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

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.