Determinacy

Definition

Let $A$ be a nonempty set and let $X \subseteq A^{\N}$ be a set of infinite sequences of elements of $A$. The game $G(A, X)$ is played by two players, I and II. Player I chooses $a_0 \in A$, then player II chooses $a_1 \in A$, then player I chooses $a_2 \in A$, and so on forever. Player I wins the play $x = (a_0, a_1, a_2, \dots)$ if $x \in X$, and player II wins it otherwise.

1. A strategy for player I is a map $\sigma$ from the finite sequences of elements of $A$ of even length to $A$. A strategy for player II is a map $\tau$ from the finite sequences of odd length to $A$.
2. A play follows $\sigma$ if $a_{2n} = \sigma(a_0, \dots, a_{2n-1})$ for all $n \ge 0$, and similarly for $\tau$ and the odd positions.
3. A strategy for a player is winning if that player wins every play that follows it.

The game $G(A, X)$, and the set $X$, is determined if player I or player II has a winning strategy.