 Solution

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.