is by contradiction. If there exists such a pmf $p_{USXY}$ then conditions 1) and 2) completely specify it's marginal on $\\StateAlphabet \\times \\InputAlphabet \\times \\OutputAlphabet$ and it maybe verified that $p_{SY}(0,0)=(1-\\epsilon)(1-\\theta), p_{SY}(0,1)=(1-\\epsilon)\\theta, p_{SY}(1,0)=\\epsilon\\theta,p_{SY}(1,1)=\\epsilon(1-\\theta)$, where $\\theta \\define \\delta(1-\\tau)+(1-\\delta)\\tau$ takes a value in $(0,1)$. Since $\\epsilon \\in (0,1)$, $p_{SY}(s,y) \\in (0,1)$ for each $(s,y) \\in \\StateAlphabet \\times \\OutputAlphabet$.\nIf we let $\\beta_{i}\\define p_{U|Y}(i|0) :i=0,1,2,3$ and $\\gamma_{j}\\define p_{U|Y}(j|1):j=0,1,2,3$, then Markov chain $U-Y-S$ implies $p_{USY}$ is as in table \\ref{Table:p_USY}.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUSY & $p_{USY}$&USY & $p_{USY}$\\\\\n\\hline\n000 & $(1-\\epsilon)(1-\\theta)\\beta_{0}$&200 & $(1-\\epsilon)(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n001 & $(1-\\epsilon)\\theta\\gamma_{0}$&201 & $(1-\\epsilon)\\theta \\gamma_{2}$ \\\\\n\\hline\n010 & $\\epsilon\\theta\\beta_{0}$&210 & $\\epsilon\\theta\\beta_{2}$ \\\\\n\\hline\n011 & $\\epsilon(1-\\theta) \\gamma_{0}$ &211 & $\\epsilon(1-\\theta)\\gamma_{2}$ \\\\\n\\hline\n100 & $(1-\\epsilon)(1-\\theta)\\beta_{1}$ &300 & $(1-\\epsilon)(1-\\theta)\\beta_{3}$ \\\\\n\\hline\n101 & $(1-\\epsilon)\\theta \\gamma_{1}$ &301 & $(1-\\epsilon)\\theta \\gamma_{3}$ \\\\\n\\hline\n110 & $\\epsilon\\theta\\beta_{1}$ &310 & $\\epsilon\\theta\\beta_{3}$ \\\\\n\\hline\n111 & $\\epsilon(1-\\theta) \\gamma_{1}$ &311 & $\\epsilon(1-\\theta)\\gamma_{3}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{USY}$} \\label{Table:p_USY} \n\\end{table} \nSince $X$ is a function of $(U,S)$\\footnote{With probability $1$}, there exist $z_{i} \\in \\left\\{ 0,1 \\right\\}:i=0,1,\\cdots,7$ such that entries of table \\ref{Table:p_USX} hold true.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|} \\hline\n$p_{USX}(0,0,0) =p_{US}(0,0)z_{0}$&$p_{USX}(0,1,0) =p_{US}(0,1)z_{4}$\\\\\n\\hline\n$p_{USX}(1,0,0) =p_{US}(1,0)z_{1}$&$p_{USX}(1,1,0) =p_{US}(1,1)z_{5}$\\\\\n\\hline\n$p_{USX}(2,0,0) =p_{US}(2,0)z_{2}$&$p_{USX}(2,1,0) =p_{US}(2,1)z_{6}$\\\\\n\\hline\n$p_{USX}(3,0,0) =p_{US}(3,0)z_{3}$&$p_{USX}(3,1,0) =p_{US}(3,1)z_{7}$\\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{USX}$} \\label{Table:p_USX} \n\\end{table} \nMoreover, condition 4) and Markov chain $X-(U,S)-Y$ implies $p_{USXY}$ is completely determined in terms of entries of table \\ref{Table:p_USY} and $z_{i}:i=0,1,\\cdots,7$. For example $p_{USXY}(3,0,1,1)=p_{USY}(3,0,1)(1-z_{3})$. This enables us compute marginal $p_{SXY}$ in terms of entries of table \\ref{Table:p_USY} and $z_{i}:i=0,1,\\cdots,7$. This marginal must satisfy conditions 1) and 2) which implies\/is equivalent to the last two columns of table \\ref{Table:P_SXY} being equal.\n\\begin{eqnarray}\n\\label{Eqn:P_SYXOf000}\np_{SYX}(0,0,0)&=&(1-\\epsilon)(1-\\theta)\\left[\\beta_0z_0+\\beta_1 z_1+\\beta_2 z_2+\\beta_{3}z_{3} \\right] =(1-\\tau)(1-\\epsilon)(1-\\delta)\\\\\n\\label{Eqn:P_SYXOf001}\np_{SYX}(0,0,1)&=& (1-\\epsilon)(1-\\theta)\\left[1-\\beta_0z_0-\\beta_1 z_1-\\beta_2 z_2-\\beta_{3}z_{3} \\right] = \\tau(1-\\epsilon)\\delta\\nonumber\\\\\n\\label{Eqn:P_SYXOf010}\np_{SYX}(0,1,0)&=& (1-\\epsilon)\\theta \\left[\\gamma_0z_0+\\gamma_1 z_1+\\gamma_2 z_2 +\\gamma_{3} z_{3} \\right]= (1-\\tau)(1-\\epsilon)\\delta\\\\\n\\label{Eqn:P_SYXOf011}\np_{SYX}(0,1,1)&=& (1-\\epsilon)\\theta\\left[1-\\gamma_0z_0-\\gamma_1 z_1-\\gamma_2 z_2-\\gamma_{3} z_{3} \\right]\n= \\tau(1-\\epsilon)(1-\\delta)\\nonumber\\\\\n\\label{Eqn:P_SYXOf100}\np_{SYX}(1,0,0)&=& \\epsilon \\theta\\left[\\beta_0z_4+\\beta_1 z_5+\\beta_2 z_6+\\beta_{3}z_{7} \\right] = (1-\\tau)\\epsilon \\delta\\\\\n\\label{Eqn:P_SYXOf101}\np_{SYX}(1,0,1)&=& \\epsilon \\theta\\left[1-\\beta_0z_4-\\beta_1 z_5-\\beta_2 z_6-\\beta_{3}z_{7} \\right] = \\tau \\epsilon(1-\\delta)\\nonumber\\\\\n\\label{Eqn:P_SYXOf110}\np_{SYX}(1,1,0)&=& \\epsilon(1-\\theta) \\left[\\gamma_0z_4+\\gamma_1 z_5+\\gamma_2 z_6+\\gamma_{3}z_{7} \\right] = (1-\\tau) \\epsilon (1-\\delta)\\\\\n\\label{Eqn:P_SYXOf111}\np_{SYX}(1,1,1)&=& \\epsilon(1-\\theta) \\left[1-\\gamma_0z_4-\\gamma_1 z_5-\\gamma_2 z_6-\\gamma_{3}z_{7} \\right]= \\tau \\epsilon \\delta\\nonumber\n\\end{eqnarray}\nSince $\\epsilon \\notin \\left\\{ 0,1\\right\\}$, (\\ref{Eqn:P_SYXOf000}),(\\ref{Eqn:P_SYXOf110}) imply \n\\begin{eqnarray}\n\\label{Eqn:EquatingP_SYX000AndP_SYX110}\n(1-\\theta)\\left[\\beta_0z_0+\\beta_1 z_1+\\beta_2\n z_2+\\beta_{3}z_{3} \\right]=(1-\\theta) \\left[\\gamma_0z_4+\\gamma_1 z_5+\\gamma_2\n z_6+\\gamma_{3}z_{7} \\right]\\nonumber\n\\end{eqnarray}\nwhich further implies\n\\begin{eqnarray}\n \\label{Eqn:ConsequenceEquatingP_SYX000AndP_SYX110}\n\\beta_0z_0+\\beta_1 z_1+\\beta_2\n z_2+\\beta_{3}z_{3} =\\gamma_0z_4+\\gamma_1 z_5+\\gamma_2\n z_6+\\gamma_{3}z_{7}=:\\psi_{1}\\nonumber\n\\end{eqnarray}\nSimilarly (\\ref{Eqn:P_SYXOf010}),(\\ref{Eqn:P_SYXOf100}) imply\n\\begin{eqnarray}\n \\label{Eqn:ConsequenceEquatingP_SYX010AndP_SYX100}\n\\gamma_0z_0+\\gamma_1 z_1+\\gamma_2\n z_2 +\\gamma_{3} z_{3} =\\beta_0z_4+\\beta_1 z_5+\\beta_2\n z_6+\\beta_{3}z_{7}=:\\psi_{2}\\nonumber\n\\end{eqnarray}\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nSYX & $p_{SYX}$ &\\\\\n\\hline\n000 & $(1-\\epsilon)(1-\\theta)\\left[\\beta_0z_0+\\beta_1 z_1+\\beta_2\n z_2+\\beta_{3}z_{3} \\right]$ & $(1-\\tau)(1-\\epsilon)(1-\\delta)$ \\\\\n\\hline\n001 & $(1-\\epsilon)(1-\\theta)\\left[1-\\beta_0z_0-\\beta_1 z_1-\\beta_2\n z_2-\\beta_{3}z_{3} \\right] $ & $\\tau(1-\\epsilon)\\delta$\\\\\n\\hline\n010 & $(1-\\epsilon)\\theta \\left[\\gamma_0z_0+\\gamma_1 z_1+\\gamma_2\n z_2 +\\gamma_{3} z_{3} \\right]$ & $(1-\\tau)(1-\\epsilon)\\delta$\\\\\n\\hline\n011 & $(1-\\epsilon)\\theta\\left[1-\\gamma_0z_0-\\gamma_1 z_1-\\gamma_2\n z_2-\\gamma_{3} z_{3} \\right]$ & $\\tau(1-\\epsilon)(1-\\delta)$\\\\\n\\hline\n100 & $ \\epsilon \\theta\\left[\\beta_0z_4+\\beta_1 z_5+\\beta_2\n z_6+\\beta_{3}z_{7} \\right] $ & $(1-\\tau)\\epsilon \\delta$\\\\\n\\hline\n101 & $\\epsilon \\theta\\left[1-\\beta_0z_4-\\beta_1 z_5-\\beta_2\n z_6-\\beta_{3}z_{7} \\right]$ & $\\tau \\epsilon(1-\\delta)$\\\\\n\\hline\n110 & $\\epsilon(1-\\theta) \\left[\\gamma_0z_4+\\gamma_1 z_5+\\gamma_2\n z_6+\\gamma_{3}z_{7} \\right]$ & $(1-\\tau) \\epsilon (1-\\delta)$\\\\\n\\hline\n111 & $\\epsilon(1-\\theta) \\left[1-\\gamma_0z_4-\\gamma_1 z_5-\\gamma_2\n z_6-\\gamma_{3}z_{7} \\right]$ & $\\tau \\epsilon \\delta$\\\\\n\\hline\n\\end{tabular} \\end{center}\n\\caption{Enforcing conditions 1) and 2) for $p_{SXY}$} \\label{Table:P_SXY}\n\\end{table}\nWe now argue there exists no choice of values for $z_{i}:i=0,1\\cdots,7$. Towards that end, we make a couple of observations. Firstly, we argue $\\psi_{1} \\neq \\psi_{2}$. Since $\\epsilon \\neq 1$ and $\\theta \\in (0,1)$, we have $\\psi_{1}=\\frac{(1-\\tau)(1-\\delta)}{(1-\\theta)}$ and $\\psi_{2} = \\frac{(1-\\tau)\\delta}{\\theta}$ from (\\ref{Eqn:P_SYXOf000}) and (\\ref{Eqn:P_SYXOf010}) respectively. Equating $\\psi_{1}$ and $\\psi_{2}$, we obtain either $\\tau=1$ or $\\tau=0$ or $\\delta=\\frac{1}{2}$. Since none of the latter conditions hold, we conclude $\\psi_{1}\\neq \\psi_{2}$. Secondly, one can verify $\\psi_{1}+\\psi_{2}-1=\\frac{\\delta (1-\\delta)(1-2\\tau)}{\\theta(1-\\theta)}$. Since $\\delta \\in (0,\\frac{1}{2}),\\theta \\in (0,1)$ and $\\tau \\in (0,\\frac{1}{2})$, $\\psi_{1}+\\psi_{2} > 1$.\nWe now eliminate the possible choices for $z_{i}:i=0,1\\cdots,7$ through the following cases. let $m \\define \\left| \\left\\{ i \\in \\left\\{ 0,1,2,3 \\right\\} :z_{i}=1\\right\\} \\right|$ and $l \\define \\left| \\left\\{ i \\in \\left\\{ 4,5,6,7 \\right\\} :z_{i}=1\\right\\} \\right|$.\n\n\\noindent \\textit{Case 1:} All of $z_{0},z_{1},z_{2},z_{3}$ or all of $z_{4},z_{5},z_{6},z_{7}$ are equal to 0, i.e., $m=0$ or $l=0$. This implies $\\psi_{1}=\\psi_{2}=0$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\n\\noindent \\textit{Case 2:} All of $z_{0},z_{1},z_{2},z_{3}$ or all of $z_{4},z_{5},z_{6},z_{7}$ are equal to 1, i.e., $m=4$ or $l=4$. This implies $\\psi_{1}=\\psi_{2}=1$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nCases 1 and 2 imply $m,l \\in \\left\\{1,2,3\\right\\}$.\n\n\\noindent \\textit{Case 3: $m=l=3$}. If $i_{1},i_{2},i_{3}$ are distinct indices in $\\left\\{ 0,1,2,3 \\right\\}$ such that $z_{i_{1}}=z_{i_{2}}=z_{i_{3}}=1$, then one among $z_{i_{1}+4},z_{i_{2}+4},z_{i_{3}+4}$ has to be $0$. Else $\\psi_{1}=\\beta_{i_{1}}+\\beta_{i_{2}}+\\beta_{i_{3}}$ and $\\psi_{2}=\\beta_{i_{1}}z_{i_{1}+4}+\\beta_{i_{2}}z_{i_{2}+4}+\\beta_{i_{3}}z_{i_{3}+4}=\\beta_{i_{1}}+\\beta_{i_{2}}+\\beta_{i_{3}}=\\psi_{1}$ contradicting $\\psi_{1} \\neq \\psi_{2}$. Let us consider the case $z_{0}=z_{1}=z_{2}=1$, $z_{3}=z_{4}=0$ and $z_{5}=z_{6}=z_{7}=1$. Table \\ref{Table:p_UXSY} tabulates $p_{USXY}$ for this case.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUXSY & $p_{UXSY}$&UXSY & $p_{UXSY}$\\\\\n\\hline\n0000 & $(1-\\epsilon)(1-\\theta)\\beta_{0}$&2000 & $(1-\\epsilon)(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n0001 & $(1-\\epsilon)\\theta\\beta_{3}$&2001 & $(1-\\epsilon)\\theta \\gamma_{2}$ \\\\\n\\hline\n0110 & $\\epsilon\\theta\\beta_{0}$&2010 & $\\epsilon\\theta\\beta_{2}$ \\\\\n\\hline\n0111 & $\\epsilon(1-\\theta) \\beta_{3}$ &2011 & $\\epsilon(1-\\theta)\\gamma_{2}$ \\\\\n\\hline\n1000 & $(1-\\epsilon)(1-\\theta)\\beta_{1}$ &3100 & $(1-\\epsilon)(1-\\theta)\\beta_{3}$ \\\\\n\\hline\n1001 & $(1-\\epsilon)\\theta \\gamma_{1}$ &3101 & $(1-\\epsilon)\\theta \\beta_{0}$ \\\\\n\\hline\n1010 & $\\epsilon\\theta\\beta_{1}$ &3010 & $\\epsilon\\theta\\beta_{3}$ \\\\\n\\hline\n1011 & $\\epsilon(1-\\theta) \\gamma_{1}$ &3011 & $\\epsilon(1-\\theta)\\beta_{0}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{UXSY}$} \\label{Table:p_UXSY} \n\\end{table}\nWe have $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{1}+\\gamma_{2}+\\gamma_{3}$ or equivalently $\\psi_{1}=1-\\beta_{3}=1-\\gamma_{0}$ and $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{3}=\\beta_{1}+\\beta_{2}+\\beta_{3}$ or equivalently $\\psi_{2}=1-\\gamma_{3}=1-\\beta_{0}$. These imply $\\gamma_{3}=\\beta_{0}$, $\\gamma_{0}=\\beta_{3}$ which further imply $\\gamma_{1}+\\gamma_{2}=\\beta_{1}+\\beta_{2}$ (since $1=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}+\\gamma_{3}=\\beta_{0}+\\beta_{1}+\\beta_{2+}\\beta_{3}$). From table \\ref{Table:p_UXSY}, one can verify \n\\begin{eqnarray}\n\\label{Eqn:P_UGivenXSYOf0Given001}\n&p_{U|XSY}(0|0,0,1)=\\frac{\\beta_{3}(1-\\epsilon)\\theta}{(1-\\epsilon)\\theta(\\beta_{3}+\\gamma_{1}+\\gamma_{2})} = \\frac{\\beta_{3}}{\\beta_{1}+\\beta_{2}+\\beta_{3}},\n\\nonumber\\\\\n\\lefteqn{p_{U|XS}(0|0,0)=\\frac{(1-\\theta)\\beta_{0}+\\theta\\beta_{3}}{(1-\\theta)(\\beta_{0}+\\beta_{1}+\\beta_{2})+\\theta(\\beta_{3}+\\gamma_{1}+\\gamma_{2})}}.\\nonumber\n\\end{eqnarray}\nThe Markov chain $U-(X,S)-Y$ implies $p_{U|XSY}(0|0,0,1)=p_{U|XS}(0|0,0)$. Equating the right hand sides of the above equations, we obtain $(1-\\theta)(\\beta_{0}-\\beta_{3})(\\beta_{1}+\\beta_{2})=0$. Since $\\theta \\neq 0$, $\\beta_{1}+\\beta_{2}=0$ or $\\beta_{0}=\\beta_{3}$. If $\\beta_{0}=\\beta_{3}$, then $1-\\beta_{3}=\\psi_{1}=\\psi_{2}=1-\\beta_{0}$ thus contradicting $\\psi_{1} \\neq \\psi_{2}$. If $\\beta_{1}+\\beta_{2}=0$, then $\\beta_{0}+\\beta_{3}=1$ implying $\\psi_{1}+\\psi_{2}=1$ contradicting $\\psi_{1}+\\psi_{2}>1$.\n\n\\noindent \\textit{Case 4: $m=3, l=2$}. Let us assume $z_{0}=z_{1}=z_{2}=z_{6}=z_{7}=1,z_{3}=z_{4}=z_{5}=0$. We then have $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{2}+\\gamma_{3}$ and $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=\\beta_{2}+\\beta_{3}$. Since $\\beta_{0}+\\beta_{1}+\\beta_{2}=1-\\beta_{3}$ and $\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=1-\\gamma_{3}$, we have $\\gamma_{2}+\\gamma_{3}=1-\\beta_{3}$ and $\\beta_{2}+\\beta_{3}=1-\\gamma_{3}$ and therefore $\\gamma_{2}=\\beta_{2}$.Table \\ref{Table:p_UXSYCasemIs3lIs2} tabulates $p_{USXY}$ for this case.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUXSY & $p_{UXSY}$&UXSY & $p_{UXSY}$\\\\\n\\hline\n0000 & $(1-\\epsilon)(1-\\theta)\\beta_{0}$&2000 & $(1-\\epsilon)(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n0001 & $(1-\\epsilon)\\theta\\gamma_{0}$&2001 & $(1-\\epsilon)\\theta \\beta_{2}$ \\\\\n\\hline\n0110 & $\\epsilon\\theta\\beta_{0}$&2010 & $\\epsilon\\theta\\beta_{2}$ \\\\\n\\hline\n0111 & $\\epsilon(1-\\theta) \\gamma_{0}$ &2011 & $\\epsilon(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n1000 & $(1-\\epsilon)(1-\\theta)\\beta_{1}$ &3100 & $(1-\\epsilon)(1-\\theta)\\beta_{3}$ \\\\\n\\hline\n1001 & $(1-\\epsilon)\\theta \\gamma_{1}$ &3101 & $(1-\\epsilon)\\theta \\gamma_{3}$ \\\\\n\\hline\n1110 & $\\epsilon\\theta\\beta_{1}$ &3010 & $\\epsilon\\theta\\beta_{3}$ \\\\\n\\hline\n1111 & $\\epsilon(1-\\theta) \\gamma_{1}$ &3011 & $\\epsilon(1-\\theta)\\gamma_{3}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{UXSY}$} \\label{Table:p_UXSYCasemIs3lIs2} \n\\end{table}\nFrom table \\ref{Table:p_UXSYCasemIs3lIs2}, one can verify \n\\begin{eqnarray}\n\\label{Eqn:P_UGivenXSYOf0Given001}\n&p_{U|XSY}(2|0,0,1)=\\frac{\\beta_{2}(1-\\epsilon)\\theta}{(1-\\epsilon)\\theta(\\beta_{2}+\\gamma_{0}+\\gamma_{1})} = \\frac{\\beta_{2}}{\\beta_{2}+\\gamma_{0}+\\gamma_{1}},\n\\nonumber\\\\\n\\lefteqn{p_{U|XS}(2|0,0)=\\frac{\\beta_{2}}{(1-\\theta)(\\beta_{0}+\\beta_{1})+\\theta(\\gamma_{0}+\\gamma_{1})+\\beta_{2}}}.\\nonumber\n\\end{eqnarray}\nThe Markov chain $U-(X,S)-Y$ implies $p_{U|XSY}(2|0,0,1)=p_{U|XS}(2|0,0)$. Equating the RHS of the above equations, we obtain $\\beta_{0}+\\beta_{1}=\\gamma_{0}+\\gamma_{1}$. This implies $\\beta_{2}+\\beta_{3}=\\gamma_{2}+\\gamma_{3}$. However $\\psi_{1}=\\beta_{2}+\\beta_{3}$ and $\\psi_{2}=\\gamma_{2}+\\gamma_{3}$, this contradicting $\\psi \\neq \\psi_{2}$.\n\nLet us assume $z_{0}=z_{1}=z_{2}=z_{5}=z_{6}=1$ and $z_{3}=z_{4}=z_{7}=0$. It can be verified that $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{1}+\\gamma_{2}$ and $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=\\beta_{1}+\\beta_{2}$. This implies $\\psi_{1}-\\psi_{2}=\\beta_{0}=-\\gamma_{0}$. Since $\\beta_{0}$ and $\\gamma_{0}$ are non-negative, $\\beta_{0}=\\gamma_{0}=0$ implying $\\psi_{1}-\\psi_{2}=0$, contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\n\\noindent \\textit{Case 5: $m=3, l=1$}. Assume $z_{0}=z_{1}=z_{2}=z_{4}=1$, $z_{3}=z_{5}=z_{6}=z_{7}=0$. It can be verified that $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{0}$ and $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=\\beta_{0}$. Therefore $\\psi_{1}-\\psi_{2}=\\beta_{1}+\\beta_{2}$ and $\\psi_{2}-\\psi_{1}=\\gamma_{1}+\\gamma_{2}$. Since $\\beta_{i},\\gamma_{i}:i \\in \\left\\{ 0,1,2,3 \\right\\}$ are non-negative, $\\psi_{1}-\\psi_{2} \\geq 0$ and $\\psi_{2}-\\psi_{1} \\geq 0$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nAssume $z_{0}=z_{1}=z_{2}=z_{7}=1$ and $z_{3}=z_{4}=z_{5}=z_{6}=0$. In this case, $\\psi_{1}=\\beta_{0}+\\beta_{1}+\\beta_{2}=\\gamma_{3}$, $\\psi_{2}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}=1-\\gamma_{3}$. We have $\\psi_{1}+\\psi_{2}=1$ contradicting $\\psi_{1}+\\psi_{2}>1$.\n\n\\noindent \\textit{Case 6: $m=2, l=2$}. Assume $z_{0}=z_{1}=z_{4}=z_{5}=1$, $z_{2}=z_{3}=z_{6}=z_{7}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{0}=\\gamma_{1}$, $\\psi_{2}=\\gamma_{0}+\\gamma_{1}=\\beta_{0}+\\beta_{1}$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nAssume $z_{0}=z_{1}=z_{6}=z_{7}=1$, $z_{2}=z_{3}=z_{4}=z_{5}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{2}+\\gamma_{3}$, $\\psi_{2}=\\gamma_{0}+\\gamma_{1}=\\beta_{2}+\\beta_{3}$ contradicting $\\psi_{1} + \\psi_{2}>1$.\n\nAssume $z_{0}=z_{1}=z_{5}=z_{6}=1$, $z_{2}=z_{3}=z_{4}=z_{7}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{1}+\\gamma_{2}$, $\\psi_{2}=\\gamma_{0}+\\gamma_{1}=\\beta_{1}+\\beta_{2}$ and therefore $\\beta_{2}+\\beta_{3}=\\gamma_{0}+\\gamma_{3}$ and $\\beta_{0}+\\beta_{3}=\\gamma_{2}+\\gamma_{3}$. We observe\n\\begin{equation}\n \\label{Eqn:CasemIs2lIs2Psi1-Psi2}\n\\psi_{1}-\\psi_{2}= \\beta_{0}-\\beta_{2}=\\gamma_{2}-\\gamma_{0}\n\\end{equation}\n\nPMF $p_{UXSY}$ is tabulated in \\ref{Table:p_UXSYCasemIs2lIs2} for this case. Table \\ref{Table:p_UXSYCasemIs2lIs2} enables us compute conditional pmf $p_{U|XSY}$ which is tabulated in table \\ref{Table:p_UGivenXSYCasemIs2lIs2}.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUXSY & $p_{UXSY}$&UXSY & $p_{UXSY}$\\\\\n\\hline\n0000 & $(1-\\epsilon)(1-\\theta)\\beta_{0}$&2100 & $(1-\\epsilon)(1-\\theta)\\beta_{2}$ \\\\\n\\hline\n0001 & $(1-\\epsilon)\\theta\\gamma_{0}$&2101 & $(1-\\epsilon)\\theta \\gamma_{2}$ \\\\\n\\hline\n0110 & $\\epsilon\\theta\\beta_{0}$&2010 & $\\epsilon\\theta\\beta_{2}$ \\\\\n\\hline\n0111 & $\\epsilon(1-\\theta) \\gamma_{0}$ &2011 & $\\epsilon(1-\\theta)\\gamma_{2}$ \\\\\n\\hline\n1000 & $(1-\\epsilon)(1-\\theta)\\beta_{1}$ &3100 & $(1-\\epsilon)(1-\\theta)\\beta_{3}$ \\\\\n\\hline\n1001 & $(1-\\epsilon)\\theta \\gamma_{1}$ &3101 & $(1-\\epsilon)\\theta \\gamma_{3}$ \\\\\n\\hline\n1010 & $\\epsilon\\theta\\beta_{1}$ &3110 & $\\epsilon\\theta\\beta_{3}$ \\\\\n\\hline\n1011 & $\\epsilon(1-\\theta) \\gamma_{1}$ &3111 & $\\epsilon(1-\\theta)\\gamma_{3}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{UXSY}$} \\label{Table:p_UXSYCasemIs2lIs2} \n\\end{table}\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\nUXSY & $p_{U|XSY}$&UXSY & $p_{U|XSY}$\\\\\n\\hline\n0000 & $\\frac{\\beta_{0}}{\\beta_{0}+\\beta_{1}}$&0001 & $\\frac{\\gamma_{0}}{\\gamma_{0}+\\gamma_{1}}$ \\\\\n\\hline\n0110 & $\\frac{\\beta_{0}}{\\beta_{0}+\\beta_{3}}$&0111 & $\\frac{\\gamma_{0}}{\\gamma_{0}+\\gamma_{3}}$ \\\\\n\\hline\n1000 & $\\frac{\\beta_{1}}{\\beta_{0}+\\beta_{1}}$ &1001 & $\\frac{\\gamma_{1}}{\\gamma_{0}+\\gamma_{1}}$ \\\\\n\\hline\n1010 & $\\frac{\\beta_{1}}{\\beta_{1}+\\beta_{2}}$ &1011 & $\\frac{\\gamma_{1}}{\\gamma_{1}+\\gamma_{2}}$ \\\\\n\\hline\n 2100& $\\frac{\\beta_{2}}{\\beta_{2}+\\beta_{3}}$&2101 & $\\frac{\\gamma_{2}}{\\gamma_{2}+\\gamma_{3}}$ \\\\\n\\hline\n 2010& $\\frac{\\beta_{2}}{\\beta_{1}+\\beta_{2}}$ &2011 & $\\frac{\\gamma_{2}}{\\gamma_{1}+\\gamma_{2}}$ \\\\\n\\hline\n 3100& $\\frac{\\beta_{3}}{\\beta_{2}+\\beta_{3}}$ &3101 & $\\frac{\\gamma_{3}}{\\gamma_{2}+\\gamma_{3}}$ \\\\\n\\hline\n3110 & $\\frac{\\beta_{3}}{\\beta_{0}+\\beta_{3}}$ &3111 & $\\frac{\\gamma_{3}}{\\gamma_{0}+\\gamma_{3}}$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{U|XSY}$} \\label{Table:p_UGivenXSYCasemIs2lIs2} \n \\end{table}\nMarkov chain $U-(X,S)-Y$ implies columns 2 and 4 of table \\ref{Table:p_UGivenXSYCasemIs2lIs2} are identical. This implies\n\\begin{eqnarray}\n \\label{Eqn:Relation1ComingOutOfU-XS-YMarkovChain}\n\\frac{\\beta_{0}}{\\gamma_{0}}\\overset{(a)}{=}\\frac{\\beta_{0}+\\beta_{1}}{\\gamma_{0}+\\gamma_{1}}\\overset{(b)}{=}\\frac{\\beta_{1}}{\\gamma_{1}}, \\frac{\\beta_{2}}{\\gamma_{2}}\\overset{(c)}{=}\\frac{\\beta_{2}+\\beta_{3}}{\\gamma_{2}+\\gamma_{3}}\\overset{(d)}{=}\\frac{\\beta_{3}}{\\gamma_{3}}, ~~\\mbox{ and }~~ \\frac{\\beta_{0}}{\\gamma_{0}}\\overset{(e)}{=}\\frac{\\beta_{0}+\\beta_{3}}{\\gamma_{0}+\\gamma_{3}}\\overset{(f)}{=}\\frac{\\beta_{3}}{\\gamma_{3}},\n\\end{eqnarray}\nwhere (a),(b),(c),(d) in (\\ref{Eqn:Relation1ComingOutOfU-XS-YMarkovChain}) is obtained by equating rows 1, 3, 5, 7 of columns 2 and 4 respectively and (e) and (f) in (\\ref{Eqn:Relation1ComingOutOfU-XS-YMarkovChain}) are obtained by equating rows 2 and 8 of columns 2 and 4 respectively. (\\ref{Eqn:Relation1ComingOutOfU-XS-YMarkovChain}), enables us conclude\n\\begin{equation}\n \\label{Eqn:BetaIsEqualToGamma}\n\\frac{\\beta_{0}}{\\gamma_{0}}=\\frac{\\beta_{1}}{\\gamma_{1}}=\\frac{\\beta_{2}}{\\gamma_{2}}=\\frac{\\beta_{3}}{\\gamma_{3}}. \\nonumber\n\\end{equation}\nSince $\\beta_{0}+\\beta_{1}+\\beta_{2}+\\beta_{3}=\\gamma_{0}+\\gamma_{1}+\\gamma_{2}+\\gamma_{3}=1$, we have $\\beta_{i}=\\gamma_{i}$ for each $i \\in \\left\\{ 0,1,2,3\\right\\}$ which yields $\\psi_{1}=\\psi_{2}$ in (\\ref{Eqn:CasemIs2lIs2Psi1-Psi2}) contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\n\\noindent \\textit{Case 7: $m=2, l=1$}. Assume $z_{0}=z_{1}=z_{4}=1, z_{2}=z_{3}=z_{5}=z_{6}=z_{7}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{0}, \\psi_{2}= \\gamma_{0}+\\gamma_{1}=\\beta_{0}$ and hence $\\psi_{1}-\\psi_{2}=\\beta_{1}$ and $\\psi_{2}-\\psi_{1}=\\gamma_{1}$. Since $\\gamma_{1}$ and $\\beta_{1}$ are non-negative, we have $\\psi_{1}=\\psi_{2}$ contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nAssume $z_{0}=z_{1}=z_{7}=1, z_{2}=z_{3}=z_{4}=z_{5}=z_{6}=0$. Note that $\\psi_{1}=\\beta_{0}+\\beta_{1}=\\gamma_{3}, \\psi_{2}=\\gamma_{0}+\\gamma_{1}=\\beta_{3}$ and hence $\\psi_{1}+\\psi_{2} = \\beta_{0}+\\beta_{1}+\\beta_{3} \\leq 1$ contradicting $\\psi_{1}+\\psi_{2} >1$.\n\n\\noindent \\textit{Case 6: $m=1, l=1$}. Assume $z_{0}=z_{4}=1, z_{1}=z_{2}=z_{3}=z_{5}=z_{6}=z_{7}=0$. Note that $\\psi_{1}=beta_{0}=\\gamma_{0},\\psi_{2}=\\gamma_{0}=\\beta_{0}$, thus contradicting $\\psi_{1} \\neq \\psi_{2}$.\n\nAssume $z_{0}=z_{5}=1, z_{1}=z_{2}=z_{3}=z_{4}=z_{6}=z_{7}=0$. Note that $\\psi_{1}=\\beta_{0}=\\gamma_{1},\\psi_{2}=\\gamma_{0}=\\beta_{1}$, and hence $\\psi_{1}+\\psi_{2} = \\beta_{0}+\\beta_{1} \\leq 1$, thus contradicting $\\psi_{1} + \\psi_{2} >1$.\n\\end{proof}\n\n\\section{Proof of lemma \\ref{Lem:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput}}\n\\label{AppSec:ForBSCMarokovChainWithOutputImpliesMarkovChainWithInput}\nSince $A-B-Y$ and $AB-X-Y$ are Markov chains, to prove $A-B-XY$ is a Markov chain, it suffices to prove $A-B-X$ is a Markov chain. We therefore need to prove $p_{XA|B}(x_{k},a_{i}|b_{j})=p_{X|B}(x_{k}|b_{j})p_{A|B}(a_{i}|b_{j})$ for every $(x_{k},a_{i},b_{j}) \\in \\left\\{ 0,1\\right\\}\\times \\mathcal{A} \\times \\mathcal{B}$ such that $p_{B}(b_{j})>0$. It suffices to prove $p_{XA|B}(0,a_{i}|b_{j})=p_{X|B}(0|b_{j})p_{A|B}(a_{i}|b_{j})$ for every $(a_{i},b_{j}) \\in \\mathcal{A} \\times \\mathcal{B}$ such that $p_{B}(b_{j})>0$.\\footnote{Indeed, $p_{XA|B}(1,a_{i}|b_{j})=p_{A|B}(a_{i}|b_{j})-p_{XA|B}(0,a_{i}|b_{j})=p_{A|B}(a_{i}|b_{j})(1-p_{X|B}(0|b_{j}))=p_{A|B}(a_{i}|b_{j})p_{X|B}(1|b_{j})$.}\n\nFix a $b_{j}$ for which $p_{B}(b_{j})>0$. Let $p_{A|B}(a_{i}|b_{j})=\\alpha_{i}$ for each $i \\in \\naturals$ and $p_{XA|B}(0,a_{i}|b_{j})=\\chi_{i}$ for each $(i,j) \\in \\naturals\\times \\naturals$. It can be verified $p_{XYA|B}(\\cdot,\\cdot,\\cdot|b_{j})$ is as in table \\ref{Table:p_XYAGivenB}. From table \\ref{Table:p_XYAGivenB}, we infer $p_{AY|B}(a_{i}0|b_{j})=\\chi_{i}(1-\\eta)+(\\alpha_{i}-\\chi_{i})\\eta=\\alpha_{i}\\eta+\\chi_{i}(1-2\\eta)$. From the Markov chain $A-B-Y$, we have $p_{AY|B}(a_{i}0|b_{j})=p_{A|B}(a_{i}|b_{j})p_{Y|B}(0|b_{j})= \\alpha_{i}p_{Y|B}(0|b_{j})$. Therefore, $ \\alpha_{i}p_{Y|B}(0|b_{j})=\\alpha_{i}\\eta+\\chi_{i}(1-2\\eta)$. Since $1-2\\eta \\neq 0$, we substitute for $\\chi_{i}$ and $\\alpha_{i}$ in terms of their definitions to conclude\n\\begin{equation}\n \\label{Eqn:ProbOfXAGivenB}\np_{XA|B}(0,a_{i}|b_{j})=\\chi_{i}= \\alpha_{i}\\cdot \\frac{p_{Y|B}(0|b_{j})-\\eta}{1-2\\eta}=p_{A|B}(a_{i}|b_{j})\\frac{p_{Y|B}(0|b_{j})-\\eta}{1-2\\eta}.\\nonumber\n\\end{equation}\nSince $\\frac{p_{Y|B}(0|b_{j})-\\eta}{1-2\\eta}$ is independent of $i$ and $b_{j}$ was an arbitrary element in $\\mathcal{B}$ that satisfies $p_{B}(b_{j})>0$, we have established Markov chain $A-B-X$.\n\\begin{table} \\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|} \\hline\nAXY & $p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$&AXY & $p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$&AXY & $p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$&AXY & $p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$\\\\\n\\hline\n$a_{i}00$& $\\chi_{i}(1-\\eta)$&$a_{i}01$ & $\\chi_{i}\\eta$&$a_{i}10$ & $(\\alpha_{i}-\\chi_{i})\\eta$&$a_{i}11$ & $(\\alpha_{i}-\\chi_{i})(1-\\eta)=$ \\\\\n\\hline\n\\end{tabular} \\end{center}\\caption{$p_{AXY|B}(\\cdot,\\cdot,\\cdot|b_{j})$} \\label{Table:p_XYAGivenB} \n\\end{table}\n\\section*{Acknowledgment}\nThis work was supported by NSF grant CCF-1116021.\n\\bibliographystyle{..\/sty\/IEEEtran}\n{\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}