Geodesic
Nash Equilibria in a class of Markov stopping games
Kybernetika, Tome 48 (2012) no. 5, pp. 1027-1044 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 91A10, 91A15
Keywords: zero-sum stopping game; equality of the upper and lower value functions; contractive operator; hitting time; stationary strategy 
@article{KYB_2012_48_5_a13,
     author = {Cavazos-Cadena, Rolando and Hern\'andez-Hern\'andez, Daniel},
     title = {Nash {Equilibria} in a class of {Markov} stopping games},
     journal = {Kybernetika},
     pages = {1027--1044},
     year = {2012},
     volume = {48},
     number = {5},
     mrnumber = {3086867},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2012_48_5_a13/}
}
TY  - JOUR
AU  - Cavazos-Cadena, Rolando
AU  - Hernández-Hernández, Daniel
TI  - Nash Equilibria in a class of Markov stopping games
JO  - Kybernetika
PY  - 2012
SP  - 1027
EP  - 1044
VL  - 48
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/KYB_2012_48_5_a13/
LA  - en
ID  - KYB_2012_48_5_a13
ER  - 
%0 Journal Article
%A Cavazos-Cadena, Rolando
%A Hernández-Hernández, Daniel
%T Nash Equilibria in a class of Markov stopping games
%J Kybernetika
%D 2012
%P 1027-1044
%V 48
%N 5
%U http://geodesic.mathdoc.fr/item/KYB_2012_48_5_a13/
%G en
%F KYB_2012_48_5_a13
Cavazos-Cadena, Rolando; Hernández-Hernández, Daniel. Nash Equilibria in a class of Markov stopping games. Kybernetika, Tome 48 (2012) no. 5, pp. 1027-1044. http://geodesic.mathdoc.fr/item/KYB_2012_48_5_a13/

[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. | MR | Zbl

[2] Atar, R., Budhiraja, A.: A stochastic differential game for the inhomogeneous infinty-Laplace equation. Ann. Probab. 2 (2010), 498-531. | DOI | MR

[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. | DOI | MR | Zbl

[4] Dynkin, E. B.: The optimum choice for the instance for stopping Markov process. Soviet. Math. Dokl. 4 (1963), 627-629.

[5] Kolokoltsov, V. N., Malafeyev, O. A.: Understanding Game Theory. World Scientific, Singapore 2010. | MR | Zbl

[6] Peskir, G.: On the American option problem. Math. Finance 15 (2010), 169-181. | DOI | MR | Zbl

[7] Peskir, G., Shiryaev, A.: Optimal Stopping and Free-Boundary Problems. Birkhauser, Boston 2010. | MR | Zbl

[8] Puterman, M.: Markov Decision Processes. Wiley, New York 1994. | MR | Zbl

[9] Shiryaev, A.: Optimal Stopping Rules. Springer, New York 1978. | MR | Zbl

[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.

[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.

[12] Shapley, L. S.: Stochastic games. Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 1095-1100. | DOI | MR | Zbl

[13] Wal, J. van der: Discounted Markov games: Successive approximation and stopping times. Internat. J. Game Theory 6 (1977), 11-22. | DOI | MR

[14] Wal, J. van der: Discounted Markov games: Generalized policy iteration method. J. Optim. Theory Appl. 25 (1978), 125-138. | DOI | MR

[15] White, D. J.: Real applications of Markov decision processes. Interfaces 15 (1985), 73-83. | DOI

[16] White, D. J.: Further real applications of Markov decision processes. Interfaces 18 (1988), 55-61. | DOI

[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. | MR | Zbl