Take
$$
\boxed{X=M^{<\omega},\qquad Y=M.}
$$
Thus $\Phi_M$ 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>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 158 / 1 / iii / Solution

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.

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 Incoming links: Choice characterization of quasideterminacy