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

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.