Browsing by Subject "Signaling games"

Now showing 1 - 10 of 10

Open Access
Dynamic signaling games with quadratic criteria under Nash and Stackelberg equilibria
(Elsevier, 2020-01) Sarıtaş, Serkan; Yüksel, Serdar; Gezici, Sinan
This paper considers dynamic (multi-stage) signaling games involving an encoder and a decoder who have subjective models on the cost functions. We consider both Nash (simultaneous-move) and Stackelberg (leader–follower) equilibria of dynamic signaling games under quadratic criteria. For the multi-stage scalar cheap talk, we show that the final stage equilibrium is always quantized and under further conditions the equilibria for all time stages must be quantized. In contrast, the Stackelberg equilibria are always fully revealing. In the multi-stage signaling game where the transmission of a Gauss–Markov source over a memoryless Gaussian channel is considered, affine policies constitute an invariant subspace under best response maps for Nash equilibria; whereas the Stackelberg equilibria always admit linear policies for scalar sources but such policies may be non-linear for multi-dimensional sources. We obtain an explicit recursion for optimal linear encoding policies for multi-dimensional sources, and derive conditions under which Stackelberg equilibria are informative.
Open Access
Hypothesis testing under subjective priors and costs as a signaling game
(IEEE, 2019) Sarıtaş, S.; Gezici, Sinan; Yüksel, S.
Many communication, sensor network, and networked control problems involve agents (decision makers) which have either misaligned objective functions or subjective probabilistic models. In the context of such setups, we consider binary signaling problems in which the decision makers (the transmitter and the receiver) have subjective priors and/or misaligned objective functions. Depending on the commitment nature of the transmitter to his policies, we formulate the binary signaling problem as a Bayesian game under either Nash or Stackelberg equilibrium concepts and establish equilibrium solutions and their properties. We show that there can be informative or non-informative equilibria in the binary signaling game under the Stackelberg and Nash assumptions, and derive the conditions under which an informative equilibrium exists for the Stackelberg and Nash setups. For the corresponding team setup, however, an equilibrium typically always exists and is always informative. Furthermore, we investigate the effects of small perturbations in priors and costs on equilibrium values around the team setup (with identical costs and priors), and show that the Stackelberg equilibrium behavior is not robust to small perturbations whereas the Nash equilibrium is.
Open Access
Quadratic multi-dimensional signaling games and affine equilibria
(Institute of Electrical and Electronics Engineers Inc., 2017) Sarıtaş, S.; Yüksel S.; Gezici, Sinan
This paper studies the decentralized quadratic cheap talk and signaling game problems when an encoder and a decoder, viewed as two decision makers, have misaligned objective functions. The main contributions of this study are the extension of Crawford and Sobel's cheap talk formulation to multi-dimensional sources and to noisy channel setups. We consider both (simultaneous) Nash equilibria and (sequential) Stackelberg equilibria. We show that for arbitrary scalar sources, in the presence of misalignment, the quantized nature of all equilibrium policies holds for Nash equilibria in the sense that all Nash equilibria are equivalent to those achieved by quantized encoder policies. On the other hand, all Stackelberg equilibria policies are fully informative. For multi-dimensional setups, unlike the scalar case, Nash equilibrium policies may be of non-quantized nature, and even linear. In the noisy setup, a Gaussian source is to be transmitted over an additive Gaussian channel. The goals of the encoder and the decoder are misaligned by a bias term and encoder's cost also includes a penalty term on signal power. Conditions for the existence of affine Nash equilibria as well as general informative equilibria are presented. For the noisy setup, the only Stackelberg equilibrium is the linear equilibrium when the variables are scalar. Our findings provide further conditions on when affine policies may be optimal in decentralized multi-criteria control problems and lead to conditions for the presence of active information transmission in strategic environments.
Open Access
Quadratic privacy-signaling games and the MMSE ınformation bottleneck problem for gaussian sources
(Institute of Electrical and Electronics Engineers Inc., 2022-05-23) Kazıklı, E.; Gezici, Sinan; Yüksel, S.
We investigate a privacy-signaling game problem in which a sender with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian. The sender aims to hide one of these random vectors and convey the other one whereas the objective of the receiver is to accurately estimate both of the random vectors. We analyze these conflicting objectives in a game theoretic framework with quadratic costs where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg (Bayesian persuasion) equilibria. We show that a payoff dominant Nash equilibrium among all admissible policies is attained by a set of explicitly characterized linear policies. We also show that a payoff dominant Nash equilibrium coincides with a Stackelberg equilibrium. We formulate the information bottleneck problem within our Stackelberg framework under the mean squared error distortion criterion where the information bottleneck setup has a further restriction that only one of the random variables is observed at the sender. We show that this MMSE Gaussian Information Bottleneck Problem admits a linear solution which is explicitly characterized in the paper. We provide explicit conditions on when the optimal solutions, or equilibrium solutions in the Nash setup, are informative or noninformative.
Open Access
Quadratic signaling with prior mismatch at an encoder and decoder: equilibria, continuity, and robustness properties
(Institute of Electrical and Electronics Engineers, 2022-01-11) Kazikli, E.; Sartas, S.; Gezici, Sinan
We consider communications through a Gaussian noise channel between an encoder and a decoder which have subjective probabilistic models on the source distribution. Although they consider the same cost function, the induced expected costs are misaligned due to their prior mismatch, which requires a game theoretic approach. We consider two approaches: a Nash setup, with no prior commitment, and a Stackelberg solution concept, where the encoder is committed to a given announced policy apriori. We show that the Stackelberg equilibrium cost of the encoder is upper semi continuous, under the Wasserstein metric, as encoder's prior approaches the decoder's prior, and it is also lower semi continuous with Gaussian priors. For the Stackelberg setup, the optimality of affine policies for Gaussian signaling no longer holds under prior mismatch, and thus team-theoretic optimality of linear/affine policies are not robust to perturbations. We provide conditions under which there exist informative Nash and Stackelberg equilibria with affine policies. Finally, we show existence of fully informative Nash and Stackelberg equilibria for the cheap talk problem under an absolute continuity condition.
Open Access
Signaling and information games with subjective costs or priors and privacy constraints
(2021-08) Kazıklı, Ertan
We investigate signaling game problems where an encoder and a decoder with misaligned objectives communicate. We consider a variety of setups involving cost criterion mismatch, prior mismatch and a particular application to privacy problems. We also consider both Nash and Stackelberg solution concepts. First, we extend the classical results on the scalar cheap talk problem which is intro-duced by Crawford and Sobel. In prior work, it is shown that the encoder must employ a quantization policy under any Nash equilibrium for arbitrary source distributions. We specifically consider sources with a log-concave density and in-vestigate properties of equilibria. For sources with two-sided unbounded support, we prove that, for any finite number of bins, there exists a unique equilibrium. If the source has semi-unbounded support, then there may exist a finite upper bound on the number of bins in equilibrium depending on certain explicit condi-tions. Moreover, we show that an equilibrium with more bins is more informative by showing that the expected costs of the encoder and the decoder in equilibrium decrease as the number of bins increases. Furthermore, for strictly log-concave sources with two-sided unbounded support, we prove that if the encoder and decoder iteratively compute their best responses starting from a given number of bins, then the resulting policies converge to the unique equilibrium with the corresponding number of bins. Second, we model a privacy problem as a signaling game between an encoder and a decoder. Given a pair of correlated observations modeled as jointly Gaus-sian random vectors, the encoder aims to hide one of them and convey the other one to the decoder. In contrast, the aim of the decoder is to accurately estimate both of the random vectors. For the resulting signaling game problem, we show that a payoff dominant Nash equilibrium among all admissible policies is attained by a set of explicitly characterized linear policies. We also show that a payoff dominant Nash equilibrium coincides with a Stackelberg equilibrium. Moreover, we formulate the information bottleneck problem within our Stackelberg frame-work under the mean squared error criterion where the information bottleneck setup has a further restriction that only one of the parameters is observed at the encoder. We show that the Gaussian information bottleneck problem admits a linear solution which is explicitly characterized. Third, we investigate communications through a Gaussian noise channel be-tween an encoder and a decoder with prior mismatch. Although they consider the same cost function, the induced expected costs as a map of their policies are misaligned due to their prior mismatch. We analyze the resulting signaling game problem under Stackelberg equilibria. We first investigate robustness of equilibria and show that the Stackelberg equilibrium cost of the encoder is upper semi continuous, under the Wasserstein metric, as the encoder’s prior approaches the decoder’s prior, and it is also lower semi continuous with Gaussian priors. In addition, we show that the optimality of affine policies for Gaussian signaling no longer holds under prior mismatch. Furthermore, we provide conditions under which there exist informative equilibria under an affine policy restriction. Fourth, we extend Crawford and Sobel’s formulation to a multidimensional source setting. We first provide a set of geometry conditions that decoder actions at a Nash equilibrium has to satisfy considering any multidimensional source. Then, we consider independent and identically distributed sources and charac-terize necessary and sufficient conditions under which an informative linear equi-librium exists. We observe that these conditions involve the bias vector that leads to misaligned costs. Depending on certain conditions on the bias vector, the existence of linear equilibria may require sources with a Gaussian or a sym-metric density. Moreover, we provide a rate-distortion theoretic formulation of the cheap talk problem and obtain achievable rate and distortion pairs for the Gaussian case. Finally, in a communication theoretic setup, we consider modulation classi-fication and symbol decoding problems jointly and propose optimal strategies under various settings. The aim is to decode a sequence of received signals with an unknown modulation scheme. First, the prior probabilities of the candidate modulation schemes are assumed to be known and a formulation is proposed under the Bayesian framework. Second, we address the case when the prior prob-abilities of the candidate modulation schemes are unknown, and provide a method under the minimax framework. Numerical simulations show that the proposed techniques improve the performance under the employed criteria compared to the conventional techniques in a variety of system configurations.
Open Access
Signaling games for log-concave distributions: Number of bins and properties of equilibria
(IEEE, 2021-11-25) Kazıklı, E.; Sarıtaş, Serkan; Gezici, Sinan; Linder, T.; Yüksel, S.
We investigate the equilibrium behavior for the decentralized cheap talk problem for real random variables and quadratic cost criteria in which an encoder and a decoder have misaligned objective functions. In prior work, it has been shown that the number of bins in any equilibrium has to be countable, generalizing a classical result due to Crawford and Sobel who considered sources with density supported on [0, 1]. In this paper, we first refine this result in the context of log-concave sources. For sources with two-sided unbounded support, we prove that, for any finite number of bins, there exists a unique equilibrium. In contrast, for sources with semi-unbounded support, there may be a finite upper bound on the number of bins in equilibrium depending on certain conditions stated explicitly. Moreover, we prove that for log-concave sources, the expected costs of the encoder and the decoder in equilibrium decrease as the number of bins increases. Furthermore, for strictly log-concave sources with two-sided unbounded support, we prove convergence to the unique equilibrium under best response dynamics which starts with a given number of bins, making a connection with the classical theory of optimal quantization and convergence results of Lloyd’s method. In addition, we consider more general sources which satisfy certain assumptions on the tail(s) of the distribution and we show that there exist equilibria with infinitely many bins for sources with two-sided unbounded support. Further explicit characterizations are provided for sources with exponential, Gaussian, and compactly-supported probability distributions.
Open Access
Signaling games in higher dimensions: geometric properties of equilibrium partitions
(IEEE, 2021-11-13) Kazıklı, Ertan; Gezici, Sinan
Signaling game problems investigate communication scenarios where encoder(s) and decoder(s) have misaligned objectives due to the fact that they either employ different cost functions or have inconsistent priors. We investigate a signaling game problem where an encoder observes a multi-dimensional source and conveys a message to a decoder, and the quadratic objectives of the encoder and decoder are misaligned due to a bias vector. For the scalar case, Crawford and Sobel in their seminal paper, show that under certain technical assumptions an encoding policy must be a quantization policy at any Nash equilibrium. We first provide a set of geometry conditions that needs to be satisfied in equilibrium considering any multi-dimensional source. Then, we consider multi-dimensional sources with independent and identically distributed components and completely characterize conditions under which a Nash equilibrium with a linear encoder exists. In particular, we show that if the components of the bias vector are not equal in magnitude, then there exists a linear equilibrium if and only if the source distribution is Gaussian. On the other hand, for a linear equilibrium to exist in the case of equal bias components, it is required that the source density is symmetric around its mean. Moreover, in the case of Gaussian sources, our results have a rate-distortion theoretic implication that achievable rates and distortions in the considered game theoretic setup can be obtained from their team theoretic counterpart.
Open Access
Signaling games in multiple dimensions: geometric properties of equilibrium solutions
(Elsevier, 2023-07-20) Kazıklı, E.; Gezici, Sinan; Yüksel, S.
Signaling game problems investigate communication scenarios where encoder(s) and decoder(s) have misaligned objectives due to the fact that they either employ different cost functions or have inconsistent priors. This problem has been studied in the literature for scalar sources under various setups. In this paper, we consider multi-dimensional sources under quadratic criteria in the presence of a bias leading to a mismatch in the criteria, where we show that the generalization from the scalar setup is more than technical. We show that the Nash equilibrium solutions lead to structural richness due to the subtle geometric analysis the problem entails, with consequences in both system design, the presence of linear Nash equilibria, and an information theoretic problem formulation. We first provide a set of geometric conditions that must be satisfied in equilibrium considering any multi-dimensional source. Then, we consider independent and identically distributed sources and characterize necessary and sufficient conditions under which an informative linear Nash equilibrium exists. These conditions involve the bias vector that leads to misaligned costs. Depending on certain conditions related to the bias vector, the existence of linear Nash equilibria requires sources with a Gaussian or a symmetric density. Moreover, in the case of Gaussian sources, our results have a rate–distortion theoretic implication that achievable rates and distortions in the considered game theoretic setup can be obtained from its team theoretic counterpart.
Open Access
Stochastic control approach to reputation games
(IEEE, 2020) Nuh Aygün, Dalkıran; Yüksel, S.
Through a stochastic-control-theoretic approach, we analyze reputation games, where a strategic long-lived player acts in a sequential repeated game against a collection of short-lived players. The key assumption in our model is that the information of the short-lived players is nested in that of the long-lived player. This nested information structure is obtained through an appropriate monitoring structure. Under this monitoring structure, we show that, given mild assumptions, the set of perfect Bayesian equilibrium payoffs coincides with Markov perfect equilibrium payoffs, and hence, a dynamic programming formulation can be obtained for the computation of equilibrium strategies of the strategic long-lived player in the discounted setup. We also consider the undiscounted average-payoff setup, where we obtain an optimal equilibrium strategy of the strategic long-lived player under further technical conditions. We then use this optimal strategy in the undiscounted setup as a tool to obtain a tight upper payoff bound for the arbitrarily patient long-lived player in the discounted setup. Finally, by using measure concentration techniques, we obtain a refined lower payoff bound on the value of reputation in the discounted setup. We also study the continuity of equilibrium payoffs in the prior beliefs.