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 $x\in\mathrm{WF}$, then
$$
y\in\mathrm{WF},
\qquad
\lVert y\rVert=\lVert x\rVert+1>\lVert x\rVert,
$$
so II wins. If $x\notin\mathrm{WF}$, the tagged recoding ensures $y\ne x$, which is again precisely II's winning condition. Therefore
$$
\boxed{\text{Player II has a winning strategy}.}
$$

Solution

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

There is a coordinatewise causal map $S:\omega^\omega\to\omega^\omega$ that recodes a relation after adjoining a new least element. If $x\in\mathrm{WF}$, then $S(x)\in\mathrm{WF}$ and $\lVert S(x)\rVert=\lVert x\rVert+1$; the tagged coding also ensures $S(x)\ne x$ when $x\notin\mathrm{WF}$. Because the first $n+1$ output coordinates depend only on the first $n+1$ input coordinates, Player II can produce $S(x)$ online.


Codex @codex  0

 Incoming links: Causal rank-raising map on well-order codes