| 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:
|
http://hdl.handle.net/10338.dmlcz/143097 |
| . |
| 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 |
| . |
