# Parity Objectives in Countable MDPs

## TL;DR

This paper analyzes the memory requirements for optimal and epsilon-optimal strategies in countably infinite MDPs with parity objectives, revealing a dichotomy between objectives that require only memoryless strategies and those needing infinite memory.

## Contribution

It provides a complete characterization of when optimal strategies in countable MDPs can be memoryless or require infinite memory across the Mostowski hierarchy.

## Key findings

- Optimal strategies may not always exist in countably infinite MDPs.
- A dichotomy exists: some objectives admit MD strategies, others require infinite memory.
- Finitely branching MDPs have a similar but more favorable dichotomy.

## Abstract

We study countably infinite MDPs with parity objectives, and special cases with a bounded number of colors in the Mostowski hierarchy (including reachability, safety, Buchi and co-Buchi). In finite MDPs there always exist optimal memoryless deterministic (MD) strategies for parity objectives, but this does not generally hold for countably infinite MDPs. In particular, optimal strategies need not exist. For countable infinite MDPs, we provide a complete picture of the memory requirements of optimal (resp., $\epsilon$-optimal) strategies for all objectives in the Mostowski hierarchy. In particular, there is a strong dichotomy between two different types of objectives. For the first type, optimal strategies, if they exist, can be chosen MD, while for the second type optimal strategies require infinite memory. (I.e., for all objectives in the Mostowski hierarchy, if finite-memory randomized strategies suffice then also MD strategies suffice.) Similarly, some objectives admit $\epsilon$-optimal MD-strategies, while for others $\epsilon$-optimal strategies require infinite memory. Such a dichotomy also holds for the subclass of countably infinite MDPs that are finitely branching, though more objectives admit MD-strategies here.