= Solution
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 when
$$
y\notin pA
\quad\text{or}\quad
(x,y)\in A.
$$
Player 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 forces
$$
(f(y),y)\in A,
$$
so $f:pA\to\mathbb R$ is the required <uniformization of a binary relation>. This is the direct game proof of <uniformization from determinacy>.
Back to article page
