Strategy Complexity of Mean Payoff, Total Payoff and Point Payoff Objectives in Countable MDPs

Authors Richard Mayr, Eric Munday

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Richard Mayr
  • University of Edinburgh, UK
Eric Munday
  • University of Edinburgh, UK


We thank an anonymous reviewer for very detailed and helpful comments.

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Richard Mayr and Eric Munday. Strategy Complexity of Mean Payoff, Total Payoff and Point Payoff Objectives in Countable MDPs. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study countably infinite Markov decision processes (MDPs) with real-valued transition rewards. Every infinite run induces the following sequences of payoffs: 1. Point payoff (the sequence of directly seen transition rewards), 2. Total payoff (the sequence of the sums of all rewards so far), and 3. Mean payoff. For each payoff type, the objective is to maximize the probability that the liminf is non-negative. We establish the complete picture of the strategy complexity of these objectives, i.e., how much memory is necessary and sufficient for ε-optimal (resp. optimal) strategies. Some cases can be won with memoryless deterministic strategies, while others require a step counter, a reward counter, or both.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Probability and statistics
  • Markov decision processes
  • Strategy complexity
  • Mean payoff


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