We analyze the strength of standard Determinacy principles as well as ones for
Turing Determinacy that are provable in (subsystems of) second order
arithmetic (equivalently ZFC$^{-}$). These are all at low levels of the
arithmetic hierarchy. We consider three notions of strength. The first is in
the sense of reverse mathematics which asks what axioms (e.g. comprehension
for $\Pi_{n}^{1}$ formulas) are needed to prove the principles. The second is
more traditionally proof theoretic in that we compare principles in terms of
consistency strength. The third is recursion or set theoretic in that we want
to determine the existence of which ordinals (or better levels of the
constructible universe $L$) are is implied by theses principles. Here the
measure is in terms of levels of admissibility ($\Sigma_{n}$ replacement) or
nonprojectability ($\Sigma_{n}$ comprehension axioms).

This is joint work with Antonio Montalban.