Working under AD, we investigate the length of prewellorderings given by the iterates of ℳ2k+1, which is the minimal proper class mouse with 2k + 1 many Woodin cardinals. In particular, we answer some questions from [4] (the discussion of the questions appears in the last section of [2]).

Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference hierarchy over $\Sigma^0_2$. This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory. We raise the problem of transfinite fixpoint hierarchies.