Correlated Equilibria in Large Anonymous Bayesian Games

TL;DR

This paper introduces the concept of Bayes correlated Wardrop equilibrium in large anonymous Bayesian games, demonstrating convergence of equilibria and costs in nonatomic limits and potential games.

Contribution

It extends the concept of Bayes correlated equilibrium to nonatomic games and proves convergence results for large finite-player games with potential functions.

Findings

Bayes correlated Wardrop equilibria are limits of finite-player game equilibria.

In convex potential games, equilibrium costs are identical across distributions.

Flow distributions from no-regret sequences converge to Wardrop equilibria.

Abstract

We consider multi-population Bayesian games with a large number of players. Each player aims at minimizing a cost function that depends on this player's own action, the distribution of players' actions in all populations, and an unknown state parameter. We study the nonatomic limit versions of these games and introduce the concept of Bayes correlated Wardrop equilibrium, which extends the concept of Bayes correlated equilibrium to nonatomic games. We prove that Bayes correlated Wardrop equilibria are limits of action flows induced by Bayes correlated equilibria of the game with a large finite set of small players. For nonatomic games with complete information admitting a convex potential, we prove that the set of correlated and of coarse correlated Wardrop equilibria coincide with the set of probability distributions over Wardrop equilibria, and that all equilibrium outcomes have the same costs. We get the following consequences. First, all flow distributions of (coarse) correlated equilibria in convex potential games with finitely many players converge to Wardrop equilibria when the weight of each player tends to zero. Second, for any sequence of flows satisfying a no-regret property, its empirical distribution converges to the set of distributions over Wardrop equilibria and the average cost converges to the unique Wardrop cost.