Correlated equilibrium

Correlated equilibrium
Name	Correlated equilibrium
Introduced	1974
Inventor	Robert Aumann
Field	Game theory
Notable work	"Subjectivity and Correlated Equilibrium"

Correlated equilibrium Correlated equilibrium is a solution concept in non-cooperative game theory that generalizes Nash equilibrium by allowing an external signal to correlate players' actions. Originating in the work of Robert Aumann and interacting with research from John Nash, Lloyd Shapley, John von Neumann, and Oskar Morgenstern, it has influenced studies at RAND Corporation, Bell Labs, Princeton University, and Hebrew University of Jerusalem.

Definition

A correlated equilibrium describes a probability distribution over the set of action profiles such that, given a private recommendation drawn from that distribution, no player can gain by unilaterally deviating; key formalizations relate to ideas in Aumann's correlated equilibrium, Nash equilibrium, Minimax theorem, Bayesian game, and Common knowledge. The definition uses expected payoffs from actions in games studied at Game Theory Society meetings and in texts by John Harsanyi, Kenneth Arrow, Robert Wilson, Thomas Schelling, and John Maynard Smith.

Formal properties and existence

Correlated equilibria form a convex polytope in the space of probability distributions over action profiles, a property analyzed in works by Michael E. Dyer, John B. Kruskal, László Lovász, and Martin Shubik. Existence follows from linear inequalities similar to those used in proofs at Carnegie Mellon University and in lectures by Drew Fudenberg and Eric Maskin; these inequalities ensure nonempty sets in finite games like those studied by Reinhard Selten and Lloyd Shapley. The set contains all Nash equilibria as extreme points in certain games, a relationship explored in papers by Robert Aumann and Mike Osborne published in journals associated with American Economic Association and Econometrica.

Examples and applications

Simple examples include coordination games and the Battle of the Sexes where a correlating device can improve expected payoffs relative to Nash predictions; similar analyses appear alongside studies of the Prisoner's Dilemma, Stag Hunt, Chicken (game), and Matching Pennies. Applications span mechanism design in contexts like Vickrey auction, Myerson auction, Federal Trade Commission policy discussions, and traffic routing problems related to work at MIT, Stanford University, and California Institute of Technology. Correlated strategies have been proposed in models of repeated interaction such as folk theorem analyses by Robert Aumann, David K. Levine, Dilip Abreu, and in experiments at University of Chicago, London School of Economics, and Yale University.

Computation and algorithms

Computing correlated equilibria reduces to solving linear programs, a fact leveraged by algorithms developed at IBM Research, Microsoft Research, and in textbooks by Vijay Vazirani and Christos Papadimitriou. Polynomial-time procedures exploit separable structures studied in combinatorial optimization by Jack Edmonds and Richard Karp, while sampling-based methods draw on techniques from Leslie Valiant and R. M. Karp. For large games, iterative learning algorithms such as regret-matching and no-regret dynamics were proposed by Sorin Hart, Andreu Mas-Colell, and Michael Kearns, and implemented in systems at Google and Facebook for multi-agent coordination.

Relationship to other solution concepts

Correlated equilibrium relates to Nash equilibrium, Bayesian equilibrium, and Stackelberg strategies; comparisons feature in foundational papers by John Nash, Harsanyi, Mailath and Samuelson, and Fudenberg and Tirole. It sits between cooperative and non-cooperative frameworks discussed by Oliver Williamson and Elinor Ostrom, and its welfare properties connect to results in social choice theory by Kenneth Arrow and Amartya Sen. In repeated games, correlating devices can implement outcomes characterized in the Folk theorem literature by Mertens and Rubinstein.

Extensions and generalizations

Generalizations include communication equilibria, mediator models, and correlated equilibria in stochastic and extensive-form games studied by Ariel Rubinstein, R. Selten, Martin Osborne, and Alvin Roth. Quantum correlated equilibrium models extend the concept into quantum game theory explored at MIT, University of Oxford, and Perimeter Institute by researchers engaging with John Preskill and Anton Zeilinger. Networked and decentralized correlating mechanisms appear in control theory and multi-agent reinforcement learning work at Carnegie Mellon University, DeepMind, and OpenAI.

Category:Game theory