Enumerating the Nash equilibria of rank 1-games

TL;DR

This paper studies the enumeration of Nash equilibria in rank 1 bimatrix games, introducing a parametric algorithm that can find all equilibria, highlighting limitations of existing path-following methods.

Contribution

It presents a novel parametric simplex-type algorithm for enumerating all Nash equilibria in non-degenerate rank 1 games, and shows not all equilibria are reachable by Lemke-Howson paths.

Findings

Not all equilibria are reachable by Lemke-Howson paths in rank 1 games.

The proposed algorithm can enumerate all Nash equilibria in such games.

The paper characterizes the structure of equilibria in rank 1 bimatrix games.

Abstract

A bimatrix game $(A,B)$ is called a game of rank $k$ if the rank of the matrix $A+B$ is at most $k$. We consider the problem of enumerating the Nash equilibria in (non-degenerate) games of rank 1. In particular, we show that even for games of rank 1 not all equilibria can be reached by a Lemke-Howson path and present a parametric simplex-type algorithm for enumerating all Nash equilibria of a non-degenerate game of rank 1.