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.