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 there can be a superexponential separation between the number of extreme Nash and extreme correlated equilibria. 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 joint moments, in marked contrast to the set of Nash equilibria which is always expressible in terms of finitely many moments.
Noah D. Stein, A. Ozdaglar, P. Parrilo