Characterization and computation of equilibria in infinite games

Abstract

Broadly, we study continuous games (those with continuous strategy spaces and utility functions) with a view towards computation of equilibria. We cover all of the game-theoretic background needed to understand these results in detail. Then we present new work, which can be divided into three parts. First, it is known that arbitrary continuous games may have arbitrarily complicated equilibria, so we investigate some properties of games with polynomial utility functions and a class of games with polynomial-like utility functions called separable games. We prove new bounds on the complexity of equilibria of separable games in terms of the complexity of the utility functions. In order to measure this complexity we propose a new definition for the rank of a continuous game; when applied to the case of finite games this improves on the results known in that setting. Furthermore, we prove a characterization theorem showing that several conditions which are necessary for a game to possess a finite-dimensional representation each define the class of separable games precisely, providing evidence that separable games are the natural class of continuous games in which to study computation. The characterization theorem also provides a natural connection between separability and the notion of the rank of a game. Second, we apply this theory to give an algorithm for computing e-Nash equilibria of two-player separable games with continuous strategy spaces. While a direct comparison to corresponding algorithms for finite games is not possible, the asymptotic running time in the complexity of the game grows slower for our algorithm than for any known algorithm for finite games.

(cont.) Nonetheless, as in finite games, computing e-Nash equilibria still appears to be difficult for infinite games. Third, we consider computing approximate correlated equilibria in polynomial games. To do so, we first prove several new characterizations of correlated equilibria in continuous games which may be of independent interest. Then we introduce three algorithms for approximating correlated equilibria of polynomial games arbitrarily accurately. These include two discretization algorithms for computing a sample correlated equilibrium: a naive linear programming approach called static discretization which operates without regard to the structure of the game, and a semidefinite programming approach called adaptive discretization which exploits the structure of the game to achieve far better performance in practice. The third algorithm consists of a nested sequence of semidefinite programs converging to a description of the entire set of correlated equilibria.