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Zero-sum risk-sensitive stochastic games

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  • Bäuerle, Nicole
  • Rieder, Ulrich

Abstract

In this paper we consider two-person zero-sum risk-sensitive stochastic dynamic games with Borel state and action spaces and bounded reward. The term risk-sensitive refers to the fact that instead of the usual risk neutral optimization criterion we consider the exponential certainty equivalent. The discounted reward case on a finite and an infinite time horizon is considered, as well as the ergodic reward case. Under continuity and compactness conditions we prove that the value of the game exists and solves the Shapley equation and we show the existence of optimal (non-stationary) strategies. In the ergodic reward case we work with a local minorization property and a Lyapunov condition and show that the value of the game solves the Poisson equation. Moreover, we prove the existence of optimal stationary strategies. A simple example highlights the influence of the risk-sensitivity parameter. Our results generalize findings in Basu and Ghosh (2014) and answer an open question posed there.

Suggested Citation

  • Bäuerle, Nicole & Rieder, Ulrich, 2017. "Zero-sum risk-sensitive stochastic games," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 622-642.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:2:p:622-642
    DOI: 10.1016/j.spa.2016.06.020
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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. 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.
    4. El-Karoui, N. & Hamadène, S., 2003. "BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 145-169, September.
    5. Ronald A. Howard & James E. Matheson, 1972. "Risk-Sensitive Markov Decision Processes," Management Science, INFORMS, vol. 18(7), pages 356-369, March.
    6. Bäuerle, Nicole & Jaśkiewicz, Anna, 2015. "Risk-sensitive dividend problems," European Journal of Operational Research, Elsevier, vol. 242(1), pages 161-171.
    7. Arnab Basu & Mrinal K. Ghosh, 2012. "Zero-Sum Risk-Sensitive Stochastic Differential Games," Mathematics of Operations Research, INFORMS, vol. 37(3), pages 437-449, August.
    8. Amogh Deshpande & Saul D. Jacka, 2015. "Game-theoretic approach to risk-sensitive benchmarked asset management," Papers 1503.01802, arXiv.org.
    9. 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.
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    Cited by:

    1. Elena Parilina & Stepan Akimochkin, 2021. "Cooperative Stochastic Games with Mean-Variance Preferences," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    2. 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.
    3. Qingda Wei & Xian Chen, 2019. "Risk-Sensitive Average Equilibria for Discrete-Time Stochastic Games," Dynamic Games and Applications, Springer, vol. 9(2), pages 521-549, June.
    4. Chen, Fang & Guo, Xianping, 2023. "Two-person zero-sum risk-sensitive stochastic games with incomplete reward information on one side," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 218-245.
    5. 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.
    6. Julio Saucedo-Zul & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2020. "A Discounted Approach in Communicating Average Markov Decision Chains Under Risk-Aversion," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 585-606, November.
    7. Subrata Golui & Chandan Pal & Subhamay Saha, 2022. "Continuous-Time Zero-Sum Games for Markov Decision Processes with Discounted Risk-Sensitive Cost Criterion," Dynamic Games and Applications, Springer, vol. 12(2), pages 485-512, June.
    8. Tomasz R. Bielecki & Igor Cialenco & Andrzej Ruszczy'nski, 2022. "Risk Filtering and Risk-Averse Control of Markovian Systems Subject to Model Uncertainty," Papers 2206.09235, arXiv.org.
    9. Hubert Asienkiewicz & Łukasz Balbus, 2019. "Existence of Nash equilibria in stochastic games of resource extraction with risk-sensitive players," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 502-518, October.

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