Strategy Complexity of Parity Objectives in Countable MDPs

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.