Markov stopping games with an absorbing state and total reward criterion

Rolando Cavazos-Cadena; Luis Rodríguez-Gutiérrez; Dulce María Sánchez-Guillermo

Kybernetika (2021)

  • Volume: 57, Issue: 3, page 474-492
  • ISSN: 0023-5954

Abstract

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This work is concerned with discrete-time zero-sum games with Markov transitions on a denumerable space. At each decision time player II can stop the system paying a terminal reward to player I, or can let the system to continue its evolution. If the system is not halted, player I selects an action which affects the transitions and receives a running reward from player II. Assuming the existence of an absorbing state which is accessible from any other state, the performance of a pair of decision strategies is measured by the total expected reward criterion. In this context it is shown that the value function of the game is characterized by an equilibrium equation, and the existence of a Nash equilibrium is established.

How to cite

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Cavazos-Cadena, Rolando, Rodríguez-Gutiérrez, Luis, and Sánchez-Guillermo, Dulce María. "Markov stopping games with an absorbing state and total reward criterion." Kybernetika 57.3 (2021): 474-492. <http://eudml.org/doc/297464>.

@article{Cavazos2021,
abstract = {This work is concerned with discrete-time zero-sum games with Markov transitions on a denumerable space. At each decision time player II can stop the system paying a terminal reward to player I, or can let the system to continue its evolution. If the system is not halted, player I selects an action which affects the transitions and receives a running reward from player II. Assuming the existence of an absorbing state which is accessible from any other state, the performance of a pair of decision strategies is measured by the total expected reward criterion. In this context it is shown that the value function of the game is characterized by an equilibrium equation, and the existence of a Nash equilibrium is established.},
author = {Cavazos-Cadena, Rolando, Rodríguez-Gutiérrez, Luis, Sánchez-Guillermo, Dulce María},
journal = {Kybernetika},
keywords = {non-expansive operator; monotonicity property; fixed point; equilibrium equation; hitting time; bounded rewards},
language = {eng},
number = {3},
pages = {474-492},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Markov stopping games with an absorbing state and total reward criterion},
url = {http://eudml.org/doc/297464},
volume = {57},
year = {2021},
}

TY - JOUR
AU - Cavazos-Cadena, Rolando
AU - Rodríguez-Gutiérrez, Luis
AU - Sánchez-Guillermo, Dulce María
TI - Markov stopping games with an absorbing state and total reward criterion
JO - Kybernetika
PY - 2021
PB - Institute of Information Theory and Automation AS CR
VL - 57
IS - 3
SP - 474
EP - 492
AB - This work is concerned with discrete-time zero-sum games with Markov transitions on a denumerable space. At each decision time player II can stop the system paying a terminal reward to player I, or can let the system to continue its evolution. If the system is not halted, player I selects an action which affects the transitions and receives a running reward from player II. Assuming the existence of an absorbing state which is accessible from any other state, the performance of a pair of decision strategies is measured by the total expected reward criterion. In this context it is shown that the value function of the game is characterized by an equilibrium equation, and the existence of a Nash equilibrium is established.
LA - eng
KW - non-expansive operator; monotonicity property; fixed point; equilibrium equation; hitting time; bounded rewards
UR - http://eudml.org/doc/297464
ER -

References

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  1. Altman, E., Shwartz, A., , In: Annals of Dynamic Games (V. Gaitsgory, J. Filar and K. Mizukami, eds. 6, Birkhauser, Boston 2000, pp. 213-221. Zbl0957.91014DOI
  2. Atar, R., Budhiraja, A., , Ann. Probab.2 (2010), 498-531. DOI
  3. Bäuerle, N., Rieder, U., , Stoch. Proc. Appl. 127 (2017), 622-642. DOI
  4. Bielecki, T., Hernández-Hernández, D., Pliska, S. R., , Math. Methods Oper. Res. 50 (1999), 167-188. Zbl0959.91029DOI
  5. Cavazos-Cadena, R., Hernández-Hernández, D., Nash equilibria in a class of Markov stopping games., Kybernetika 48 (2012), 1027-1044. 
  6. Filar, J. A., Vrieze, O. J., Competitive Markov Decision Processes., Springer, Berlin 1996. 
  7. Hernández-Lerma, O., Adaptive Markov Control Processes., Springer, New York 1989. Zbl0677.93073
  8. Hernández-Lerma, O., Lasserre, J. B., Discrete-Time Markov Control Processes: Basic Optimality Criteria., Springer, New York 1996. Zbl0840.93001
  9. Kolokoltsov, V. N., Malafeyev, O. A., Understanding Game Theory., World Scientific, Singapore 2010. Zbl1189.91001
  10. Martínez-Cortés, V. M., , Kybernetika 57 (2021), 1-14. DOI
  11. Peskir, G., , Math. Finance 15 (2005), 169-181. Zbl1109.91028DOI
  12. Peskir, G., Shiryaev, A., Optimal Stopping and Free-Boundary Problems., Birkhauser, Boston 2010. Zbl1115.60001
  13. Piunovskiy, A. B., Examples in Markov Decision Processes., Imperial College Press, London 2013. 
  14. Puterman, M., Markov Decision Processes., Wiley, New York 1994. Zbl1184.90170
  15. Shapley, L. S., 10.1073/pnas.39.10.1953, Proc. Natl. Acad. Sci. USA 39 (1953), 1095-1100. Zbl1180.91042DOI10.1073/pnas.39.10.1953
  16. Shiryaev, A., Optimal Stopping Rules., Springer, New York 1978. Zbl1138.60008
  17. 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. 
  18. Sladký, K., Risk-sensitive Ramsey growth model., In: Proc. 28th International Conference on Mathematical Methods in Economics (M. Houda and J. Friebelová, eds.) České Budějovice 2010, pp. 560-565. 
  19. Sladký, K., , Kybernetika 54 (2018), 1218-1230. DOI
  20. White, D. J., , Interfaces 15 (1985), 73-83. DOI
  21. White, D. J., , Interfaces 18 (1988), 55-61. DOI
  22. White, D. J., , J. Opl. Res. Soc. 44 (1993), 1073-1096. DOI
  23. Zachrisson, L. E., Markov Games., In: Advances in Game Theory (M. Dresher, L. S. Shapley and A. W. Tucker, eds.), Princeton Univ. Press, Princeton N.J. 1964, pp. 211-253. 

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