# One-to-Two-Player Lifting for Mildly Growing Memory

We investigate so-called "one-to-two-player lifting" theorems for infinite-duration two-player games on graphs with zero-sum objectives. These theorems are concerned with questions of the following form. If that much memory is sufficient to play optimally in one-player games, then how much memory is needed to play optimally in two-player games? In 2005, Gimbert and Zielonka (CONCUR 2005) have shown that if no memory is needed in the one-player games, then the same holds for the two-player games. Building upon their work, Bouyer et al. (CONCUR 2020) have shown that if some constant amount of memory (independent of the size of a game graph) is sufficient in the one-player games, then exactly the same constant is sufficient in the two-player games. They also provide an example in which every one-player game requires only a finite amount of memory (now this amount depends on the size of a game) while some two-player game requires infinite memory. Our main result states the following. If the memory grows just a bit slower (in the one-player games) than in the example of Bouyer et al., then in every two-player game it is sufficient to have finite memory. Thus, our work identifies the exact barrier for the one-to-two-player lifting theorems in a context of finite-memory strategies.

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