We study countably infinite MDPs with parity objectives. Unlike in finite MDPs, optimal strategies need not exist, and may require infinite memory if they do. We provide a complete picture of the exact strategy complexity of $\varepsilon$-optimal strategies (and optimal strategies, where they exist) for all subclasses of parity objectives in the Mostowski hierarchy. Either MD-strategies, Markov strategies, or 1-bit Markov strategies are necessary and sufficient, depending on the number of colors, the branching degree of the MDP, and whether one considers $\varepsilon$-optimal or optimal strategies. In particular, 1-bit Markov strategies are necessary and sufficient for $\varepsilon$-optimal (resp. optimal) strategies for general parity objectives.

中文翻译：

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可数 MDP 中奇偶目标的策略复杂性

我们研究了具有同等目标的可数无限 MDP。与有限 MDP 不同，最优策略不需要存在，如果存在，可能需要无限内存。我们提供了对 Mostowski 层次结构中奇偶目标的所有子类的 $\varepsilon$-最优策略（和最优策略，如果存在）的确切策略复杂性的完整描述。MD-strategies、Markov 策略或 1-bit Markov 策略是必要和充分的，这取决于颜色的数量、MDP 的分支程度以及是否考虑 $\varepsilon$-最优或最优策略。特别是，对于一般奇偶目标的 $\varepsilon$-optimal (resp.optimal) 策略来说，1 位马尔可夫策略是必要和充分的。