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Discrete-time zero-sum games for Markov chains with risk-sensitive average cost criterion

Author

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  • Ghosh, Mrinal K.
  • Golui, Subrata
  • Pal, Chandan
  • Pradhan, Somnath

Abstract

We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable/compact state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded for countable state space case and for compact state space case it is a real-valued and bounded function. For countable state space case, under a certain Lyapunov type stability assumption on the dynamics we establish the existence of the value and a saddle point equilibrium. For compact state space case we establish these results without any Lyapunov type stability assumptions. Using the stochastic representation of the principal eigenfunction of the associated optimality equation, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Also, we present and analyze an illustrative example.

Suggested Citation

  • Ghosh, Mrinal K. & Golui, Subrata & Pal, Chandan & Pradhan, Somnath, 2023. "Discrete-time zero-sum games for Markov chains with risk-sensitive average cost criterion," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 40-74.
    • Handle: RePEc:eee:spapps:v:158:y:2023:i:c:p:40-74
    DOI: 10.1016/j.spa.2022.12.009
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    References listed on IDEAS

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    1. 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.
    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 & Raúl Montes-de-Oca & Karel Sladký, 2014. "A Counterexample on Sample-Path Optimality in Stable Markov Decision Chains with the Average Reward Criterion," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 674-684, November.
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    5. Balaji, S. & Meyn, S. P., 2000. "Multiplicative ergodicity and large deviations for an irreducible Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 123-144, November.
    6. Qingda Wei & Xian Chen, 2021. "Nonzero-sum Risk-Sensitive Average Stochastic Games: The Case of Unbounded Costs," Dynamic Games and Applications, Springer, vol. 11(4), pages 835-862, December.
    7. Bäuerle, Nicole & Rieder, Ulrich, 2017. "Zero-sum risk-sensitive stochastic games," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 622-642.
    8. Arapostathis, Ari & Biswas, Anup, 2018. "Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1485-1524.
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    10. 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.
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