Abstract
In 1976, Bewley and Kohlberg proved that the discounted values v λ of finite zero-sum stochastic games have a limit, as λ tends to 0, using the Tarski–Seidenberg elimination theorem from real algebraic geometry. This was a fundamental step in the development of the theory of stochastic games. The current paper provides a new and direct proof for this result, relying on the explicit description of asymptotically optimal strategies. Both approaches can be used to obtain the existence of the uniform value using the construction from Mertens and Neyman (1981).
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
References
T. Bewley and E. Kohlberg, “The asymptotic theory of stochastic games”, Mathematics of Operation Research 1 (1976), 197–208.
J. Bolte, S. Gaubert, and G. Vigeral, “Definable zero-sum stochastic games”, Math. Oper. Res. 40 (1) (2015), 171–191.
J.F. Mertens and A. Neyman, “Stochastic games”, International Journal of Game Theory 10 (1981), 53–66.
J.F. Mertens, A. Neyman, and D. Rosenberg, “Absorbing games with compact action spaces”, Mathematics of Operation Research 34 (2009), 257–262.
L.S. Shapley, “Stochastic games”, Proc. Nat. Acad. Sci. 39 (1953), 1095–1100.
E. Solan and N. Vieille, “Computing uniformly optimal strategies in two-player stochastic games”, Econ. Theory 42 (2010), 237–253.
G. Vigeral, “A zero-sum stochastic game with compact action sets and no asymptotic value”, Dynamic Games and Applications 3 (2) (2013), 172–186.
B. Ziliotto, “Zero-sum repeated games: counterexamples to the existence of the asymptotic value and the conjecture \(\mathop{\mathrm{maxmin}}\nolimits =\mathop{ \mathrm{lim}}\nolimits v_{n}\)”, Ann. Probab. 44 (2) (2016), 1107–1133.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Oliu-Barton, M. (2017). The Asymptotic Value in Finite Stochastic Games. In: Díaz, J., Kirousis, L., Ortiz-Gracia, L., Serna, M. (eds) Extended Abstracts Summer 2015. Trends in Mathematics(), vol 6. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-51753-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-51753-7_14
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-51752-0
Online ISBN: 978-3-319-51753-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)