IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v81y2015i3p269-298.html
   My bibliography  Save this article

Continuity of the optimal average cost in Markov decision chains with small risk-sensitivity

Author

Listed:
  • Selene Chávez-Rodríguez
  • Rolando Cavazos-Cadena
  • Hugo Cruz-Suárez

Abstract

This note concerns discrete-time controlled Markov chains driven by a decision maker with constant risk-sensitivity $$\lambda $$ λ . Assuming that the system evolves on a denumerable state space and is endowed with a bounded cost function, the paper analyzes the continuity of the optimal average cost with respect to the risk-sensitivity parameter, a property that is promptly seen to be valid at each no-null value of $$\lambda $$ λ . Under standard continuity-compactness conditions, it is shown that a general form of the simultaneous Doeblin condition allows to establish the continuity of the optimal average cost at $$\lambda=0$$ λ = 0 , and explicit examples are given to show that, even if every state is positive recurrent under the action of any stationary policy, the above continuity conclusion can not be ensured under weaker recurrence requirements, as the Lyapunov function condition. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Selene Chávez-Rodríguez & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2015. "Continuity of the optimal average cost in Markov decision chains with small risk-sensitivity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 81(3), pages 269-298, June.
    • Handle: RePEc:spr:mathme:v:81:y:2015:i:3:p:269-298
    DOI: 10.1007/s00186-015-0496-y
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00186-015-0496-y
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00186-015-0496-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Rolando Cavazos-Cadena, 2003. "Solution to the risk-sensitive average cost optimality equation in a class of Markov decision processes with finite state space," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 57(2), pages 263-285, May.
    2. Rolando Cavazos-Cadena & Emmanuel Fernández-Gaucherand, 1999. "Controlled Markov chains with risk-sensitive criteria: Average cost, optimality equations, and optimal solutions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(2), pages 299-324, April.
    3. Rolando Cavazos-Cadena & Daniel Hernández-Hernández, 2003. "Solution to the risk-sensitive average optimality equation in communicating Markov decision chains with finite state space: An alternative approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(3), pages 473-479, January.
    4. Ronald A. Howard & James E. Matheson, 1972. "Risk-Sensitive Markov Decision Processes," Management Science, INFORMS, vol. 18(7), pages 356-369, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Rolando Cavazos-Cadena, 2009. "Solutions of the average cost optimality equation for finite Markov decision chains: risk-sensitive and risk-neutral criteria," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(3), pages 541-566, December.
    2. Daniel Hernández Hernández & Diego Hernández Bustos, 2017. "Local Poisson Equations Associated with Discrete-Time Markov Control Processes," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 1-29, April.
    3. Rolando Cavazos-Cadena & Daniel Hernández-Hernández, 2011. "Discounted Approximations for Risk-Sensitive Average Criteria in Markov Decision Chains with Finite State Space," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 133-146, February.
    4. Rolando Cavazos-Cadena & Raúl Montes-de-Oca, 2003. "The Value Iteration Algorithm in Risk-Sensitive Average Markov Decision Chains with Finite State Space," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 752-776, November.
    5. Karel Sladký, 2013. "Risk-Sensitive and Mean Variance Optimality in Markov Decision Processes," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 7(3), pages 146-161, November.
    6. Basu, Arnab & Ghosh, Mrinal Kanti, 2014. "Zero-sum risk-sensitive stochastic games on a countable state space," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 961-983.
    7. V. S. Borkar & S. P. Meyn, 2002. "Risk-Sensitive Optimal Control for Markov Decision Processes with Monotone Cost," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 192-209, February.
    8. Rolando Cavazos-Cadena, 2010. "Optimality equations and inequalities in a class of risk-sensitive average cost Markov decision chains," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 47-84, February.
    9. Gustavo Portillo-Ramírez & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2023. "Contractive approximations in average Markov decision chains driven by a risk-seeking controller," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 98(1), pages 75-91, August.
    10. Jaśkiewicz, Anna & Nowak, Andrzej S., 2014. "Stationary Markov perfect equilibria in risk sensitive stochastic overlapping generations models," Journal of Economic Theory, Elsevier, vol. 151(C), pages 411-447.
    11. Arnab Basu & Mrinal K. Ghosh, 2018. "Nonzero-Sum Risk-Sensitive Stochastic Games on a Countable State Space," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 516-532, May.
    12. Özlem Çavuş & Andrzej Ruszczyński, 2014. "Computational Methods for Risk-Averse Undiscounted Transient Markov Models," Operations Research, INFORMS, vol. 62(2), pages 401-417, April.
    13. Selene Chávez-Rodríguez & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2016. "Controlled Semi-Markov Chains with Risk-Sensitive Average Cost Criterion," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 670-686, August.
    14. Pestien, Victor & Wang, Xiaobo, 1998. "Markov-achievable payoffs for finite-horizon decision models," Stochastic Processes and their Applications, Elsevier, vol. 73(1), pages 101-118, January.
    15. Lucy Gongtao Chen & Daniel Zhuoyu Long & Melvyn Sim, 2015. "On Dynamic Decision Making to Meet Consumption Targets," Operations Research, INFORMS, vol. 63(5), pages 1117-1130, October.
    16. Bäuerle, Nicole & Rieder, Ulrich, 2017. "Zero-sum risk-sensitive stochastic games," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 622-642.
    17. Guglielmo D’Amico & Fulvio Gismondi & Jacques Janssen & Raimondo Manca, 2015. "Discrete Time Homogeneous Markov Processes for the Study of the Basic Risk Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(4), pages 983-998, December.
    18. Zeynep Erkin & Matthew D. Bailey & Lisa M. Maillart & Andrew J. Schaefer & Mark S. Roberts, 2010. "Eliciting Patients' Revealed Preferences: An Inverse Markov Decision Process Approach," Decision Analysis, INFORMS, vol. 7(4), pages 358-365, December.
    19. Nicole Bäuerle & Ulrich Rieder, 2014. "More Risk-Sensitive Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 105-120, February.
    20. Carlos Camilo-Garay & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2022. "Contractive Approximations in Risk-Sensitive Average Semi-Markov Decision Chains on a Finite State Space," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 271-291, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:mathme:v:81:y:2015:i:3:p:269-298. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.