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  • $\begingroup$ Thank you for your answer! I think the Kechris-Solovay argument part also works over $\mathsf{ACA}_0 + \mathbf{\Delta}^1_2\text{-}\mathsf{Det}$ since $\mathbf{\Delta}^1_2\text{-}\mathsf{Det}$ bootstraps the background theory to $\Pi^1_1\text{-}\mathsf{CA}_0$, which is enough to prove Shoenfield absoluteness. I guess the role of $\mathsf{ZFC}_N$ in your proof is to guarantee Shoenfield absoluteness, so we should replace it with $\Pi^1_1\text{-}\mathsf{CA}_0$. $\endgroup$ Commented Nov 18 at 17:14
  • $\begingroup$ I am a little lost in the second part of the proof. showing $L[a]$ is a model of $\mathsf{ZFC}$. The proof uses $\bigcup_{n<\omega}\omega\cdot n \text{-}\mathbf{\Pi}^1_1$-Determinacy, but does it follow from $\mathbf{\Delta}^1_2$-Determinacy even over $\mathsf{Z}_2$? $\endgroup$ Commented Nov 18 at 17:26
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    $\begingroup$ Yes, can't we just directly express the payoff in boldface-$\Sigma^1_2$ and in boldface-$\Pi^1_2$ form? (We can in fact do it for any countable length sequence in the difference hierarchy over boldface-$\Pi^1_1$. Given a sequence $\left<A_\alpha\right>_{\alpha<\beta}$, where $\beta$ is an ordinal of our $Z_2$ universe (that is, the sequence is coded by some real), player I wins a run $(x,y)$ iff there is an ordinal $\alpha<\beta$, which is even (or maybe odd, however one defines it...), $(x,y)\in A_\gamma$ for all $\gamma<\alpha$, and $(x,y)\notin A_\alpha$. $\endgroup$ Commented Nov 18 at 20:39