González-Padilla, Francisco J.; Montes-de-Oca, Raúl Nash \(\epsilon\)-equilibria for stochastic games with total reward functions: an approach through Markov decision processes. (English) Zbl 1449.91018 Kybernetika 55, No. 1, 152-165 (2019). 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. MSC: 91A15 Stochastic games, stochastic differential games 91A05 2-person games 90C40 Markov and semi-Markov decision processes Keywords:stochastic games; Nash equilibrium; Markov decision processes; total rewards PDF BibTeX XML Cite \textit{F. J. González-Padilla} and \textit{R. Montes-de-Oca}, Kybernetika 55, No. 1, 152--165 (2019; Zbl 1449.91018) Full Text: DOI OpenURL References: [1] Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis., Springer 2006 [2] Ash, R. B., Real Analysis and Probability., Academic Press, New York 1972 [3] Bartle, R., The Elements of Real Analysis., John Wiley and Sons, Inc. 1964 [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 [5] Filar, J.; Vrieze, K., Competitive Markov Decision Processes., Springer-Verlag, New York 1997 [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.) [7] Hernández-Lerma, O.; Lasserre, J. B., Discrete-Time Markov Control Processes: Basic Optimality Criteria., Springer-Verlag, New York 1996 [8] Hordijk, A., Dynamic Programming and Markov Potential Theory., Mathematical Centre Tracts 51, Amsterdam 1974 [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 [10] Kakutani, S., A generalization of Brouwer’s fixed point theorem., Duke Math. J. 8 (1942), 457-459 [11] Kelley, J. L., General Topology., Springer, New York 1955 [12] Köthe, G., Topological Vector Spaces I., Springer-Verlag, 1969 [13] Puterman, M., Markov Decision Processes., John Wiley and Sons, Inc. New Jersey 1994 [14] Shapley, L. S., Stochastic games., Proc. Nat. Acad. Sci. U. S. A. 39 (1953), 1095-1100 [15] Thuijsman, F., Optimality and Equilibria in Stochastic Games., CW1 Tract-82, Amsterdam 1992 [16] Zeidler, E., Nonlinear Functional Analysis and its Applications., Springer-Verlag, New York Inc. 1988 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.