Pure-strategy Nash equilibria in nonatomic games with infinite-dimensional action spaces∗

This paper studies the existence of pure-strategy Nash equilibria for nonatomic games where players take actions in an infinite-dimensional Banach space. For any infinitedimensional Banach space, we construct a nonatomic game with actions in this space which has no pure-strategy Nash equilibria, provided that the player space is modeled by the usual Lebesgue unit interval. We also show that if the player space is modeled by a saturated probability space, every nonatomic game has a pure-strategy Nash equilibrium. On the other hand, if every game with a fixed nonatomic player space and a fixed infinitedimensional action space has a pure-strategy Nash equilibrium, then the underlying player space must be saturated. (110 words) JEL classification: C62, C72.