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Title: Nash Equilibria in a class of Markov stopping games (English)
Author: Cavazos-Cadena, Rolando
Author: Hernández-Hernández, Daniel
Language: English
Journal: Kybernetika
ISSN: 0023-5954
Volume: 48
Issue: 5
Year: 2012
Pages: 1027-1044
Summary lang: English
Category: math
Summary: This work concerns a class of discrete-time, zero-sum games with two players and Markov transitions on a denumerable space. At each decision time player II can stop the system paying a terminal reward to player I and, if the system is no halted, player I selects an action to drive the system and receives a running reward from player II. Measuring the performance of a pair of decision strategies by the total expected discounted reward, under standard continuity-compactness conditions it is shown that this stopping game has a value function which is characterized by an equilibrium equation, and such a result is used to establish the existence of a Nash equilibrium. Also, the method of successive approximations is used to construct approximate Nash equilibria for the game. (English)
Keyword: zero-sum stopping game
Keyword: equality of the upper and lower value functions
Keyword: contractive operator
Keyword: hitting time
Keyword: stationary strategy
MSC: 91A10
MSC: 91A15
idMR: MR3086867
Date available: 2012-12-17T13:44:33Z
Last updated: 2013-09-24
Stable URL:
Reference: [1] Altman, E., Shwartz, A.: Constrained Markov Games: Nash Equilibria..In: Annals of Dynamic Games (V. Gaitsgory, J. Filar and K. Mizukami, eds.) 6 (2000), pp. 213-221, Birkhauser, Boston. Zbl 0957.91014, MR 1764491
Reference: [2] Atar, R., Budhiraja, A.: A stochastic differential game for the inhomogeneous infinty-Laplace equation..Ann. Probab. 2 (2010), 498-531. MR 2642884, 10.1214/09-AOP494
Reference: [3] Bielecki, T., Hernández-Hernández, D., Pliska, S. R.: Risk sensitive control of finite state Markov chains in discrete time, with applications to portfolio management..Mathe. Methods Oper. Res. 50 (1999), 167-188. Zbl 0959.91029, MR 1732397, 10.1007/s001860050094
Reference: [4] Dynkin, E. B.: The optimum choice for the instance for stopping Markov process..Soviet. Math. Dokl. 4 (1963), 627-629.
Reference: [5] Kolokoltsov, V. N., Malafeyev, O. A.: Understanding Game Theory..World Scientific, Singapore 2010. Zbl 1189.91001, MR 2666863
Reference: [6] Peskir, G.: On the American option problem..Math. Finance 15 (2010), 169-181. Zbl 1109.91028, MR 2116800, 10.1111/j.0960-1627.2005.00214.x
Reference: [7] Peskir, G., Shiryaev, A.: Optimal Stopping and Free-Boundary Problems..Birkhauser, Boston 2010. Zbl 1115.60001, MR 2256030
Reference: [8] Puterman, M.: Markov Decision Processes..Wiley, New York 1994. Zbl 1184.90170, MR 1270015
Reference: [9] Shiryaev, A.: Optimal Stopping Rules..Springer, New York 1978. Zbl 1138.60008, MR 0468067
Reference: [10] Sladký, K.: Ramsey Growth model under uncertainty..In: Proc. 27th International Conference Mathematical Methods in Economics (H. Brozová, ed.), Kostelec nad Černými lesy 2009, pp. 296-300.
Reference: [11] Sladký, K.: Risk-sensitive Ramsey Growth model..In: Proc. of 28th International Conference on Mathematical Methods in Economics (M. Houda and J. Friebelová, eds.) České Budějovice 2010.
Reference: [12] Shapley, L. S.: Stochastic games..Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 1095-1100. Zbl 1180.91042, MR 0061807, 10.1073/pnas.39.10.1095
Reference: [13] Wal, J. van der: Discounted Markov games: Successive approximation and stopping times..Internat. J. Game Theory 6 (1977), 11-22. MR 0456797, 10.1007/BF01770870
Reference: [14] Wal, J. van der: Discounted Markov games: Generalized policy iteration method..J. Optim. Theory Appl. 25 (1978), 125-138. MR 0526244, 10.1007/BF00933260
Reference: [15] White, D. J.: Real applications of Markov decision processes..Interfaces 15 (1985), 73-83. 10.1287/inte.15.6.73
Reference: [16] White, D. J.: Further real applications of Markov decision processes..Interfaces 18 (1988), 55-61. 10.1287/inte.18.5.55
Reference: [17] Zachrisson, L. E.: Markov games..In: Advances in Game Theory (M. Dresher, L. S.Shapley and A. W. Tucker, eds.), Princeton Univ. Press, Princeton 1964, pp. 211-253. Zbl 0126.36507, MR 0170729


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