Regularized Bayesian best response learning in finite games

TL;DR

This paper introduces a regularized Bayesian best response dynamic in finite population games, establishing existence, approximation, and convergence properties of Bayesian equilibria under perturbations.

Contribution

It develops a new RBBR learning dynamic for Bayesian games, proving existence of equilibria, solution properties, and applying the framework to potential and negative semidefinite games.

Findings

Existence of regularized Bayesian equilibrations

Unique continuous solutions from arbitrary initial conditions

Convergence results for Bayesian potential and negative semidefinite games

Abstract

We introduce the notion of regularized Bayesian best response (RBBR) learning dynamic in heterogeneous population games. We obtain such a dynamic via perturbation by an arbitrary lower semicontinuous, strongly convex regularizer in Bayesian population games with finitely many strategies. We provide a sufficient condition for the existence of rest points of the RBBR learning dynamic, and hence the existence of regularized Bayesian equilibrium in Bayesian population games. These equilibria are shown to approximate the original Bayesian equilibria for vanishingly small perturbations. We also explore the fundamental properties of the RBBR learning dynamic, which includes the existence of unique continuous solutions from arbitrary initial conditions, as well as the continuity of the solution trajectories thus obtained with respect to the initial conditions. Finally, as application the the theory we introduce the notions of Bayesian potential and Bayesian negative semidefinite games and provide convergence results for such games.