Resumen de Nash equilibria in normal form games

Hannu Salonen

• We show that there exists a Nash equilibrium in mixed strategies in all normal form games such that pure strategy sets are nonempty compact metric spaces and utility functions are continuous. The player set can be arbitrary nonempty set. The utility functions of the players are defined on the set of pure strategy profiles, the product space of individul players� pure strategy sets.

  This product space is equipped with the product topology and the product sigma-algebra, so the product space is a compact Hausdorff space, and the continuous utility functions are also measurable. Continuity and compactness ensure that each utility function depends only on countably many coordinates.

  This suggests that the existence of a Nash equilibrium in the general case follows actually from the existence of equilibrium in games with countably many players. This indeed is the case. In the proof of the main result the player set is equipped with a well-ordering, so the Axiom of Choice is assumed. However, both the Tychonoff theorem and the product measure existence theorem (or the Kolmogorov�s existence theorem) already depend on the Axiom of Choice. The former ensures that the set of strategy profiles is a nonempty compact space, and the latter ensures that players can �calculate� expected utilities associated with mixed strategy profiles. Therefore we need the Axiom of Choice (at least implicitly) already if we want to talk about non-trivial normal form games with arbitrary player sets.