On the Approximate Purification of Mixed Strategies in Games with Infinite Action Sets

TL;DR

This paper proves that in games with uncountably infinite action sets, any mixed strategy can be approximated by a pure strategy, leading to the existence of pure strategy approximate Nash equilibria under weak assumptions.

Contribution

It introduces a method to approximate mixed strategies with pure strategies in infinite action set games, extending purification results under weak conditions.

Findings

Any mixed strategy has an approximately equivalent pure strategy.

Existence of pure strategy approximate Nash equilibria under weak assumptions.

Pure strategies can be finite-action strategies.

Abstract

We consider a game in which the action set of each player is uncountable, and show that, from weak assumptions on the common prior, any mixed strategy has an approximately equivalent pure strategy. The assumption of this result can be further weakened if we consider the purification of a Nash equilibrium. Combined with the existence theorem for a Nash equilibrium, we derive an existence theorem for a pure strategy approximated Nash equilibrium under sufficiently weak assumptions. All of the pure strategies we derive in this paper can take a finite number of possible actions.