A Stable-Set Bound and Maximal Numbers of Nash Equilibria in Bimatrix Games

TL;DR

This paper proves that the maximum number of Nash equilibria in a 5x5 non-degenerate bimatrix game is at most 31, resolving an open conjecture for this case by linking game theory to polytope combinatorics.

Contribution

It introduces a novel combinatorial obstruction based on equilibrium indices and verifies the bound using classification of 5-dimensional polytopes, solving an open problem.

Findings

Maximum Nash equilibria in 5x5 games is 31.

The bound is verified through polytope classification.

The approach links game theory with polytope combinatorics.

Abstract

Quint and Shubik (1997) conjectured that a non-degenerate n-by-n game has at most 2^n-1 Nash equilibria in mixed strategies. The conjecture is true for n at most 4 but false for n=6 or larger. We answer it positively for the remaining case n=5, which had been open since 1999. The problem can be translated to a combinatorial question about the vertices of a pair of simple n-polytopes with 2n facets. We introduce a novel obstruction based on the index of an equilibrium, which states that equilibrium vertices belong to two equal-sized disjoint stable sets of the graph of the polytope. This bound is verified directly using the known classification of the 159,375 combinatorial types of dual neighborly polytopes in dimension 5 with 10 facets. Non-neighborly polytopes are analyzed with additional combinatorial techniques where the bound is used for their disjoint facets.