Convergence and Correlation in Large Games

We consider multi-population Bayesian games with a large number of players. Each player aims at minimizing a cost function that depends on her own action, the distribution of players' actions in all populations, and an unknown state parameter. We study the nonatomic limit versions of these games. We 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, are reduced to 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 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.