# Fast Algorithms for Rank-1 Bimatrix Games

## TL;DR

This paper introduces efficient algorithms for analyzing rank-1 bimatrix games, demonstrating polynomial-time solutions for some equilibria and revealing the complex structure and potential exponential number of equilibria.

## Contribution

It provides the first polynomial-time algorithm for finding an equilibrium in rank-1 bimatrix games and characterizes their equilibrium set structure.

## Key findings

- One equilibrium can be found in polynomial time.
- All equilibria can be traced via a piecewise linear path.
- Number of equilibria may be exponential.

## Abstract

The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. This paper comprehensively analyzes games of rank one, and shows the following: (1) For a game of rank r, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank r-1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear path. In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank. It is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component.