Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 158 / 4 / i / c / Solution

Let $S$ be the causal rank-raising map on well-order codes obtained by tagging a coded relation and adjoining a new least element. Player II follows the online rule $y=S(x)$. If $x\in \mathrm{WF}$, thenso II wins. If $x\notin \mathrm{WF}$, the tagged recoding ensures $y\ne x$, which is again precisely II's winning condition. Therefore