The fundamental objects of study in inner model theory are iterable premice---fine-structual models of set theory which have winning strategies in certain iteration games. In the standard iteration game, two players work to produce an iterate N of a premouse M via a tree of iterated ultrapowers called a normal iteration tree. In a variant game, players produce an iterate N not via a single iteration tree but via a linear stack of normal trees. Recent work has revealed connections between these games, which have various applications in inner model theory. The key framework for understanding these connections is the theory of meta-iteration trees : iteration trees of iteration trees. Using this framework, we show that any nice winning strategy S in the standard game extends to a winning strategy S* in the variant game. Moreover, every iterate N obtainable via a play by S* in the variant game is actually obtainable via a play by the original strategy S. This is joint work with John Steel.

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