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

2024-11-19

arXiv.org

AI Summary

This paper resolves the long-standing open case of the Quint–Shubik (1997) conjecture on the maximum number of mixed-strategy Nash equilibria in nondegenerate (n imes n) bimatrix games for (n = 5). Methodologically, it introduces a novel graph-theoretic obstruction—rooted in equilibrium indices—that mandates all equilibrium vertices lie in one of two disjoint, equal-sized stable sets of the polyhedral graph. Leveraging this, the authors systematically classify and analyze all 159,375 combinatorial types of 5-dimensional dual-adjacent polyhedra. Their analysis yields the first rigorous proof that the equilibrium upper bound for (n = 5) is (2^5 - 1 = 31). The approach integrates combinatorial polytope theory, stable-set characterizations, geometric representations of Nash equilibria, and non-adjacency decomposition techniques. The results fully confirm the Quint–Shubik conjecture and establish a general stable-set-based framework for deriving equilibrium upper bounds—providing a new paradigm for analyzing equilibrium structure in higher-dimensional games.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Resolving Quint and Shubik's conjecture about maximum Nash equilibria in 5×5 games
• Developing a novel obstruction method using equilibrium index and stable sets
• Analyzing combinatorial properties of dual neighborly polytopes with 10 facets

Innovation

• Index-based obstruction for equilibrium vertices
• Stable set bound on polytope graph
• Combinatorial classification of dual neighborly polytopes