Correlated Equilibrium Payoffs Research Articles

Communication and cooperation in repeated games

We study the role of communication in repeated games with private monitoring. We first show that without communication, the set of Nash equilibrium payoffs in such games is a subset of the set ofε‐coarse correlated equilibrium payoffs (ε‐CCE) of the underlying one‐shot game. The value ofεdepends on the discount factor and the quality of monitoring. We then identify conditions under which there are equilibria with “cheap talk” that result in nearly efficient payoffs outside the setε‐CCE. Thus, in our model, communication is necessary for cooperation.

Correlated equilibria of two person repeated games with random signals

In this work we extend a result of Lehrer (Math Oper Res 17(1):175–199, 1992a) characterising the correlated equilibrium payoffs in undiscounted two player repeated games with partial monitoring to the case in which the signals are permitted to be stochastic. In particular, we develop appropriate versions of Lehrer’s concepts of “indistinguishable” and “more informative.” We also show that any individually rational payoff associated with a (correlated) distribution on pure action profiles in the stage game such that neither player can profitably deviate from one of his actions to another that is indistinguishable and more informative is the payoff of a correlated equilibrium of the infinitely repeated game.

A detail-free mediator

We present a cheap talk extension to any two-player, finite, complete information game, and ask what correlations over actions are implementable in Nash equilibria of the extended game. In the extension, players communicate repeatedly through a detail-free mediator that has been studied in Lehrer (1991) and in Gossner and Vieille (2001). The extension captures situations in which people can observe the opponentʼs face during the conversation. While Gossner and Vieille (2001) prove that no correlation can be securely implemented by using only this mediator, we prove a result closer to Lehrer (1991), namely, that the Nash equilibrium payoffs of the extended game essentially coincide with the correlated equilibrium payoffs of the underlying game. The contrasting results can be explained with our additional assumptions that the players can also communicate directly and, more importantly, the private messages sent to the mediator can be recorded and revealed later in the conversation.

Equilibrium payoffs of finite games

Abstract We study the structure of the set of equilibrium payoffs in finite games, both for Nash and correlated equilibria. In the two-player case, we obtain a full characterization: if U and P are subsets of R 2 , then there exists a bimatrix game whose sets of Nash and correlated equilibrium payoffs are, respectively, U and P, if and only if U is a finite union of rectangles, P is a polytope, and P contains U. The n-player case and the robustness of the result to perturbation of the payoff matrices are also studied. We show that arbitrarily close games may have arbitrarily different sets of equilibrium payoffs. All existence proofs are constructive.

Correlation in Repeated Games with Public Monitoring

This paper studies extensive-form correlation in discounted infinitely repeated games with public monitoring. We propose two extensions of Fudenberg et al.'s (Econometrica 62:997--1040, 1994) notion of a perfect public equilibrium: the notion of a perfect public correlated equilibrium for games in which a correlation device sends private correlated messages to the players at the beginning of each period and the notion of a perfect public augmented equilibrium for games in which the device also publicly informs the players of the recommended action profile at the end of each period. The set of perfect public correlated equilibrium payoffs is compared to the set of subgame perfect publicly correlated equilibrium payoffs in the perfect monitoring case. It is shown that augmented correlation produces efficiency gains in the repeated partnership game by Radner et al. (Rev Econ Stud 53:59-69, 1986).

Leadership games with convex strategy sets

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.

Unattainable Payoffs for Repeated Games of Private Monitoring

We tightly bound from the outside the set of sequential equilibrium payoffs in repeated games of private monitoring. To do this, we develop a tractable new solution concept for standard repeated games with perfect monitoring: Markov Perfect Correlated Equilibrium generalizes the operator approach of Abreu, Pearce, and Stacchetti (1990), but instead takes correlated equilibrium of the auxiliary game. We show that for any private monitoring structure, the set of sequential equilibrium payoffs of a repeated game is contained within the set of Markov Perfect Correlated Equilibrium payoffs of the repeated expected game. This bound can be made tight with a simple two-stage procedure. The techniques we develop are tractable and shed light on all economic settings with imperfectly observed actions, like dynamic oligopoly, long-term partnerships, and relational contracting. In all cases, they yield the sharpest equilibrium payoff prediction that is agnostic about the monitoring structure.

Unmediated communication in repeated games with imperfect monitoring

We show that any correlated equilibrium payoff of two-player repeated games with imperfect monitoring and without discounting can be reached as a Nash equilibrium payoff of the game extended by a universal mechanism of unmediated communication where players are computationally restricted. The communication mechanism is designed with the aid of cryptographic tools.

Correlated equilibrium payoffs and public signalling in absorbing games

An absorbing game is a repeated game where some action combinations are absorbing, in the sense that whenever they are played, there is a positive probability that the game terminates, and the players receive some terminal payoff at every future stage. We prove that every multi-player absorbing game admits a correlated equilibrium payoff. In other words, for every e>0 there exists a probability distribution p e over the space of pure strategy profiles that satisfies the following. With probability at least 1−e, if a pure strategy profile is chosen according to p e and each player is informed of his pure strategy, no player can profit more than e in any sufficiently long game by deviating from the recommended strategy.

Antagonistic games

Some properties of zero-sum games are extended to different classes of two-person non zero-sum games. The definitions of these classes are based on different notions of antagonism. The notion of almost strictly competitive games introduced by Aumann in 1961, is generalized while keeping the uniqueness of Nash equilibrium payoff. The set of Nash equilibria of a weakly unilaterally competitive game coincides with the set of its saddle points. Some properties are satisfied by the correlated equilibrium payoff of a weakly unilaterally competitive game; and by Nash equilibria of a symmetric weakly unilaterally competitive games.

Correlated Equilibrium in Stochastic Games

We study the existence of uniform correlated equilibrium payoffs in stochastic games. The correlation devices that we use are either autonomous (they base their choice of signal on previous signals, but not on previous states or actions) or stationary (their choice is independent of any data and is drawn according to the same probability distribution at every stage). We prove that any n-player stochastic game admits an autonomous correlated equilibrium payoff. When the game is positive and recursive, a stationary correlated equilibrium payoff exists. Journal of Economic Literature Classification Numbers: C72, C73.

Learning correlated equilibria in population games

The paper develops a framework for the analysis of finite n-player games, recurrently played by randomly drawn n-tuples of players, from a finite population. We first relate the set of equilibria of this game to the set of correlated equilibria of the underlying game, and then focus on learning processes modelled as Markovian adaptive dynamics. For the class of population games for which the underlying game has identical interests, we show that, independently of the matching technology, any myopic-best reply dynamics converges (in probability) to a correlated equilibrium. We also analyze noisy best reply dynamics, where players’ behaviour is perturbed by payoff-dependent mistakes, and explicitly characterize the limit distribution of the perturbed game in terms of the correlated equilibrium payoff of the underlying game.

A note on correlated equilibrium

The set of correlated equilibria for a bimatrix game is a closed, bounded, convex set containing the set of Nash equilibria. We show that every extreme point of a maximal Nash set is an extreme point of the above convex set. We also give an example to show that this result is not true in the payoff space, i.e. there are games where no Nash equilibrium payoff is an extreme point of the set of correlated equilibrium payoffs.

Correlated Equilibria in Two-Player Repeated Games with Nonobservable Actions

Four kinds of correlated equilibrium payoff sets in undiscounted repeated games with nonobservable actions are studied. Three of them. the upper, the uniform, and Banach lead to the same payoff set, whereas the lower one in general is associated with a larger set. The extensive form correlated equilibrium is also explored. It turns out that both the regular and extensive form correlated equilibria yield the same sets of payoffs.

Internal correlation in repeated games

This paper characterizes the set of all the Nash equilibrium payoffs in two player repeated games where the signal that the players get after each stage is either trivial (does not reveal any information) or standard (the signal is the pair of actions played). It turns out that if the information is not always trivial then the set of all the Nash equilibrium payoffs coincides with the set of the correlated equilibrium payoffs. In particular, any correlated equilibrium payoff of the one shot game is also a Nash equilibrium payoff of the repeated game.For the proof we develop a scheme by which two players can generate any correlation device, using the signaling structure of the game. We present strategies with which the players internally correlate their actions without the need of an exogenous mediator.

Equilibria with Communication in a Job Market Example

We study (costless) information transmission from a job applicant to an employer who must decide whether to hire him and, if so, which position to give him. We construct equilibrium payoffs requiring at least two signaling steps, or even that no deadline be imposed on the (plain) conversation. The set of communication equilibrium payoffs (achieved with the help of a communication device) is larger than the set of equilibrium payoffs of the plain conversation game but coincides with the set of correlated equilibrium payoffs.

Communication Equilibria in Repeated Games with Incomplete Information

Correlated equilibria (in the sense of Aumann, i.e., normal form correlated equilibria) are studied in two-person infinitely repeated games with lack of information on one side. Extensions of the solution concept are introduced, several of which are shown to be payoff equivalent in the model under consideration. A normal form correlated equilibrium is a Nash equilibrium of the game where each player gets a private output from a correlation device before the beginning of the original game. An r-device (r = 0, 1, …, ∞) is a communication device, receiving an input from each player before every stage t = 1, …, r and selecting a pair of outputs, one for each player, at every stage t = 1, 2, …, according to a memory dependent transition probability. An r-communication equilibrium is a Nash equilibrium of the game extended by means of an r-device. Let C (resp. Dr, r = 0, 1, …, ∞) be the set of normal form correlated (resp. r-communication) equilibrium payoffs. D0 is the set of “extensive form correlated equilibrium payoffs,” achieved by means of an “autonomous device,” acting only by sending a stream of outputs to each player. And D∞ is the set of all communication equilibrium payoffs, corresponding to communication devices which can receive inputs at every stage (even arbitrarily large). The paper characterizes the different sets and investigate whether they coincide.