Quasidetermined infinite game

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.