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

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.

Consider the following game on the real numbers. On their first moves, Player I plays $y\in \mathbb{R}$ and Player II replies with $x\in \mathbb{R}$; later moves are ignored. Declare Player II the winner whenPlayer I cannot have a winning strategy in an infinite game: its first move is some fixed $y$, and if $y\notin pA$ then II wins automatically, while if $y\in pA$ then II can reply with an $x$ satisfying $(x,y)\in A$.

The axiom of determinacy for games on $\mathbb{R}$ therefore gives Player II a winning strategy $\tau$. For every $y\in pA$, define $f(y)$ to be II's first response to the move $y$. The winning condition forcesso $f:pA\to \mathbb{R}$ is the required uniformization of a binary relation. This is the direct game proof of uniformization from determinacy.

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.