Existence of equilibria in countable games: an algebraic approach

TL;DR

This paper proves the existence of equilibria in countable and certain uncountable games using algebraic methods, allowing finitely additive strategies and providing conditions for equilibrium existence.

Contribution

It introduces an algebraic approach to establish equilibrium existence in countable games with group-structured strategies, including finitely additive strategies and a selection method independent of the measure choice.

Findings

Existence of equilibria in countable games with group-structured strategies.

Wald's game admits an equilibrium when finitely additive strategies are allowed.

Equilibria exist for extensions of matching-pennies and rock-scissors-paper.

Abstract

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Wald's pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are rarely applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group and the payoffs are functions of the group operation. In order to obtain the existence of equilibria, finitely additive mixed strategies have to be allowed. This creates a problem of selection of a product measure of mixed strategies. We propose a family of such selections and prove existence of an equilibrium that does not depend on the selection. As a byproduct we show that if finitely additive mixed strategies are allowed, then Wald's game admits an equilibrium. We also prove existence of equilibria for nontrivial extensions of matching-pennies and rock-scissors-paper. Finally we extend the main results to uncountable games.