Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$ Let $Z_2$ and $Z_3$ be second and third order arithmetics respectively.
In $Z_2$'s language, $\text{AD}$ (the axiom of determinacy) and $\text{PD}$ (projective determinacy) are stated the same way (since projective sets are precisely the ones defined by formulas in $Z_2$'s language). But in $Z_3$'s language we can state $\text{AD}$ and $\text{PD}$ separately.
$Z_2+\text{PD}$, $Z_3+\text{PD}$ and in fact $\text{ZFC+PD}$ are all equiconsistent, and weaker than $\text{ZF+AD / ZFC+AD}^{L(\mathbb{R})}$. I was wondering if $Z_3+\text{AD}$ was also equiconsistent with those, or if it has the full consistency strength of $\text{ZF+AD / ZFC+AD}^{L(\mathbb{R})}$ (or falls somewhere inbetween). I would appreciate a reference or a proof sketch.
 A: The only reference I know for precisely these matters is the handbook chapter 

MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. 

(Particularly, section 7.)
For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harrington's principle. 
First of all, $\mathsf{ZFC}+\mathsf{PD}$ proves the consistency of $Z_3+\mathsf{PD}$, and similarly $\mathsf{ZF}+\mathsf{AD}$ and $\mathsf{ZFC}+\mathsf{AD}^{L(\mathbb R)}$ prove the consistency of $Z_3+\mathsf{AD}$: (depending on your precise formulation of second- and third-order arithmetic), $(V_\omega,V_{\omega+1},V_{\omega+2},\in)^{L(\mathbb R)}$ is a model of the latter.
That said, however, (as pointed out by Andreas in a comment) there is a subtlety here in that $\mathsf{PD}$ is a single sentence. If instead we consider the infinite collection of axioms, the $n$th of which states the determinacy of boldface $\Sigma^1_n$-games, we should probably get equiconsistency, perhaps after some minor tweaking of the theories. 
Section 7 of the reference above addresses these matters in reasonable detail for $\mathsf{ZFC}+$"there is a Woodin cardinal" and close variants, and $Z_2+$ lightface $\Delta^1_2$-determinacy, and close variants. For instance, from $Z_2+$ boldface $\Delta^1_2$-determinacy, they obtain (something stronger than) the consistency of $\mathsf{ZFC}+$"There is a Woodin cardinal". Note that the latter theory is equiconsistent with $\mathsf{ZFC}+$ lightface $\Delta^1_2$-determinacy which, in turn, implies the consistency of $Z_2+$ lightface $\Delta^1_2$-determinacy, so one needs to be careful here. They conjecture that $Z_2+$ lightface $\Delta^1_2$-determinacy is equiconsistent with $\mathsf{ZFC}+$"$\mathrm{ORD}$ is Woodin". In any case, from these results and the techniques in that chapter (particularly in section 6) one should obtain the equiconsistency results I suggested in the previous paragraph.
