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Partially Observable Risk-Sensitive Markov Decision Processes

Author

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  • Nicole Bäauerle

    (Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany)

  • Ulrich Rieder

    (University of Ulm, D-89069 Ulm, Germany)

Abstract

We consider the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite time horizon that is generated by a partially observable Markov decision process (POMDP). In contrast to a risk-neutral decision maker, this optimization criterion takes the variability of the cost into account. It contains as a special case the classical risk-sensitive optimization criterion with an exponential utility. We show that this optimization problem can be solved by embedding the problem into a completely observable Markov decision process with extended state space and give conditions under which an optimal policy exists. The state space has to be extended by the joint conditional distribution of current unobserved state and accumulated cost. In case of an exponential utility, the problem simplifies considerably and we rediscover what in previous literature has been named information state . However, since we do not use any change of measure techniques here, our approach is simpler. A simple example, namely, a risk-sensitive Bayesian house selling problem, is considered to illustrate our results.

Suggested Citation

  • Nicole Bäauerle & Ulrich Rieder, 2017. "Partially Observable Risk-Sensitive Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1180-1196, November.
  • Handle: RePEc:inm:ormoor:v:42:y:2017:i:4:p:1180-1196
    DOI: 10.1287/moor.2016.0844
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    References listed on IDEAS

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    1. Jun-Yi Fu & An-Hua Wan, 2002. "Generalized vector equilibrium problems with set-valued mappings," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(2), pages 259-268, November.
    2. Nicole Bäuerle & Ulrich Rieder, 2014. "More Risk-Sensitive Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 105-120, February.
    3. Rolando Cavazos-Cadena & Daniel Hernández-Hernández, 2016. "A Characterization of the Optimal Certainty Equivalent of the Average Cost via the Arrow-Pratt Sensitivity Function," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 224-235, February.
    4. Ronald A. Howard & James E. Matheson, 1972. "Risk-Sensitive Markov Decision Processes," Management Science, INFORMS, vol. 18(7), pages 356-369, March.
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    Cited by:

    1. Rasouli, Mohammad & Saghafian, Soroush, 2018. "Robust Partially Observable Markov Decision Processes," Working Paper Series rwp18-027, Harvard University, John F. Kennedy School of Government.
    2. Randall Martyr & John Moriarty & Magnus Perninge, 2019. "Discrete-time risk-aware optimal switching with non-adapted costs," Papers 1910.04047, arXiv.org, revised Sep 2021.
    3. Jingnan Fan & Andrzej Ruszczyński, 2018. "Risk measurement and risk-averse control of partially observable discrete-time Markov systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(2), pages 161-184, October.
    4. Tomasz Kosmala & Randall Martyr & John Moriarty, 2023. "Markov risk mappings and risk-sensitive optimal prediction," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 97(1), pages 91-116, February.

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