For every countable wellordering $\alpha $ greater than $\omega $ , it is shown that clopen determinacy for games of length $\alpha $ with moves in $\mathbb {N}$ is equivalent to determinacy for a class of shorter games, but with more complicated payoff. In particular, it is shown that clopen determinacy for games of length $\omega ^2$ is equivalent to $\sigma $ -projective determinacy for games of length $\omega $ and that clopen determinacy for games of length $\omega ^3$ is equivalent to determinacy for games of length $\omega ^2$ in the smallest $\sigma $ -algebra on $\mathbb {R}$ containing all open sets and closed under the real game quantifier.

We show that for an uncountable κ in a suitable Cohen real model for any family {Av}v<κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets Av, into the algebra of Baire subsets of 2κ modulo meager sets such that for all Borel B,

The proof is uniform, works also for random reals and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property.