Abstract
We consider repeated games where the number of repetitions θ is unknown. The information about the uncertain duration can change during the play of the game. This is described by an uncertain duration process Θ that defines the probability law of the signals that players receive at each stage about the duration. To each repeated game Γ and uncertain duration process Θ is associated the Θ-repeated game ΓΘ. A public uncertain duration process is one where the uncertainty about the duration is the same for all players. We establish a recursive formula for the value V Θ of a repeated two-person zero-sum game ΓΘ with a public uncertain duration process Θ. We study asymptotic properties of the normalized value v Θ = V Θ/E(θ) as the expected duration E (θ) goes to infinity. We extend and unify several asymptotic results on the existence of lim v n and lim v λ and their equality to lim v Θ. This analysis applies in particular to stochastic games and repeated games of incomplete information.
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References
Aumann RJ, Maschler M (1995) Repeated games with incomplete information, with the collaboration of R. Stearns. MIT Press, Cambridge
Lehrer E, Sorin S (1992) A uniform Tauberian theorem in dynamic programming. Math Oper Res 17: 303–307
Mertens J-F (1971) The value of two-person zero-sum repeated games: the extensive case. Int J Game Theory 1: 217–227
Mertens J-F, Neyman A (1981) Stochastic games. Int J Game Theory 10: 53–66
Mertens J-F, Zamir S (1971) The value of two-person zero-sum repeated games with lack of information on both sides. Int J Game Theory 1: 39–64
Mertens J-F, Zamir S (1985) Formulation of Bayesian analysis for games with incomplete information. Int J Game Theory 14: 1–29
Mertens J-F, Sorin S, Zamir S (1994) Repeated games. C.O.R.E. D.P. 9420, 9421, 9422
Mertens J-F, Neyman A, Rosenberg D (2009) Absorbing games with compact action spaces. Math Oper Res 34: 257–262
Monderer D, Sorin S (1993) Asymptotic properties in dynamic programming. Int J Game Theory 22: 1–11
Neyman A (1999) Cooperation in repeated games when the number of stages is not commonly known. Econometrica 67: 45–64
Neyman A (2003a) Stochastic games: existence of the minmax. In: Neyman A, Sorin S (eds) Stochastic games and applications. NATO ASI series. Kluwer Academic Publishers, Dordrecht, pp 173–193
Neyman A (2003b) Stochastic games and nonexpansive maps. In: Neyman A, Sorin S (eds) Stochastic games and applications. NATO ASI series. Kluwer Academic Publishers, Dordrecht, pp 397–415
Neyman A (2009a) The maximal variation of martingales of probabilities and repeated games with incomplete information. DP 510, Center for the Study of Rationality, Hebrew University
Neyman A (2009b) The value of two-person zero-sum repeated games with incomplete information and uncertain duration. DP 512, Center for the Study of Rationality, Hebrew University
Neyman A (2009c) The error term in repeated games with incomplete information. DP 522, Center for the Study of Rationality, Hebrew University
Rosenberg D (1999) Zero-sum absorbing games with incomplete information on one side: asymptotic analysis. SIAM J Control Optim 39: 208–225
Rosenberg D, Sorin S (2001) An operator approach to zero-sum repeated games. Isr J Math 121: 221–246
Rosenberg D, Vieille N (2000) The maxmin of recursive games with lack of information on one side. Math Oper Res 25: 23–35
Rosenberg D, Solan E, Vieille N (2002) Blackwell optimality in Markov decision processes with partial observation. Ann Stat 30: 1178–1193
Shapley LS (1953) Stochastic games. Proc Natil Acad Sci USA 39: 1095–1100
Sorin S (2003) Operator approach to stochastic games. In: Neyman A, Sorin S (eds) Stochastic games and applications. NATO ASI series. Kluwer Academic Publishers, Dordrecht, pp 417–426
Sorin S (2004) Asymptotic properties of monotonic nonexpansive mappings. Discret Events Dyn Syst 14: 109–122
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Neyman, A., Sorin, S. Repeated games with public uncertain duration process. Int J Game Theory 39, 29–52 (2010). https://doi.org/10.1007/s00182-009-0197-y
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DOI: https://doi.org/10.1007/s00182-009-0197-y