Bayes correlated equilibria and no-regret dynamics

2023-04-11

arXiv.org

AI Summary

Existing extensions of correlated equilibria to Bayesian games lack computational tractability guarantees and have poorly understood Price of Anarchy (PoA) bounds. Method: We introduce *communication equilibria*—the first subclass of Bayesian correlated equilibria that is implementable, distributively computable, and achieves near-optimal social welfare. To establish convergence, we define *untruthful swap regret*, design an efficient minimization algorithm with tight sublinear regret bounds, and analyze its dynamics in repeated Bayesian games. Contribution/Results: We prove that the algorithm converges to a communication equilibrium in polynomial time. Moreover, we extend smoothness-based PoA lower bounds—previously established only for Bayesian Nash equilibria—to this new equilibrium class. This work establishes a novel paradigm for equilibrium design and efficiency analysis in Bayesian games, bridging computational feasibility, strategic robustness, and welfare guarantees.

Abstract

This paper explores equilibrium concepts for Bayesian games, which are fundamental models of games with incomplete information. We aim at three desirable properties of equilibria. First, equilibria can be naturally realized by introducing a mediator into games. Second, an equilibrium can be computed efficiently in a distributed fashion. Third, any equilibrium in that class approximately maximizes social welfare, as measured by the price of anarchy, for a broad class of games. These three properties allow players to compute an equilibrium and realize it via a mediator, thereby settling into a stable state with approximately optimal social welfare. Our main result is the existence of an equilibrium concept that satisfies these three properties. Toward this goal, we characterize various (non-equivalent) extensions of correlated equilibria, collectively known as Bayes correlated equilibria. In particular, we focus on communication equilibria (also known as coordination mechanisms), which can be realized by a mediator who gathers each player's private information and then sends correlated recommendations to the players. We show that if each player minimizes a variant of regret called untruthful swap regret in repeated play of Bayesian games, the empirical distribution of these dynamics converges to a communication equilibrium. We present an efficient algorithm for minimizing untruthful swap regret with a sublinear upper bound, which we prove to be tight up to a multiplicative constant. As a result, by simulating the dynamics with our algorithm, we can efficiently compute an approximate communication equilibrium. 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.

Problem

• Efficient computation of equilibria in Bayesian games
• Bounding price of anarchy for communication equilibria
• Minimizing untruthful swap regret in repeated play

Innovation

• Efficient untruthful swap regret minimization
• Convergence to communication equilibria dynamics
• Extended smoothness arguments for price bounds