# Is this determinacy principle consistent?

Let $$\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$$ be the following principle ("determinacy for simple open length-$$\omega_1$$ games"):

If $$\kappa$$ is any ordinal and $$X\subseteq \kappa^{<\omega_1}$$ is in $$L(\mathbb{R})$$, then the game of length $$\omega_1$$ on $$\kappa$$ with payoff set $$[X]:=\{f\in\kappa^{\omega_1}: \exists\alpha\in\omega_1(f\upharpoonright\alpha\not\in X)\}$$ is determined in $$V$$.

My question is whether $$\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$$ is consistent relative to large cardinals. I recall seeing a while ago a proof that the answer is yes, relative to something like a proper class of Woodins, but I can't track it down. (I'm pretty sure I can prove that this principle is equivalent to its clopen game analogue, but that's no easier to prove consistent as far as I can tell.) Neeman's work on long games is the most relevant I can find, but doesn't seem to directly address this unless I'm missing something.

• I'm confused. Doesn't your principle imply $\text{AD}_{\omega_1}$ for games of length $\omega$ using countable ordinal moves? If so, it is inconsistent. That is, for $\kappa=\omega_1$, I seem to be able to mimic games on countable ordinals of length $\omega$ in your context. But there is a nondetermined such game. May 7 at 1:17
• @JoelDavidHamkins Doesn't such a game have payoff set not in $L(\mathbb{R})$? May 7 at 1:19
• $\text{AD}_{\omega_1}$ is inconsistent with ZF, which holds in $L(\mathbb{R})$. May 7 at 1:20
• Ah, you mean the strategy might not be in $L(\mathbb{R})$. I see. May 7 at 1:20
• @JoelDavidHamkins Yes, the game's payoff set needs to be in $L(\mathbb{R})$, but the strategy just has to live in $V$. May 7 at 1:21