= 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 $\lVert x\rVert<\lVert y\rVert$. 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
$$
\boxed{\text{Player II}.}
$$
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