The generalization of this for longer games is also known: Neeman and Trang--Woodin's results say that for any $2\leq\alpha<\omega_1$, the analytic determinacy of games of length $\omega^{\alpha}$ is equivalent to the sharp for an inner model with $\omega^{-1+\alpha}$ many Woodin cardinals.

We show that such equivalence also holds for games of variable countable length, in which the length of a play is still countable but determined by the moves.

A game introduced in the talk can be viewed as the "diagonalization" of all games of fixed countable length and its analytic determinacy turns out to be equivalent to the sharp for an inner model with ?$\lambda$ many Woodin cardinals, where ?$\lambda$ is the order type of Woodin cardinals below $\lambda$?.

This is an ongoing joint work with Sandra Müller.