Determinacy axioms are mathematical principles which assert that various infinite games are determined. In this article, we prove three general meta-theorems on the logical strength of determinacy axioms. These allow us to reduce a metamathematical analysis of the principle of $\Gamma $ -determinacy over a weak base theory to an analysis of a principle $\Gamma '$ -determinacy, where $\Gamma '$ is a strictly smaller complexity class than $\Gamma $ . The meta-theorems are proved in the weak theory ${\mathsf{RCA_{0}}}$ . However, they are formulated in a general way and also have applications in the context of ${\mathsf{ZFC}}$ , eventually leading to an optimal generalization of Martin’s Borel determinacy theorem and optimal strengthenings of the transfer theorems of Martin-Harrington, Kechris-Woodin, and Neeman. As the main application of the meta-theorems, we carry out all the reverse-mathematical analyses of theories of determinacy below $T = \boldsymbol{\Pi }^{1}_{1}{-}{\mathsf{CA_{0}}}+ \Pi ^{1}_{4}{-}{ \mathsf{CA_{0}}}$ which are missing from the literature. More precisely, let $\Gamma \subset \mathcal{P}(\mathbb{R})$ be called a Wadge class if $\Gamma $ is closed under continuous preimages. For each Wadge class $\Gamma $ such that the consistency of $\Gamma $ -Determinacy is provable in $T$ , we reduce the principle of $\Gamma $ -Determinacy to a combination of Comprehension, Monotone Induction, and $\beta $ -Reflection axioms. It follows from our results that these classes $\Gamma $ are precisely those which satisfy \( o(\Gamma ) < \omega _{1}^{\omega _{1}^{2}} \text{ and $T\vdash $``$\Gamma $ is a Wadge class.''} \) Our work extends and generalizes results of Friedman, Hachtman, Heinatsch, Martin, MedSalem, Montalbán, Möllerfeld, Nemoto, Shore, Steel, Tanaka, Welch, and others, and concludes the project of metamathematical analysis of determinacy principles (cf. e.g., Montalbán’s “Open Questions in Reverse Mathematics”, Bull. Symb. Log. 16:431–454, 2011), as far as subsystems of $T$ are concerned.
