We locate winning strategies for various Σ⁰₃-games in the L-hierarchy in order to prove the following:
Theorem 1. KP + Σ₂-Comprehension ⊢ ∃ α Lα ⊨“Σ₂-KP + Σ03-Determinacy.”
Alternatively: Π¹₃-CA₀ ⊢“there is a β-model of Δ¹₃-CA₀ + Σ03-Determinacy.” The implication is not reversible. (The antecedent here may be replaced with Π¹₃(Π¹₃)-CA₀: Π¹₃ instances of Comprehension with only Π¹₃-lightface definable parameters—or even weaker theories.)
Theorem 2. KP + Δ₂-Comprehension + Σ₂-Replacement + AQI ⊬ Σ⁰₃-Determinacy.
(Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively:
Δ¹₃CA₀ + AQI ⊬ Σ⁰₃-Determinacy.
Hence the theories: Π¹₃-CA₀, Δ¹₃-CA₀+ Σ⁰₃-Det, Δ¹₃-CA₀+AQI, and Δ¹₃-CA₀ are in strictly descending order of strength.
"Weak systems of determinacy and arithmetical quasi-inductive definitions." J. Symbolic Logic 76 (2) 418 - 436, June 2011. https://doi.org/10.2178/jsl/1305810756