  We explore a version of the minimax theorem for two-person win-lose games
with infinitely many pure strategies. In the countable case, we give a
combinatorial condition on the game which implies the minimax property. In the
general case, we prove that a game satisfies the minimax property along with
all its subgames if and only if none of its subgames is isomorphic to the
"larger number game." This generalizes a recent theorem of Hanneke, Livni and
Moran. We also propose several applications of our results outside of game
theory.
