The consistency strength of long projective determinacy

We determine the consistency strength of determinacy for projective games of length omega(2). Our main theorem is that Pi(1)(n+1)-determinacy for games of length omega(2) implies the existence of a model of set theory with omega + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M-n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies A = R and the Axiom of Determinacy. Then we argue how to obtain a model with omega + n Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length omega(2) with payoff in (sic)(R)Pi(1)(1) or with sigma-projective payoff