 Solution

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