All Games Have Equilibria
Provides a unified existence result for Nash equilibria in all infinite games, eliminating the need for technical preconditions that previously limited equilibrium analysis.
This paper proves that every game with a nonempty set of players, nonempty action sets, and bounded utility functions has a Nash equilibrium in finitely additive mixed strategies, resolving a long-standing open problem in game theory.
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.