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TakeThus ${\Phi }_M$ is equivalent in ZF to ${\mathsf{AC}}_{M^{<\omega }}(M)$. Indeed, this choice principle chooses one move from every nonempty value of a winning quasistrategy, turning it into a winning strategy in an infinite game. The converse encodes an arbitrary $M^{<\omega }$-indexed family of nonempty subsets of $M$ into a quasidetermined game. This is the choice characterization of quasideterminacy.

For a payoff set $A\subseteq M^{\omega }$, a quasistrategy for a player assigns a nonempty subset of $M$ to every finite position at which that player moves. A play is consistent with it when each of that player's moves belongs to the assigned set. The set $A$, or equivalently the infinite game of perfect information $G(A)$, is quasidetermined when one player has a quasistrategy under which every consistent play 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.