Constrained Correlated Equilibria

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.