We investigate a phenomenon of "one-to-two-player lifting" in infinite-duration two-player games on graphs with zero-sum objectives. More specifically, let $C$ be a class of strategies. It turns out that in many cases, to show that all two-player games on graphs with a given payoff function are determined in $C$, it is sufficient to do so for one-player games. That is, in many cases the determinacy in $C$ can be "lifted" from one-player games to two-player games. Namely, Gimbert and Zielonka~(CONCUR 2005) have shown this for the class of positional strategies. Recently, Bouyer et al. (CONCUR 2020) have extended this to the classes of arena-independent finite-memory strategies. Informally, these are finite-memory strategies that use the same way of storing memory in all game graphs. In this paper, we put the lifting technique into the context of memory complexity. The memory complexity of a payoff function measures, how many states of memory we need to play optimally in game graphs with up to $n$ nodes, depending on $n$. Now, assume that we know the memory complexity of our payoff function in one-player games. Then what can be said about its memory complexity in two-player games? In particular, when is it finite? Previous one-to-two-player lifting theorems only cover the case when the memory complexity is $O(1)$. In turn, we obtain the following results. Assume that the memory complexity in one-player games is sublinear in $n$ on some infinite subsequence. Then the memory complexity in two-player games is finite. We provide an example in which (a) the memory complexity in one-player games is linear in $n$; (b) the memory complexity in two-player games is infinite. Thus, we obtain the exact barrier for the one-to-two-player lifting theorems.

翻译：在具有零和目的的图表中,我们调查了“一到二玩游戏提升”的游戏现象。更具体地说,让美元成为策略的一类。在许多情况下,显示带有给付功能的图形上的所有两个玩家游戏都以$C美元确定,对于一个玩家游戏就足够这样了。这就是,在许多情况下,用一玩游戏的复杂程度游戏可以“提升”到两个玩家的游戏。也就是说,Gimbert和Zielonka~(CONCUR 2005)已经为定位策略的类别展示了这一点。最近,Bouyer 和 Al. (CONCUR 2020) 已经将这个游戏扩展到了竞技场上的所有两个玩家游戏的游戏类别,这些游戏是使用同样的方式将记忆保存在所有游戏的图表中。在这个游戏中,我们把提升的技巧放到了记忆的复杂程度。在一次游戏中,一个偿还的复杂程度是一次记忆的复杂程度,我们从一个记忆中要拿到一个。