Abstract
In this paper we study the nonzero-sum constrained stochastic games for continuous-time jump processes with denumerable states and possibly unbounded transition rates. The optimality criterion under consideration is the expected average payoff criterion and the payoff functions of the players are allowed to be unbounded. Under the reasonable conditions, we introduce an approximating sequence of the auxiliary game models and show the existence of stationary constrained Nash equilibria for these approximating game models via employing the average occupation measures and constructing a suitable multifunction. Moreover, we obtain that any limit point of the stationary constrained Nash equilibria for the approximating sequence of the game models is a constrained Nash equilibrium for the original game model. Furthermore, we use a controlled birth and death system to illustrate our main results.
Similar content being viewed by others
References
Aliprantis, C., Border, K.: Infinite Dimensional Analysis. Springer, New York (2006)
Altman, E., Shwartz, A.: Constrained Markov Games: Nash Equilibria. Ann Int Soc Dynam Games, vol. 5, pp. 213–221. Birkhäuser, Boston (2000)
Alvarez-Mena, J., Hernández-Lerma, O.: Convergence of the optimal values of constrained Markov control processes. Math. Methods Oper. Res. 55, 461–484 (2002)
Alvarez-Mena, J., Hernández-Lerma, O.: Existence of Nash equilibria for constrained stochastic games. Math. Methods Oper. Res. 63, 261–285 (2006)
Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific, Belmont (1996)
Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter, Berlin (2004)
Guo, X.P., Hernández-Lerma, O.: Continuous-Time Markov Decision Processes: Theory and Applications. Springer, Berlin (2009)
Guo, X.P., Song, X.Y.: Discounted continuous-time constrained Markov decision processes in Polish spaces. Ann. Appl. Probab. 21, 2016–2049 (2011)
Guo, X.P., Huang, Y.H., Song, X.Y.: Linear programming and constrained average optimality for general continuous-time Markov decision processes in history-dependent policies. SIAM J. Control Optim. 50, 23–47 (2012)
Hernández-Lerma, O., Lasserre, J.B.: Discrete-Time Markov Control Processes: Basic Optimality Criteria. Springer, New York (1996)
Kitaev, M.Y., Rykov, V.V.: Controlled Queueing Systems. CRC Press, Boca Raton (1995)
Piunovskiy, A., Zhang, Y.: Discounted continuous-time Markov decision processes with unbounded rates: the convex analytic approach. SIAM J. Control Optim. 49, 2032–2061 (2011)
Prieto-Rumeau, T., Hernández-Lerma, O.: Ergodic control of continuous-time Markov chains with pathwise constraints. SIAM J. Control Optim. 47, 1888–1908 (2008)
Wei, Q.D., Chen, X.: Strong average optimality criterion for continuous-time Markov decision processes. Kybernetika 50, 950–977 (2014)
Wei, Q.D., Chen, X.: Constrained stochastic games with the average payoff criteria. Oper. Res. Lett. 43, 83–88 (2015)
Wei, Q.D., Chen, X.: Stochastic games for continuous-time jump processes under finite-horizon payoff criterion. Appl. Math. Optim. 74, 273–301 (2016)
Zhang, W.Z.: Continuous-time constrained stochastic games under the discounted cost criteria. Appl. Math. Optim. 77, 275–296 (2018)
Zhang, W.Z., Guo, X.P.: Nonzero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates. Sci. China Math. 55, 2405–2416 (2012)
Zhang, W.Z., Huang, Y.H., Guo, X.P.: Nonzero-sum constrained discrete-time Markov games: the case of unbounded costs. TOP 22, 1074–1102 (2014)
Zhang, W.Z., Wang, B.F., Chen, D.W.: Continuous-time constrained stochastic games with average criteria. Oper. Res. Lett. 46, 109–115 (2018)
Acknowledgements
I am greatly indebted to the referees for the valuable comments and suggestions which have greatly improved the presentation. The research was supported by National Natural Science Foundation of China (Grant No. 11601166).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
I declare that no conflict of interest exists in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wei, Q. Constrained Expected Average Stochastic Games for Continuous-Time Jump Processes. Appl Math Optim 83, 1277–1309 (2021). https://doi.org/10.1007/s00245-019-09588-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-019-09588-9
Keywords
- Nonzero-sum games
- Expected average payoff criterion
- Constrained Nash equilibrium
- Continuous-time jump process