Moschovakis goes over various theorems proving the Determinacy of closed/open games, and I am reading into various papers regarding the characterization of ordinals where winning strategies for the players are definable over. My question is as follows:
Let $\gamma$ denote the least ordinal such that $\omega_{\gamma}^{CK} = \gamma.$ In other words, it is $\kappa$ as in Moschovakis' Descriptive Set Theory on page 221, where this inductive definition of $W$ closes off. There are various other characterizations of this ordinal, like it being the first $\Pi_1^{1}$ gap-reflecting ordinal, or the least non-Gandy ordinal, etc, but these are not important. Consider a computable open game relative to a parameter which is definable over $L_{\gamma}$. Where must the strategies for the open player show up?
My attempt? It seems pretty obvious that they will show up by $L_{\gamma^{+}}$, where $\gamma^{+}$ denotes the next admissible ordinal after $\gamma.$ But is this upper bound sharp? Will the winning strategies show up much before then?