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.

翻译：本文探讨了贝叶斯博弈中的均衡概念，这类博弈是不完全信息博弈的基本模型。我们追求均衡的三个理想性质。第一，均衡可以通过引入中介者自然实现。第二，均衡能够以分布式方式高效计算。第三，该类别中的任何均衡对于广泛类型的博弈，都能以无政府代价衡量近似最大化社会福利。这三个性质使得参与者能够计算均衡并通过中介者实现，从而进入具有近似最优社会福利的稳定状态。我们的主要结果是证明存在同时满足这三个性质的均衡概念。为此，我们刻画了相关均衡的各种（不等价）扩展，统称为贝叶斯相关均衡。特别地，我们重点关注沟通均衡（也称为协调机制），它可通过收集每个参与者私人信息并向其发送相关建议的中介者实现。我们证明，若每个参与者在重复进行贝叶斯博弈时最小化一种称为不诚实交换遗憾的遗憾变体，这些动态过程的经验分布会收敛至沟通均衡。我们提出一种高效算法，可实现不诚实交换遗憾的最小化并给出次线性上界，该上界被证明在乘法常数意义下是最优的。因此，通过使用我们的算法模拟动态过程，可以高效计算近似沟通均衡。此外，我们基于平滑性论证，将贝叶斯纳什均衡的无政府代价下界扩展至由所提动态过程获得的均衡。