For $\varphi$ a first-order sentence in the language of set theory and $\kappa$ an ordinal, let $G_\varphi^{\kappa}$ be the game of length $\kappa$ in which players $1$ and $2$ alternately play elements of $\{0,1\}$ (with $1$ going first at all limit stages) with payoff set $$\{r\in 2^\kappa: \exists\alpha<\kappa(L_\kappa[r\upharpoonright\alpha]\models\varphi)\}.$$ Relative to large cardinal hypotheses, the $G_\varphi^{\omega_1}$s are consistently determined (this follows from work of Steel if I have things right). However, it's not obvious to me what happens beyond $\omega_1$:
Question: Is $\mathsf{ZFC}$ + "Every $G_\varphi^{\omega_2}$ is determined" consistent relative to large cardinals?
I suspect the answer is negative; however, it's not even clear to me that $\mathsf{ZFC}$ proves "There are $\varphi,\kappa$ such that $G_\varphi^\kappa$ is not determined."
