All Games Have Equilibria

2026-07-16

AI Summary

This work addresses the long-standing challenge of Nash equilibrium existence in infinite games, which traditionally relies on strong assumptions and lacks a unified framework. By abandoning countable additivity and introducing finitely additive mixed strategies, the paper establishes—under the most general setting—that a Nash equilibrium exists for any nonempty set of players and any bounded utility functions. This result unifies existing equilibrium existence theorems and demonstrates that the equilibrium correspondence is nonempty, compact-valued, and upper hemicontinuous. The proof synthesizes tools from finitely additive measure theory, analysis of upper hemicontinuous correspondences, and finite approximation techniques, thereby enabling direct equilibrium analysis of infinite games previously considered intractable.

Abstract

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same is true for equilibria obtained as limits of finite approximations. Techniques developed in this paper show that infinite games long treated as intractable become amenable to direct equilibrium analysis.

Problem

• Nash equilibrium
• infinite games
• mixed strategies
• equilibrium existence
• game theory

Innovation

• finitely additive strategies
• Nash equilibrium existence
• infinite games
• equilibrium correspondence
• mixed strategies