# Article

 Title: Nash {\Large $\epsilon$}-equilibria for stochastic games with total reward functions: an approach through Markov decision processes (English) Author: González-Padilla, Francisco J. Author: Montes-de-Oca, Raúl Language: English Journal: Kybernetika ISSN: 0023-5954 (print) ISSN: 1805-949X (online) Volume: 55 Issue: 1 Year: 2019 Pages: 152-165 Summary lang: English . Category: math . Summary: The main objective of this paper is to find structural conditions under which a stochastic game between two players with total reward functions has an $\epsilon$-equilibrium. To reach this goal, the results of Markov decision processes are used to find $\epsilon$-optimal strategies for each player and then the correspondence of a better answer as well as a more general version of Kakutani's Fixed Point Theorem to obtain the $\epsilon$-equilibrium mentioned. Moreover, two examples to illustrate the theory developed are presented. (English) Keyword: stochastic games Keyword: Nash equilibrium Keyword: Markov decision processes Keyword: total rewards MSC: 90C40 MSC: 91A15 MSC: 91A50 idZBL: Zbl 07088883 idMR: MR3935419 DOI: 10.14736/kyb-2019-1-0152 . Date available: 2019-05-07T11:15:02Z Last updated: 2020-02-27 Stable URL: http://hdl.handle.net/10338.dmlcz/147710 . Reference: [1] Aliprantis, C. D., Border, K. C.: Infinite Dimensional Analysis..Springer 2006. Zbl 1156.46001, MR 2378491 Reference: [2] Ash, R. B.: Real Analysis and Probability..Academic Press, New York 1972. MR 0435320 Reference: [3] Bartle, R.: The Elements of Real Analysis..John Wiley and Sons, Inc. 1964. MR 0393369, 10.1002/zamm.19650450519 Reference: [4] Cavazos-Cadena, R., Montes-de-Oca, R.: Optimal and nearly optimal policies in Markov decision chains with nonnegative rewards and risk-sensitive expected total-reward criterion..In: Markov Processes and Controlled Markov Chains 2002 (Z. Hou, J. A. Filar and A. Chen, eds.), Kluwer Academic Publishers, pp. 189-221. MR 2022426, 10.1007/978-1-4613-0265-0_11 Reference: [5] Filar, J., Vrieze, K.: Competitive Markov Decision Processes..Springer-Verlag, New York 1997. MR 1418636 Reference: [6] Habil, E. D.: Double sequences and double series..The Islamic Univ. J., Series of Natural Studies and Engineering 14 (2006), 1-32. (This reference is available at the Islamic University Journal's site: http://journal.iugaza.edu.ps/index.php/IUGNS/article/view/1594/1525.) Reference: [7] Hernández-Lerma, O., Lasserre, J. B.: Discrete-Time Markov Control Processes: Basic Optimality Criteria..Springer-Verlag, New York 1996. Zbl 0840.93001, MR 1363487, 10.1007/978-1-4612-0729-0 Reference: [8] Hordijk, A.: Dynamic Programming and Markov Potential Theory..Mathematical Centre Tracts 51, Amsterdam 1974. MR 0432227 Reference: [9] Jaśkiewicz, A., Nowak, A. S.: Stochastic games with unbounded payoffs: Applications to robust control in Economics..Dyn. Games Appl. 1 (2011), 2, 253-279. MR 2804096, 10.1007/s13235-011-0013-8 Reference: [10] Kakutani, S.: A generalization of Brouwer's fixed point theorem..Duke Math. J. 8 (1942), 457-459. MR 0004776, 10.1215/s0012-7094-41-00838-4 Reference: [11] Kelley, J. L.: General Topology..Springer, New York 1955. MR 0070144 Reference: [12] Köthe, G.: Topological Vector Spaces I..Springer-Verlag, 1969. MR 0248498 Reference: [13] Puterman, M.: Markov Decision Processes..John Wiley and Sons, Inc. New Jersey 1994. Zbl 1184.90170, MR 1270015 Reference: [14] 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: [15] Thuijsman, F.: Optimality and Equilibria in Stochastic Games..CW1 Tract-82, Amsterdam 1992. MR 1171220 Reference: [16] Zeidler, E.: Nonlinear Functional Analysis and its Applications..Springer-Verlag, New York Inc. 1988. MR 0816732 .

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