  We determine the consistency strength of determinacy for projective games of
length $\omega^2$. Our main theorem is that
$\boldsymbol\Pi^1_{n+1}$-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 $A = \mathbb{R}$ 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 $\Game^\mathbb{R} \boldsymbol\Pi^1_1$ or with $\sigma$-projective
payoff.

Logic