## Signaling and information games with subjective costs or priors and privacy constraints

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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.