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Entropic Risk for Turn-Based Stochastic Games

Authors Christel Baier , Krishnendu Chatterjee , Tobias Meggendorfer , Jakob Piribauer

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Christel Baier
  • Technische Universität Dresden, Germany
Krishnendu Chatterjee
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Tobias Meggendorfer
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
  • Technische Universität München, Germany
Jakob Piribauer
  • Technische Universität Dresden, Germany
  • Technische Universität München, Germany

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Christel Baier, Krishnendu Chatterjee, Tobias Meggendorfer, and Jakob Piribauer. Entropic Risk for Turn-Based Stochastic Games. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 15:1-15:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Entropic risk (ERisk) is an established risk measure in finance, quantifying risk by an exponential re-weighting of rewards. We study ERisk for the first time in the context of turn-based stochastic games with the total reward objective. This gives rise to an objective function that demands the control of systems in a risk-averse manner. We show that the resulting games are determined and, in particular, admit optimal memoryless deterministic strategies. This contrasts risk measures that previously have been considered in the special case of Markov decision processes and that require randomization and/or memory. We provide several results on the decidability and the computational complexity of the threshold problem, i.e. whether the optimal value of ERisk exceeds a given threshold. In the most general case, the problem is decidable subject to Shanuel’s conjecture. If all inputs are rational, the resulting threshold problem can be solved using algebraic numbers, leading to decidability via a polynomial-time reduction to the existential theory of the reals. Further restrictions on the encoding of the input allow the solution of the threshold problem in NP∩coNP. Finally, an approximation algorithm for the optimal value of ERisk is provided.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Stochastic games
  • risk-aware verification


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