Constrained Correlated Equilibria

TL;DR

This paper defines and analyzes constrained correlated equilibria in finite non-cooperative games, exploring their existence, properties, and relation to classical correlated equilibria, with theoretical results supported by numerical examples.

Contribution

It introduces the concept of constrained correlated equilibrium, characterizes their existence conditions, and shows their relation to classical correlated equilibria in extended games.

Findings

Canonical correlation devices suffice for characterization.

Constrained equilibria of the extended game do not add new distributions.

Such equilibria may fall outside the classical correlated equilibrium polytope.

Abstract

This paper introduces constrained correlated equilibrium, a solution concept combining correlation and coupled constraints in finite non-cooperative games. In the general case of an arbitrary correlation device and coupled constraints in the extended game, we study the conditions for equilibrium. In the particular case of constraints induced by a feasible set of probability distributions over action profiles, we first show that canonical correlation devices are sufficient to characterize the set of constrained correlated equilibrium distributions and provide conditions of their existence. Second, it is shown that constrained correlated equilibria of the mixed extension of the game do not lead to additional equilibrium distributions. Third, we show that the constrained correlated equilibrium distributions may not belong to the polytope of correlated equilibrium distributions. Finally, we illustrate these results through numerical examples.