Correlated Equilibrium Distributions Research Articles

Two definitions of correlated equilibrium

Correlated equilibrium constitutes one of the basic solution concepts for static games with complete information. Actually two variants of correlated equilibrium are in circulation and have been used interchangeably in the literature. Besides the original notion due to Aumann (1974), there exists a simplified definition typically called canonical correlated equilibrium or correlated equilibrium distribution. It is known that the original and the canonical version of correlated equilibrium are equivalent from an ex-ante perspective. However, we show that they are actually distinct – both doxastically as well as behaviourally – from an interim perspective. An elucidation of this difference emerges in the reasoning realm: while Aumann’s correlated equilibrium can be epistemically characterized by common belief in rationality and a common prior, canonical correlated equilibrium additionally requires the condition of one-theory-per-choice. Consequently, the application of correlated equilibrium requires a careful choice of the appropriate variant.

A note on perfect correlated equilibria

This note studies perfect correlated equilibrium introduced by Dhillon and Mertens (1996). We show that the standard revelation principle for perfect correlated equilibrium distributions holds in two-by-two games. We then show that in N-player two-action games, although the revelation principle fails, a canonical correlated device for any perfect correlated equilibrium distribution involves two copies of action sets as the message space.

Can quantum entanglement implement classical correlated equilibria?

We ask whether players of a classical game can partition a pure quantum state to implement classical correlated equilibrium distributions. The main contribution of this work is an impossibility result: we provide an example of a classical correlated equilibrium that cannot be securely implemented without useful information leaking outside the system. We study the model where players of a classical complete information game initially share an entangled pure quantum state. Players may perform arbitrary local operations on their subsystems, but no direct communication (either quantum or classical) is allowed. We explain why, for the purpose of implementing classical correlated equilibria, it is desirable to restrict the initial state to be pure and to restrict communication. In this framework, we define the concept of pure quantum correlated equilibrium (PQCE) and show that in a normal form game, any outcome distribution implementable by a PQCE can also be implemented by a classical correlated equilibrium (CE), but that the converse is false. We extend our analysis to extensive form games, and compare the power of PQCE to extensive form classical correlated equilibria (EFCE) and immediate-revelation extensive form correlated equilibria (IR-EFCE).

Can quantum entanglement implement classical correlated equilibria

We ask whether players of a classical game can partition a pure quantum state to implement classical correlated equilibrium distributions. The main contribution of this work is an impossibility res...

Correlated equilibrium and higher order beliefs about play

We study a refinement of correlated equilibrium in which playersʼ actions are driven by their beliefs and higher order beliefs about the play of the game (beliefs over what other players will do, over what other players believe others will do, etc.). For any finite, complete-information game, we characterize the behavioral implications of this refinement with and without a common prior, and up to any a priori fixed depth of reasoning. In every finite game “most” correlated equilibrium distributions are consistent with this refinement; as a consequence, this refinement gives a classification of “most” correlated equilibrium distributions based on the maximum order of beliefs used by players in the equilibrium. On the other hand, in a generic two-player game any non-degenerate mixed-strategy Nash equilibrium is not consistent with this refinement.

Correlated Equilibrium in Games with Incomplete Information

We de…ne a notion of correlated equilibrium for games with incomplete in a general setting with …nite players, …nite actions, and …nite states. We refer to this solution concept as Bayes correlated equilibrium. For a given common prior over the payorelevant states and types, we show that the set of Bayes correlated equilibrium probability distributions equals the set of probability distributions over actions, states and types that might arise in any Bayes Nash equilibrium consistent with the given common prior over states and types. We de…ne a game of incomplete in terms of a payoenvironment, or the basic game, and a belief environment, or the information structure. We show how the structure aects the set of predictions that can be made about the Bayes correlated equilibrium distribution. We show that a more informed structure reduces the set of Bayes equilibrium distributions as it imposes additional incentive constraints.

Structure of extreme correlated equilibria: a zero-sum example and its implications

We exhibit the rich structure of the set of correlated equilibria by analyzing the simplest of polynomial games: the mixed extension of matching pennies. We show that while the correlated equilibrium set is convex and compact, the structure of its extreme points can be quite complicated. In finite games the ratio of extreme correlated to extreme Nash equilibria can be greater than exponential in the size of the strategy spaces. In polynomial games there can exist extreme correlated equilibria which are not finitely supported; we construct a large family of examples using techniques from ergodic theory. We show that in general the set of correlated equilibrium distributions of a polynomial game cannot be described by conditions on finitely many moments (means, covariances, etc.), in marked contrast to the set of Nash equilibria which is always expressible in terms of finitely many moments.

Correlated Equilibrium and Higher Order Beliefs about Play

We study a refinement of correlated equilibrium in which players' actions are driven by their beliefs and higher order beliefs about the play of the game (beliefs over what other players will do, over what other players believe others will do, etc.). For any finite, complete-information game, we characterize the behavioral implications of this refinement with and without a common prior, and up to any a priori fixed depth of reasoning. In every finite game most correlated equilibrium distributions are consistent with this refinement; as a consequence, this refinement gives a classification of most correlated equilibrium distributions based on the maximum order of beliefs used by players in the equilibrium. On the other hand, in a generic two-player game any non-degenerate mixed-strategy Nash equilibrium is not consistent with this refinement.

A difficulty in implementing correlated equilibrium distributions

We view achieving a particular correlated equilibrium distribution for a normal form game as an implementation problem. We show, using a parametric version of the two-person Chicken game, that a social choice function that chooses any particular correlated equilibrium distribution does not satisfy Maskin monotonicity and therefore cannot be fully implemented in Nash equilibrium. Thus, no mechanism that aims to understand a correlated distribution as the unique outcome of Nash equilibrium play can be found.

Message spaces for perfect correlated equilibria

We show that a perfect correlated equilibrium distribution of an N-person game, as defined by Dhillon and Mertens (1996) can be achieved using a finite number of copies of the strategy space as the message space.

Multiple Equilibria as a Difficulty in Understanding Correlated Distributions

We view achieving a particular correlated equilibrium distribution for a normal form game as an implementation problem. We show, using a parametric version of the two-person Chicken game and a wide class of correlated equilibrium distributions, that a social choice function that chooses a particular correlated equilibrium distribution from this class does not satisfy the Maskin monotonicity condition and therefore can not be fully implemented in Nash equilibrium.

On the geometry of Nash equilibria and correlated equilibria

It is well known that the set of correlated equilibrium distributions of an n-player noncooperative game is a convex polytope that includes all the Nash equilibrium distributions. We demonstrate an elementary yet surprising result: the Nash equilibria all lie on the boundary of the polytope.

Comparison of Information Structures

We introduce the notion of an information structure I as being richer than another J when for every game G, all correlated equilibrium distributions of G induced by J are also induced by I. In particular, if I is richer than J then I can make all agents as well off as J in any game. We also define J to be faithfully reproducible from I when all the players can compute from their information in I “new information” that reproduces what they could have received from J. Our main result is that I is richer than J if and only if J is faithfully reproducible from I. Journal of Economic Literature Classification Number: C72.

Correlation without Mediation: Expanding the Set of Equilibrium Outcomes by “Cheap” Pre-play Procedures

LetPbe a correlated equilibrium distribution on the set of outcomes of a gameG. CanPbe implemented by some “cheap” pre-play procedure that does not involve a mediator? It is shown that if there are two distinct Nash equilibrium payoffs for each player inGand ifPis rational (that is, consists of probabilities which are rational numbers) and generates for each playerian expected payoff which is above her worst Nash equilibrium payoff, thenPcan be virtually implemented in a sequential equilibrium of an extended game which is generated by adding a “cheap” pre-play phase.Journal of EconomicLiterature Classification Number: C72.

Fondements des concepts de solution en théorie des jeux

In this paper we study the player's reasoning in normal form games. To this end we generalize the concept of objective correlated equilibrium distributions introduced by Aumann [1987] to the subjective case. We propose simple and easily interpretable axioms of the rational behavior dealing with these distributions of subjective correlated equilibrium. We then offer a common basis to different usual solution concepts and to the less known concept of strong rationalisability (Stalnaker [1994]).

Computation as a correlation device

The definition of a correlated equilibrium takes as given the information possessed by the players. Some correlated equilibria rely on information acquisition decisions that are difficult to rationalize. We study the correlated equilibria which can arise when agents choose their information and information acquisition costs are (ordinally) symmetric. The set of correlated equilibria which are attainable in this sense is much smaller than the full set of correlated equilibria. In two-player games, the only attainable correlated equilibria are randomizations over pure strategy Nash equilibria. More generally, cyclic monotonicity is necessary and sufficient for a correlated equilibrium distribution to be attainable.

Correlated Equilibrium in Two-Person Zero-Sum Games

but any convex combination of pairs of optimal strategies such that p(2, 2) = 0 satisfies p(1, 1) > 2 (with the obvious notation p(i, j) for the induced probability of row i and column j). However, the following is easily checked. Let I and J be the sets of pure strategies of player 1 and player 2 respectively in a zero-sum game G with value v. Then p = [p(i, i)I(, J) IXJ is a correlated equilibrium distribution for G if and only if for every E J such that p(jo) > 0, the conditional probability of player 2 over player l's actions given jo' [p (iIjo)]I , is an optimal strategy for player 1, yielding exactly v against jo and similarly for [p(jlio)]jEj, io E , p(io) > 0. Hence as conjectured by R. Aumann, if a pure strategy pair occurs with positive probability in a correlated equilibrium, then it occurs with positive probability in a pair of optimal strategies. Also, if one of the players has a unique optimal strategy, then every correlated equilibrium distribution concentrates on a pair of optimal strategies.