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Mixing Any Cocktail with Limited Ingredients: On the Structure of Payoff Sets in Multi-Objective POMDPs and Its Impact on Randomised Strategies

Authors James C. A. Main , Mickael Randour

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ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Logic and verification
Keywords
  • partially observable Markov decision process
  • multiple objectives
  • randomised strategies
  • mixed strategies
  • strategy complexity

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Abstract

We consider multi-dimensional payoff functions in partially observable Markov decision processes. We study the structure of the set of expected payoff vectors of all strategies (policies) and study what kind are needed to achieve a given expected payoff vector. In general, pure strategies (i.e., not resorting to randomisation) do not suffice for this problem.
We prove that for any payoff function for which the expectation is well-defined under all strategies, it is sufficient to mix (i.e., randomly select a pure strategy at the start of a play and committing to it for the rest of the play) finitely many pure strategies to approximate any expected payoff vector up to any precision. Furthermore, for any payoff function for which the expected payoff is finite under all strategies, any expected payoff vector can be obtained exactly by mixing finitely many strategies. These results are already novel in the context of (perfect information) Markov decision processes.

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James C. A. Main and Mickael Randour. Mixing Any Cocktail with Limited Ingredients: On the Structure of Payoff Sets in Multi-Objective POMDPs and Its Impact on Randomised Strategies. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 68:1-68:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.68

Author Details

James C. A. Main
  • University of Oxford, UK
  • UMONS - Université de Mons, Belgium
Mickael Randour
  • F.R.S.-FNRS and UMONS - Université de Mons, Belgium

Funding

This work has been supported by EPSRC project EP/Z536179/1 and by the Fonds de la Recherche Scientifique - FNRS (F.R.S.-FNRS) under Grant n° T.0188.23 (PDR ControlleRS).
  • Main, James C. A.: Supported by a Research Fellowship of the F.R.S.-FNRS during the preparation of this work.
  • Randour, Mickael: F.R.S.-FNRS Senior Research Associate and member of the TRAIL institute.

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