Correlated equilibria and communication in games.

Analyse bayésienne; Théorie des jeux; Information privée;

Belief-Invariant and Quantum Equilibria in Games of Incomplete Information

Drawing on ideas from game theory and quantum physics, we investigate nonlocal correlations from the point of view of equilibria in games of incomplete information. These equilibria can be classified in decreasing power as general communication equilibria, belief-invariant equilibria and correlated equilibria, all of which contain the familiar Nash equilibria. The notion of belief-invariant equilibrium has appeared in game theory before, in the 1990s. However, the class of non-signalling correlations associated to belief-invariance arose naturally already in the 1980s in the foundations of quantum mechanics. Here, we explain and unify these two origins of the idea and study the above classes of equilibria, and furthermore quantum correlated equilibria, using tools from quantum information but the language of game theory. We present a general framework of belief-invariant communication equilibria, which contains (quantum) correlated equilibria as special cases. It also contains the theory of Bell inequalities, a question of intense interest in quantum mechanics, and quantum games where players have conflicting interests, a recent topic in physics. We then use our framework to show new results related to social welfare. Namely, we exhibit a game where belief-invariance is socially better than correlated equilibria, and one where all non-belief-invariant equilibria are socially suboptimal. Then, we show that in some cases optimal social welfare is achieved by quantum correlations, which do not need an informed mediator to be implemented. Furthermore, we illustrate potential practical applications: for instance, situations where competing companies can correlate without exposing their trade secrets, or where privacy-preserving advice reduces congestion in a network. Along the way, we highlight open questions on the interplay between quantum information, cryptography, and game theory

A Detail-free Mediator and the 3 Player Case

Two players can make use of a trusted third party who mediates and partially resolves their conflict. Usually, the mediator should be aware of the situation and give suggestions to the players accordingly. However, a corrupt mediator can have a big influence on the outcome of the game. We single out a transparent mediator which can be safely applied in any two player game without loss of efficiency. That is, the mediator is independent of the game and the desired outcome. Technically, we show that any correlated equilibrium of any two player game can be obtained as Nash equilibria of the game, extended with cheap, pre-play communication, where players can communicate through the proposed mediator. The key idea is that after the mediated communication the players can have a plain conversation. In particular, the mediating communication device is transparent, controllable and is the same for all games and for all equilibrium distributions. We extend the result to three player games and show that one of the players can play the role of the mediator. We implement the set of correlated equilibrium in Nash equilibria of an extended game where the players have a plain conversation. The central assumption is that players can be invited to eavesdrop a private conversation. We extend the analysis to games with incomplete information and to the set of communication equilibria.cheap talk, communication device, correlated equilibrium, communi- cation equilibrium, detail-free mechanism, mediator

Bayes correlated equilibria and no-regret dynamics

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

Implementability of Correlated and Communication Equilibrium Outcomes in Incomplete Information Games

In a correlated equilibrium, the players’ choice of actions is affected by random, correlated messages that they receive from an outside source, or mechanism. This allows for more equilibrium outcomes than without such messages (pure-strategy equilibrium) or with statistically independent ones (mixed-strategy equilibrium). In an incomplete information game, the messages may also convey information about the types of the other players, either because they reflect extraneous events that affect the types (correlated equilibrium) or because the players themselves report their types to the mechanism (communication equilibrium). Thus, mechanisms can be classified by the connections between the messages that the players receive and their own and the other players’ types, the dependence or independence of the messages, and whether randomness is involved. These properties may affect the achievable equilibrium outcomes, i.e., the payoffs and joint distributions of type and action profiles. Whereas for complete information games there are only three classes of equilibrium outcomes, with incomplete information the number is 14–15 for correlated equilibria and 15–17 for communication equilibria. Each class is characterized by the properties of the mechanisms that implement its members. The majority of these classes have not been described before.Correlated equilibrium, Communication equilibrium, Incomplete information, Bayesian games, Mechanism, Correlation device, Implementation

Unmediated Communication in Games with Complete and Incomplete Information

In this paper we study the effects of adding unmediated communication to static, finite games of complete and incomplete information. We characterize S^{U}(G), the set of outcomes of a game G, that are induced by sequential equilibria of cheap talk extensions. A cheap talk extension of G is an extensive-form game in which players communicate before playing G. A reliable mediator is not available and players exchange private or public messages that do not affect directly their payoffs. We first show that if G is a game of complete information with five or more players and rational parameters, then S^{U}(G) coincides with the set of correlated equilibria of G. Next, we demonstrate that if G is a game of incomplete information with at least five players, rational parameters and full support (i.e. all profiles of types have positive probability), then S^{U}(G) is equal to the set of communication equilibria of G.Communication, Correlated equilibrium, Communication equilibrium, Sequential equilibrium, Mechanism design, Revelation principle

Computing Optimal Equilibria and Mechanisms via Learning in Zero-Sum Extensive-Form Games

We introduce a new approach for computing optimal equilibria via learning in games. It applies to extensive-form settings with any number of players, including mechanism design, information design, and solution concepts such as correlated, communication, and certification equilibria. We observe that optimal equilibria are minimax equilibrium strategies of a player in an extensive-form zero-sum game. This reformulation allows to apply techniques for learning in zero-sum games, yielding the first learning dynamics that converge to optimal equilibria, not only in empirical averages, but also in iterates. We demonstrate the practical scalability and flexibility of our approach by attaining state-of-the-art performance in benchmark tabular games, and by computing an optimal mechanism for a sequential auction design problem using deep reinforcement learning