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

TL;DR

This paper analyzes quadratic signaling games with prior mismatch between encoder and decoder, exploring equilibrium properties, continuity, and robustness, and demonstrating how prior mismatch affects optimal policies and equilibrium existence.

Contribution

It introduces a game-theoretic framework for quadratic signaling with prior mismatch, analyzing equilibrium continuity, robustness, and conditions for affine policy existence.

Findings

Stackelberg equilibrium cost is upper semi continuous with prior mismatch.

Affine policies are not robust under prior mismatch.

Existence of informative equilibria under certain conditions.

Abstract

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.