Infinite game of perfect information

For $A\subseteq M^{\omega }$, the game $G(A)$ has two players alternately choose elements of the move set $M$, producing $x\in M^{\omega }$. Player I wins exactly when $x\in A$. Both players see the whole finite position before making each move.

A strategy assigns one legal move to every finite position at which its player is to move. A play follows the strategy when every move of that player is the assigned move.

A strategy is winning when every play that follows it is won by its player.

An infinite game of perfect information is determined when one of its two players has a winning strategy in an infinite game.

The axiom of determinacy states that every game $G(A)$ with $A\subseteq {\omega }^{\omega }$ is determined. Its version for a move set $M$ is denoted ${\mathsf{AD}}_M$.

Projective determinacy states that every infinite game of perfect information whose payoff is a projective set is determined.

A quasistrategy assigns a nonempty set of allowed moves to every position at which its player moves. It is winning when every play consistent with those allowed moves is won by that player.

An infinite game of perfect information is quasidetermined when one of the players has a winning quasistrategy. Choosing one allowed move at every relevant position refines a winning quasistrategy to a winning strategy in an infinite game whenever the required restricted choice principle holds.

For every move set $M$, the statement that every quasidetermined subset of $M^{\omega }$ is determined is equivalent in ZF to ${\mathsf{AC}}_{M^{<\omega }}(M)$. The forward direction encodes an arbitrary family of nonempty subsets of $M$ into a quasidetermined game; the reverse direction chooses one move from each value of a winning quasistrategy.