Parameterized games of perfect information

Considered are perfect information games with a borel measurable payoff function that is parameterized by points of a polish space. The existence domain of such a parameterized game is the set of parameters for which the game admits a subgame perfect equilibrium. We show that the existence domain of a parameterized stopping game is a borel set. In general, however, the existence domain of a parameterized game need not be borel, or even an analytic or co-analytic set. We show that the family of existence domains coincides with the family of game projections of borel sets. Consequently, we obtain an upper bound on the set-theoretic complexity of the existence domains, and show that the bound is tight.