  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.
