Abstract
The finite state space stochastic game model by Shapley [31] covered in [33] was generalized among others by Maitra and Parthasarathy [17], [33] who considered compact metric state spaces. They imposed rather strong regularity conditions on the reward and transition structure in the game and considered the discounted payoff criterion only. Their results have been generalized by many authors. For a good survey of the results which are not reported here the reader is referred to [14], [23], [25], [27], [13].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ash, R.B. (1972)Real Analysis and ProbabilityAcademic Press, New York.
Berge, C. (1963)Topological SpacesMcMillan, New York.
Bertsekas, D.P. and Shreve, S.E. (1978)Stochastic Optimal Control: The Discrete Time CaseAcademic Press, New York.
Blackwell, D., Freedman, D. and Orkin, M. (1974) The optimal reward operator in dynamic programmingAnnals of Probability2, 926–941.
Brown, L.D. and Purves, R. (1973) Measurable selections of extremaAnnals of Statistics 1902–912.
Couwenbergh, H.A.M. (1980) Stochastic games with metric state spaceInternational Journal of Game Theory9, 25–36.
Fan, K. (1953) Minimax theoremsProceedings of the National Academy of Sciences of the U.S.A.39, 42–47.
Gödel, K. (1938) The consistency of the axiom of choice and of the generalized continuum hypothesisProceedings of the National Academy of Sciences of the U.S.A.24, 556–557.
Himmelberg, C.J. (1975) Measurable relationsFundamenta Mathematicae87, 5372.
Himmelberg, C.J. and Van Vleck, F.S. (1975) Multifunctions with values in a space of probability measuresJournal of Mathematical Analysis and Applications50, 108–112.
Himmelberg, C.J., Parthasarathy, T. and Van Vleck, F.S. (1976) Optimal plans for dynamic programming problemsMathematics of Operations Research 1390–394.
Jaskiewicz, A. (2000) On strong 1-optimal policies in Markov control processes with Borel state spacesBulletin of the Polish Academy of Sciences (ser. Mathematics)48,439–450.
Jaskiewicz, A. (2002) Zero-sum semi-Markov gamesSIAM Journal on Control and Optimization41, 723–739.
Jaskiewicz, A. and Nowak, A.S. (2001) On the optimality equation for zero-sum ergodic stochastic gamesMathematical Methods of Operations Research54, 291–301.
Küenle, H.-U. (1986)Stochastische Spiele und EntscheidungsmodelleTeubnerTexte zur Mathematik 89, Teubner-Verlag, Leipzig.
Leese, S.J. (1978) Measurable selections and the uniformization of the Souslin setsAmerican Journal of Mathematics 10019–41.
Maitra, A. and Parthasarathy, T. (1970) On stochastic gamesJournal of Optimization Theory and Applications5, 289–300.
Maitra, A. and Parthasarathy, T. (1970) On stochastic games IIJournal of Optimization Theory and Applications8, 154–160.
Maitra, A. and Sudderth, W.D. (1993) Borel stochastic games with limsup payoffAnnals of Probability21, 861–885.
Maitra, A. and Sudderth, W.D. (1993) Finitely additive and measurable stochastic gamesInternational Journal of Game Theory22, 201–223.
Maitra, A. and Sudderth, W.D. (1996)Discrete Gambling and Stochastic GamesSpringer—Verlag, New York.
Nowak, A.S. (1985) Measurable selection theorems for minimax stochastic optimization problemsSIAM Journal on Control and Optimization23, 466–476.
Nowak, A.S. (1985) Universally measurable strategies in zero-sum stochastic gamesAnnals of Probability13, 269–287.
Nowak, A.S. (1986) Semicontinuous nonstationary stochastic gamesJournal of Mathematical Analysis and Applications117, 84–99.
Nowak, A.S. (1990) Semicontinuous nonstationary stochastic games IIJournal of Mathematical Analysis and Applications148, 22–43.
Nowak, A.S. (1999) Sensitive equilibria for ergodic stochastic games with countable state spacesMathematical Methods of Operations Research50, 65–76.
Nowak, A.S. (1999) Optimal strategies in a class of zero-sum ergodic stochastic gamesMathematical Methods of Operations Research50, 399–419.
Nowak, A.S. and Raghavan, T.E.S. (1991) Positive stochastic games and a theorem of Ornstein, in T.E.S. Raghavan et al. (eds.)Stochastic Games and Related Topics Essays in Honor of L.S. Shapley Kluwer Academic Publishers, Dordrecht, pp. 127–134.
Parthasarathy, K.R. (1967)Probability Measures on Metric SpacesAcademic Press, New York.
Rieder, U. (1978) On semicontinuous dynamic games, Technical Report, Department of Mathematics, University of Karlsruhe, Germany.
Shapley, L.S. (1953) Stochastic gamesProceedings of the National Academy of Sciences of the U.S.A.39, 1095–1100 (Chapter 1 in this volume).
Sorin, S. (2003) Classification and basic tools, in A. Neymanand S. Sorin (eds.),Stochastic Games and Applications, NATO Science Series C, Mathematical and Physical Sciences, Vol. 570, Kluwer Academic Publishers, Dordrecht, Chapter 3, pp. 27–35.
Sorin, S. (2003) Discounted stochastic games: The finite case, in A. Neyman and S. Sorin (eds.)Stochastic Games and ApplicationsNATO Science Series C, Mathematical and Physical Sciences, Vol. 570, Kluwer Academic Publishers, Dordrecht, Chapter 5, pp. 51–55.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this paper
Cite this paper
Nowak, A.S. (2003). Zero-Sum Stochastic Games with Borel State Spaces. In: Neyman, A., Sorin, S. (eds) Stochastic Games and Applications. NATO Science Series, vol 570. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0189-2_7
Download citation
DOI: https://doi.org/10.1007/978-94-010-0189-2_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-1493-2
Online ISBN: 978-94-010-0189-2
eBook Packages: Springer Book Archive