Noah D. Stein

Separable and low-rank continuous games

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game.

Correlated equilibria in continuous games: Characterization and computation

We present several new characterizations of correlated equilibria in games with continuous utility functions. These have the advantage of being more computationally and analytically tractable than the standard definition in terms of departure functions. We use these characterizations to construct effective algorithms for approximating a single correlated equilibrium or the entire set of correlated equilibria of a game with polynomial utility functions.

Separable and Low-Rank Continuous Games

Separable games are a structured subclass of continuous games whose payoffs take a sum-of-products form; the zero-sum case has been studied in earlier work. Included in this subclass are all finite games and polynomial games. Separable games provide a unified framework for analyzing and generating results about the structural properties of low rank games. This work extends previous results on low-rank finite games by allowing for multiple players and a broader class of payoff functions. We also discuss computation of exact and approximate equilibria in separable games. We tie these results together with alternative characterizations of separability which show that separable games are the largest class of continuous games to which low-rank arguments apply.

Characterization and computation of correlated equilibria in infinite games

Motivated by work on computing Nash equilibria in two-player zero-sum games with polynomial payoffs by semidefinite programming and in arbitrary polynomial-like games by discretization techniques, we consider the problems of characterizing and computing correlated equilibria in games with infinite strategy sets. We prove several characterizations of correlated equilibria in continuous games which are more analytically tractable than the standard definition and may be of independent interest. Then we use these to construct algorithms for approximating correlated equilibria of polynomial games with arbitrary accuracy, including a sequence of semidefinite programming relaxation algorithms and discretization algorithms.

Computing correlated equilibria of polynomial games via adaptive discretization

We construct a family of iterative discretization algorithms for computing sequences of finitely-supported ¿-correlated equilibria of n-player games with polynomial utility functions. These algorithms can be implemented efficiently using semidefinite programming and sum of squares techniques. They converge in the sense that they drive ¿ to zero in the limit as points are added to the discretization. We show how a natural discretization scheme proposed previously can be viewed as a limiting case of this new family of algorithms. Finally we provide a counterexample showing that this limiting case is singular, i.e., ¿ need not converge to zero.

Structure of extreme correlated equilibria: a zero-sum example and its implications

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.