Abstract
We survey old and new results concerning stochastic games with signals and finitely many states, actions, and signals. We state Mertens’ conjectures regarding the existence of the asymptotic value and its characterization, and present Ziliotto’s (Ann Probab, 2013, to appear) counter, example for these conjectures.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aumann RJ, Maschler M (1995) Repeated games with incomplete information. MIT Press, Cambridge
Bewley T, Kohlberg E (1976) The asymptotic theory of stochastic games. Math Oper Res 1(3):197–208
Blackwell D, Ferguson TS (1968) The big match. Ann Math Stat 39(1):159–163
Coulomb J-M (2003) Stochastic games without perfect monitoring. Int J Game Theory 32(1):73–96
Dutta P (1995) A folk theorem for stochastic games. J Econ Theory 66(1):1–32
Flesch J, Thuijsman F, Vrieze K (1997) Cyclic Markov equilibria in stochastic games. Int J Game Theory 26(3):303–314,
Fudenberg D, Levine D (1991) An approximate folk theorem with imperfect private information. J Econ Theory 54(1):26–47
Fudenberg D, Yamamoto Y (2011) The folk theorem for irreducible stochastic games with imperfect public monitoring. J Econ Theory 146(4):1664–1683
Gensbittel F, Oliu-Barton M, Venel X (2014) Existence of the uniform value in repeated games with a more informed controller. J Dyn Games 1:411–445
Gillette D (1957) Stochastic games with zero stop probabilities. Contrib Theory Games 3:179–187
Hörner J, Sugaya T, Takahashi S, Vieille N (2011) Recursive methods in discounted stochastic games: an algorithm for δ → 1 and a folk theorem. Econometrica 79(4):1277–1318
Lehrer E, Sorin S (1992) A uniform Tauberian theorem in dynamic programming. Math Oper Res 17(2):303–307
Mertens JF (1986) Repeated games. In: Proceedings of the international congress of mathematicians, Berkeley, CA, pp 1528–1577
Mertens JF, Neyman A (1981) Stochastic games. Int J Game Theory 10(2):53–66
Mertens JF, Zamir S (1971) The value of two-person zero-sum repeated games with lack of information on both sides. Int J Game Theory 1(1):39–64
Mertens JF, Sorin S, Zamir S (1994) Repeated games. CORE DP 9420–9422
Neyman A (2008) Existence of optimal strategies in markov games with incomplete information. Int J Game Theory 37(4):581–596
Neyman A, Sorin S (2010) Repeated games with public uncertain duration process. Int J Game Theory 39(1):29–52
Renault J (2006) The value of markov chain games with lack of information on one side. Math Oper Res 31(3):490–512
Renault J (2012) The value of repeated games with an informed controller. Math Oper Res 37(1):154–179
Renault J, Ziliotto B (2014) Hidden stochastic games and limit equilibrium payoffs. arXiv preprint arXiv:1407.3028
Rosenberg D (2000) Zero-sum absorbing games with incomplete information on one side: asymptotic analysis. SIAM J Control Optim 39(1):208–225
Rosenberg D, Vieille N (2000) The maxmin of recursive games with incomplete information on one side. Math Oper Res 25(1):23–35
Rosenberg D, Solan E, Vieille N (2002) Blackwell optimality in markov decision processes with partial observation. Ann Stat 30:1178–1193
Rosenberg D, Solan E, Vieille N (2003) The maxmin value of stochastic games with imperfect monitoring. Int J Game Theory 32(1):133–150
Rosenberg D, Solan E, Vieille N (2004) Stochastic games with a single controller and incomplete information. SIAM J Control Optim 43(1):86–110
Shapley LS (1953) Stochastic games. Proc Natl Acad Sci USA 39(10):1095–1100
Solan E (1999) Three-player absorbing games. Math Oper Res 24(3):669–698
Solan E, Vieille N (2002) Correlated equilibrium in stochastic games. Games Econ Behav 38(2):362–399
Sorin S (1984) Big match with lack of information on one side (part i). Int J Game Theory 13(4):201–255
Sorin S (1985) Big match with lack of information on one side (part ii). Int J Game Theory 14(3):173–204
Sorin S (2002) A first course on zero-sum repeated games. In: Mathématiques et applications, vol 37. Springer, Berlin
Venel X (2014) Commutative stochastic games. Math Oper Res 40(2):403–428
Vieille N (2000a) Two-player stochastic games i: a reduction. Isr J Math 119(1):55–91
Vieille N (2000b) Two-player stochastic games ii: the case of recursive games. Isr J Math 119(1):93–126
von Neumann J (1928) Zur theorie der gesellschaftsspiele. Math Ann 100:295–320
Vrieze OJ, Thuijsman F (1989) On equilibria in repeated games with absorbing states. Int J Game Theory 18(3):293–310
Yamamoto Y (2015) Stochastic games with hidden states, preprint, PIER working paper
Ziliotto B (2013) Zero-sum repeated games: counterexamples to the existence of the asymptotic value and the conjecture maxmin= lim v (n). Ann Probab. arXiv preprint arXiv:1305.4778 (to appear)
Ziliotto B (2015) A tauberian theorem for nonexpansive operators and applications to zero-sum stochastic games. arXiv preprint arXiv:1501.06525, Mathematics of Operations Research (to appear)
Acknowledgements
Solan acknowledges the support of grant #323/13 of the Israel Science Foundation. Ziliotto acknowledges the support of the ANR Jeudy (ANR-10-BLAN 0112) and the GDR 2932.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Solan, E., Ziliotto, B. (2016). Stochastic Games with Signals. In: Thuijsman, F., Wagener, F. (eds) Advances in Dynamic and Evolutionary Games. Annals of the International Society of Dynamic Games, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28014-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-28014-1_4
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-28012-7
Online ISBN: 978-3-319-28014-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)