  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.

Computer Science and Game Theory;Theoretical Economics