Rank-1 Bi-matrix Games: A Homeomorphism and a Polynomial Time Algorithm

TL;DR

This paper introduces a polynomial time algorithm for exactly computing Nash equilibria in rank-1 bimatrix games by establishing a homeomorphism between the game space and its equilibrium correspondence, and extends the approach to fixed rank games.

Contribution

It provides the first polynomial time algorithm for rank-1 games, constructs a homeomorphism for the equilibrium correspondence, and extends the method to fixed rank games.

Findings

Polynomial time algorithm for rank-1 bimatrix games

Homeomorphism between game space and equilibrium set

Enumeration method for all Nash equilibria

Abstract

Given a rank-1 bimatrix game (A,B), i.e., where rank(A+B)=1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed in Kannan and Theobald (SODA 2007) and Theobald (2007). In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. This technique also proves the existence, oddness and the index theorem of Nash equilibria in a bimatrix game. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on $[0,1]^k$ for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions.