A Stable-Set Bound and Maximal Numbers of Nash Equilibria in Bimatrix Games
Constantin Ickstadt, Thorsten Theobald, Bernhard von Stengel
TL;DR
The paper resolves the open $5\times5$ case of the Quint-Shubik conjecture by showing a non-degenerate $5\times5$ bimatrix game has at most $31$ Nash equilibria. It translates equilibria into vertex configurations of two simple best-response polytopes and introduces an equilibrium index-based stable-set bound, strengthened by a facet-wise version. For dual-neighborly polytopes with 10 facets, exhaustive computer verification using a comprehensive polytope catalog confirms the bound, while non-neighborly cases are handled via obstruction-vertex analysis and ILP computations on facet structures, including the special 4-cube and semi-cube configurations. The result closes a long-standing open case and demonstrates a powerful polyhedral approach to enumerating Nash equilibria in bimatrix games, with implications for algorithmic strategies and complexity in equilibrium computation.
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.