If $G\subseteq\omega^{<\omega}$ is a computable clopen game, then $G$ has a winning strategy which is hyperarithmetic $(\Delta^1_1)$, by an inductive ranking process. The key observation here is that the length of this induction is bounded above by the length of the Kleene-Brouwer ordering $G_{KB}$, which is a computable ordinal and hence $<\omega_1^{CK}$, and that each successive stage of the induction can be achieved by one application of the jump operator, so there is a winning strategy with complexity at most $0^{(\vert G_{KB}\vert)}$.

(An annoying subtlety here is that the theory $\Delta^1_1-CA_0$, which amounts to closure under hyperarithmeticity, does *not* prove determinacy of clopen games, since there are games which are not actually clopen but have no hyperarithmetic witnesses to their ill-foundedness.)

My question is whether a version of this result is also true for open games. Specifically, let $T\subseteq\omega^{<\omega}$ be an open game in which the "Open" player (i.e., the player trying to fall off the tree) has a winning strategy; do they necessarily have a winning strategy hyperarithmetic in $T$?

I'm asking this question because I was looking through my notes from a previous class, and I ran across the assertion that "a similar ranking argument" shows that the answer is 'yes;' however, I can't reconstruct this argument, and I'm wondering whether I (or the lecturer) was simply incorrect; or whether there's a basic argument I'm not seeing.