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 $boldsymbolPi^1_{n+1}$-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 $A = mathbb{R}$ 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 $Game^mathbb{R} boldsymbolPi^1_1$ or with $sigma$-projective payoff.