Optimal Signaling with Mismatch in Priors of an Encoder and Decoder
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We consider communications between an encoder and a decoder, viewed as two decision makers, which have subjective beliefs on the probabilistic model of the source distribution. Even though the decision makers employ the same cost function, induced expected costs are different from the perspective of the encoder and decoder due to their subjective probabilistic beliefs, which requires a game theoretic treatment. Depending on the commitment nature of the encoder to its policies, we analyze this signaling game problem under Nash and Stackelberg equilibrium concepts. In particular, we consider a communication scenario through a Gaussian noise channel with a power constrained encoder. We show that the Stackelberg equilibrium cost of the encoder is upper semi continuous, under the Wasserstein metric, as encoder's prior approaches to decoder's prior and in the particular case of Gaussian subjective priors it is also lower semi continuous, which proves the robustness of the equilibrium around the team setup. We further prove that the optimality of affine policies for Gaussian signaling under Stackelberg equilibria breaks down due to the presence of prior mismatch. We also investigate the informativeness of Stackelberg equilibria under affine policy restriction when there is prior mismatch and show that under certain conditions the equilibria become non-informative, that is information transmission ceases to exist. For the Nash setup, we provide necessary and sufficient conditions under which there exist informative affine Nash equilibria. Furthermore, we show that there exist fully informative Nash and Stackelberg equilibria for the cheap talk problem (i.e., no additive noise term and no power constraint at the encoder) as in the team theoretic setup under an absolute continuity condition.
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