Determinacy in third order arithmetic

We investigate the strength of open and clopen determinacy of perfect information games with real number moves in the context of third order arithmetic. This work is conducted in a framework developed in 2015 by Schweber [12], who showed that in this setting, open determinacy (Σ1R-DET) is not implied by clopen determinacy (Δ1R-DET). We give a new forcing-free proof of this result by isolating a level of L witnessing this separation. We give a notion of β-absoluteness in the context of third-order arithmetic, and show that this level of L is a β-model; combining this with our previous results on the strength of Borel determinacy, we show that Σ40-DET, determinacy for games on ω with Σ40 payoff, is sandwiched between Σ1R-DET and Δ1R-DET in terms of β-consistency strength.