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

Player I cannot have a winning strategy. This is the Solovay rank-comparison game: any proposed strategy for I can be challenged by a well-order code $y$ whose rank lies beyond the bound obtainable from that strategy, so that either I produces $x\notin \mathrm{WF}$ or produces $x\in \mathrm{WF}$ with $\Vert x\Vert <\Vert y\Vert$. In either case Player II wins.

The axiom of determinacy says that the game is determined. Since Player I has no winning strategy, the winner is therefore

Player II can force both parts of II's winning condition. First choose one coordinate $n$ reserved outside the coding of a fixed infinite descending chain and play $y_n>x_n$. On the remaining reserved coordinates, play values that make the relation coded by $y$ contain that descending chain. Thus $y\notin \mathrm{WF}$ while $y_n\nleqq x_n$ for the chosen $n$. Hence

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