The Evolutionary Biology thing is interesting. Has determinacy been helpfun in theoretical computer science (AI) then, especially with regards to the Church-Turing thesis?
In an infinite games, Adam would try to play to come up with a sequence outside A, and Eve would play to hit A. But isn't it true that the sequence s is either in A or in U\A (the complement) and no matter how Adam or Eve plays, with the production of a sequence, we'd know whether $s\in A$ or otherwise, whence the infinitary de Morgan's law? (Hence, the game is determined.)