Solvability of Two-Player Game Forms with Infinite Sets of Strategies

A game form is N-solvable for a class of payoff functions. If for every pair of payoff functions of that class, the associated game in strategic form has a Nash equilibrium. A finite game form is N-solvable (for the universal class of preferences) if and only if it is tight—that is if its α-effectivity function and its β-effectivity function are equal. We extend this result to various models of two-player game forms with infinite sets of strategies and/or alternatives. This is done by an appropriate definition of tightness relative to the underlying structure (topology, Boolean algebra, σ-algebra). We apply the current results along with well-known results on the determinacy of games with perfect information to infinitely repeated game forms. We prove that a repeated tight game form is tight on Bore1 sets.