We study countably infinite stochastic 2-player games with reachability objectives. Our results provide a complete picture of the memory requirements of $\varepsilon$-optimal (resp. optimal) strategies. These results depend on the size of the players' action sets and on whether one requires strategies that are uniform (i.e., independent of the start state). Our main result is that $\varepsilon$-optimal (resp. optimal) Maximizer strategies require infinite memory if Minimizer is allowed infinite action sets. This lower bound holds even under very strong restrictions. Even in the special case of infinitely branching turn-based reachability games, even if all states allow an almost surely winning Maximizer strategy, strategies with a step counter plus finite private memory are still useless. Regarding uniformity, we show that for Maximizer there need not exist positional (i.e., memoryless) uniformly $\varepsilon$-optimal strategies even in the special case of finite action sets or in finitely branching turn-based games. On the other hand, in games with finite action sets, there always exists a uniformly $\varepsilon$-optimal Maximizer strategy that uses just one bit of public memory.

翻译：我们研究具有可达性目标的可数无限随机双人博弈。我们的结果为$\varepsilon$-最优（或最优）策略的记忆需求提供了完整的刻画。这些结果取决于玩家动作集的规模，以及是否要求策略具有一致性（即与起始状态无关）。我们的主要结论是：若允许最小化玩家使用无限动作集，则最大化玩家的$\varepsilon$-最优（或最优）策略需要无限记忆。这一下界即使在极强限制条件下依然成立。即使在无限分支回合制可达性博弈这一特殊情形中，即使所有状态都允许最大化玩家采用几乎必然获胜的策略，仅使用步数计数器加有限私有记忆的策略仍然无效。关于一致性，我们证明即使限制在有限动作集或有限分支回合制博弈的特殊情况下，最大化玩家仍可能不存在位置性（即无记忆）的一致$\varepsilon$-最优策略。另一方面，在有限动作集博弈中，总存在仅需1比特公共记忆的一致$\varepsilon$-最优最大化策略。