Researchers prove all games have Nash equilibria with finitely additive strategies

A trio of economists and mathematicians—M. Ali Khan, Arthur Paul Pedersen, and Maxwell B. Stinchcombe—have published a landmark paper titled "All Games Have Equilibria" on arXiv. They show that every game, regardless of the number of players or the size of action sets, admits a Nash equilibrium when players are allowed to use finitely additive mixed strategies (as opposed to the standard countably additive probability measures). This eliminates a long-standing patchwork of technical preconditions and counterexamples that plagued infinite game theory. The result applies to any nonempty set of players with bounded von Neumann-Morgenstern utility functions. Moreover, the equilibrium correspondence is proven to be nonempty, compact-valued, and upper hemicontinuous, and equilibria emerge as limits of finite approximations, making infinite games directly amenable to equilibrium analysis.

The implications span theoretical economics, computer science game theory, functional analysis, and optimization. By revising the foundational model of mixed strategies, the paper unlocks analysis of games previously considered intractable—such as those with infinite action spaces or continuum of players. For AI and multi-agent systems, this means a firmer theoretical basis for modeling strategic interactions where agents face infinite choices or continuous parameter spaces. The work influences algorithmic game theory, mechanism design, and reinforcement learning in environments with continuous action spaces. As arXiv metadata notes cross-disciplinary relevance (cs.GT, math.FA, math.OC, math.PR), the paper bridges pure mathematics and applied AI, promising new tools for equilibrium computation and strategic reasoning in complex systems.