Separable and Low-Rank Continuous Games

TL;DR

This paper analyzes the structure of equilibria in nonzero-sum continuous games with a sum-of-products payoff form, establishing conditions for finite support equilibria and providing an efficient algorithm for their computation.

Contribution

It introduces the concept of game rank, characterizes separable games via this rank, and develops a polynomial-time algorithm for approximate equilibria in two-player cases.

Findings

Separable games admit finitely supported Nash equilibria.

The rank of a game characterizes its separability.

An efficient algorithm computes approximate equilibria based on game rank.

Abstract

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.