Quadratic Signaling With Prior Mismatch at an Encoder and Decoder: Equilibria, Continuity, and Robustness Properties

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
a priori
. We show that the Stackelberg equilibrium cost of the encoder is upper semicontinuous, under the Wasserstein metric, as encoder’s prior approaches the decoder’s prior, and it is also lower semicontinuous 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.