We investigate the mismatched distortion-rate problem formulated by Lapidoth in 1997. The single-letter characterization of the distortion-rate tradeoff is an open problem. When the rate is large enough, the mismatched distortion-rate problem is deeply related to the mechanism design problem formulated by Jackson and Sonnenschein in 2007. Inspired by this solution, we formulate a new converse bound that matches the achievability bound in four special cases: 1) when the rate is large enough, 2) when the source is binary and uniform and the decoder output is binary, 3) when the encoder distortion measure does not depend on the source symbols, 4) when the sum of the two distortion measures is zero. An important feature of the model is the stochastic post-processing of the codewords. By carefully selecting the randomness of the decoding function, it is optimal for the encoder to correctly transmit the source.

In this paper, we provide a geometric characterization of the set of interim equilibrium payoffs in a general class of signaling games. To obtain a tractable characterization, we use the belief based approach found in the literature on repeated games with incomplete information, cheap talk and Bayesian persuasion. This approach avoids to specify the prior, the strategies of the sender and receiver, and the belief system. The key ingredient is to consider Bayes-plausible belief distributions that are incentive-compatible for the sender. Geometrically, this leads to a constrained convexification of the graphs of the interim value correspondences. Our characterization extends the analogous result for sender-receiver cheap talk games. We illustrate the results with some classical signaling games. We derive the best equilibrium payoff for the sender when his preferences are type-independent. For zero-sum preferences, we obtain an explicit formula for the ex-ante equilibrium payoff and establish a simple condition for the uniqueness of interim equilibrium payoffs.

The strategic communication problem consists of a joint source-channel coding problem in which the encoder and the decoder optimize two arbitrary distinct distortion functions. This problem lies on the bridge between Information Theory and Game Theory. As in the persuasion game of Kamenica and Gentzkow, we consider that the encoder commits to an encoding strategy, then the decoder selects the optimal output symbol based on its Bayesian posterior belief. The informational content of the source affects differently the two distinct distortion functions, therefore each symbol is encoded in a specific way. In this work, we consider that the decoder has side information. Accordingly, we reformulate the Bayesian update of the decoder posterior beliefs and the optimal information disclosure policy of the encoder. We provide four different expressions of the solution, in terms of the expected encoder distortion optimized under an information constraint, and it in terms of convex closures of auxiliary distortion functions. We compute the encoder optimal distortion for the doubly symmetric binary source example.

We investigate a strategic formulation of the joint source-channel coding problem in which the encoder and the decoder are endowed with distinct distortion functions. We provide the solutions in four different scenarios. First, we assume that the encoder and the decoder cooperate in order to achieve a certain pair of distortion values. Second, we suppose that the encoder commits to a strategy whereas the decoder implements a best response, as in the persuasion game where the encoder is the Stackelberg leader. Third, we consider that the decoder commits to a strategy, as in the mismatched rate-distortion problem or as in the mechanism design framework. Fourth, we study the cheap talk game in which the encoding and the decoding strategies form a Nash equilibrium.