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

TL;DR

This paper explores the complex structure of correlated equilibria in simple polynomial games, revealing that their extreme points can be highly intricate and not finitely describable, contrasting with Nash equilibria.

Contribution

It demonstrates the rich and complex structure of extreme correlated equilibria in polynomial games, including examples with infinitely supported equilibria and non-finitely describable distributions.

Findings

Extreme correlated equilibria can be highly complex and not finitely supported.

The ratio of extreme correlated to Nash equilibria can grow exponentially.

Correlated equilibrium distributions in polynomial games cannot be characterized by finitely many moments.

Abstract

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 the ratio of extreme correlated to extreme Nash equilibria can be greater than exponential in the size of the strategy spaces. 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 moments (means, covariances, etc.), in marked contrast to the set of Nash equilibria which is always expressible in terms of finitely many moments.