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arXiv · 2411.12385

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

Constantin Ickstadt·Thorsten Theobald·Bernhard von Stengel
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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.

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Constantin Ickstadt, Thorsten Theobald, Bernhard von Stengel. 2024-11-19. A Stable-Set Bound and Maximal Numbers of Nash Equilibria in Bimatrix Games. https://doi.org/10.1287/moor.2024.0809

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