Let denote the minimal active iterable extender model which has Woodin cardinals and contains all reals, if it exists, in which case we denote by the class-sized model obtained by iterating the topmost measure of class-many times. We characterize the sets of reals which are -definable from over , under the assumption that projective games on reals are determined:
1. for even , ;
2. for odd , .
This generalizes a theorem of Martin and Steel for , that is, the case . As consequences of the proof, we see that determinacy of all projective games with moves in is equivalent to the statement that exists for all , and that determinacy of all projective games of length with moves in is equivalent to the statement that exists and satisfies for all .
"Projective Games on the Reals." Notre Dame J. Formal Logic 61 (4) 573 - 589, November 2020. https://doi.org/10.1215/00294527-2020-0027