Games of fixed rank: A hierarchy of bimatrix games

TL;DR

This paper introduces a hierarchy of bimatrix games based on the rank of the sum of payoff matrices, revealing complexity insights and providing polynomial-time algorithms for approximate solutions.

Contribution

It establishes a new hierarchy of fixed-rank bimatrix games and offers polynomial-time algorithms for approximate Nash equilibria, advancing understanding of game complexity.

Findings

Nash equilibria can have many connected components even for rank-1 games.

Polynomial-time algorithms exist for finding approximate Nash equilibria in fixed-rank games.

The hierarchy generalizes zero-sum games and highlights complexity challenges.

Abstract

We propose a new hierarchical approach to understand the complexity of the open problem of computing a Nash equilibrium in a bimatrix game. Specifically, we investigate a hierarchy of bimatrix games $(A,B)$ which results from restricting the rank of the matrix $A+B$ to be of fixed rank at most $k$. For every fixed $k$, this class strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games. We show that even for $k=1$ the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we can provide polynomial time algorithms for finding an $\epsilon$-approximation.