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

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Ronald A. Howard & James E. Matheson, 1972. "Risk-Sensitive Markov Decision Processes," Management Science, INFORMS, vol. 18(7), pages 356-369, March.
    4. 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.
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