Finding Nash Equilibria of Two-Player Games

TL;DR

This paper explains algorithms for finding Nash equilibria in two-player bimatrix games, focusing on the Lemke-Howson algorithm and the structure of equilibrium sets using geometric and combinatorial methods.

Contribution

It provides a detailed exposition of algorithms for computing one or all Nash equilibria, including new insights into the path-following and geometric properties of these algorithms.

Findings

Lemke-Howson algorithm finds one equilibrium using labeled best-response polytopes.

The path of the algorithm has a direction with endpoints of opposite index.

All equilibria can be characterized via complementary pairs of faces of best-response polytopes.

Abstract

This paper is an exposition of algorithms for finding one or all equilibria of a bimatrix game (a two-player game in strategic form) in the style of a chapter in a graduate textbook. Using labeled "best-response polytopes", we present the Lemke-Howson algorithm that finds one equilibrium. We show that the path followed by this algorithm has a direction, and that the endpoints of the path have opposite index, in a canonical way using determinants. For reference, we prove that a number of notions of nondegeneracy of a bimatrix game are equivalent. The computation of all equilibria of a general bimatrix game, via a description of the maximal Nash subsets of the game, is canonically described using "complementary pairs" of faces of the best-response polytopes.