Bayes correlated equilibria, no-regret dynamics in Bayesian games, and the price of anarchy

This paper investigates equilibrium computation and the price of anarchy for Bayesian games, which are the fundamental models of games with incomplete information. In normal-form games with complete information, it is known that efficiently computable no-regret dynamics converge to correlated equilibria, and the price of anarchy for correlated equilibria can be bounded for a broad class of games called smooth games. However, in Bayesian games, as surveyed by Forges (1993), several non-equivalent extensions of correlated equilibria exist, and it remains unclear whether they can be efficiently computed or whether their price of anarchy can be bounded. In this paper, we identify a natural extension of correlated equilibria that can be computed efficiently and is guaranteed to have bounds on the price of anarchy in various games. First, we propose a variant of regret called untruthful swap regret. If each player minimizes it in repeated play of Bayesian games, the empirical distribution of these dynamics is guaranteed to converge to communication equilibria, which is one of the extensions of correlated equilibria proposed by Myerson (1982). We present an efficient algorithm for minimizing untruthful swap regret with a sublinear upper bound, which we prove to be tight in terms of the number of types. As a result, by simulating the dynamics with our algorithm, we can approximately compute a communication equilibrium in polynomial time. Furthermore, we extend existing lower bounds on the price of anarchy based on the smoothness arguments from Bayes--Nash equilibria to equilibria obtained by the proposed dynamics.