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We exhibit the rich structure of the set of correlated equilibria byanalyzing the simplest of polynomial games: the mixed extension of matchingpennies. We show that while the correlated equilibrium set is convex andcompact, the structure of its extreme points can be quite complicated. Infinite games the ratio of extreme correlated to extreme Nash equilibria can begreater than exponential in the size of the strategy spaces. In polynomialgames there can exist extreme correlated equilibria which are not finitelysupported; we construct a large family of examples using techniques fromergodic theory. We show that in general the set of correlated equilibriumdistributions of a polynomial game cannot be described by conditions onfinitely many moments (means, covariances, etc.), in marked contrast to the setof Nash equilibria which is always expressible in terms of finitely manymoments.

Structure of Extreme Correlated Equilibria: a Zero-Sum Example and its Implications