A stable-set bound and maximal numbers of Nash equilibria in bimatrix games

Ickstadt, Constantin, Theobald, Thorsten and Von Stengel, Bernhard ORCID: 0000-0002-3488-8322 (2025) A stable-set bound and maximal numbers of Nash equilibria in bimatrix games. Mathematics of Operations Research. ISSN 0364-765X

Text (ssb2) - Accepted Version Available under License Creative Commons Attribution. Download (386kB)

Abstract

Quint and Shubik conjectured that a nondegenerate [Formula: see text] game has at most [Formula: see text] Nash equilibria in mixed strategies. The conjecture is true for [Formula: see text] but false for [Formula: see text]. We answer it positively for the remaining case [Formula: see text], 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 five with 10 facets. Nonneighborly polytopes are analyzed with additional combinatorial techniques where the bound is used for their disjoint facets. Funding: This work was supported by the Deutsche Forschungsgemeinschaft Priority Program “Combinatorial Synergies” [Grant 539847176].

Item Type:	Article
Additional Information:	© 2025 INFORMS
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited:	14 Oct 2025 00:18
Last Modified:	01 Jan 2026 04:01
URI:	http://eprints.lse.ac.uk/id/eprint/129789

Actions (login required)

View Item