Let o(κ) denote the Mitchell order of κ. We show how to reduce long games which run to the first ordinal admissible in the play, to iteration games on models with a cardinal κ so that (1) κ is a limit of Woodin cardinals; and (2) o(κ)=κ++. We use the reduction to derive several optimal determinacy results on games which run to the first admissible in the play.
"Determinacy for games ending at the first admissible relative to the play." J. Symbolic Logic 71 (2) 425 - 459, June 2006. https://doi.org/10.2178/jsl/1146620151