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Int J Game Theory (2010) 39:2952
DOI 10.1007/s00182-009-0197-y
Accepted: 27 October 2009 / Published online: 13 November 2009 Springer-Verlag 2009
Abstract We consider repeated games where the number of repetitions is unknown. The information about the uncertain duration can change during the play of the game. This is described by an uncertain duration process that denes the probability law of the signals that players receive at each stage about the duration. To each repeated game and uncertain duration process is associated the -repeated game . 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 of a repeated two-person zero-sum game with a public uncertain duration process . We study asymptotic properties of the normalized value v = V /E()
as the expected duration E() goes to innity. We extend and unify several asymptotic results on the existence of lim vn and lim v and their equality to lim v . This analysis applies in particular to stochastic games and repeated games of incomplete information.
Keywords Repeated games Uncertain duration Recursive formula Asymptotic
analysis Stochastic games Incomplete information
A. Neyman
Institute of Mathematics, Center for the Study of Rationality,The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel e-mail: [email protected]
S. Sorin (B)
Equipe Combinatoire and Optimisation, CNRS FRE 3232, Facult de Mathmatiques, Universit P. et M. Curie-Paris 6, 175 rue du Chevaleret, 75013 Paris, Francee-mail: [email protected]
S. Sorin
Laboratoire dEconomtrie, Ecole Polytechnique, Paris, France
Repeated games with public uncertain duration process
Abraham Neyman Sylvain Sorin
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30 A. Neyman, S. Sorin
1 Introduction
We consider repeated games with an uncertain number of stages. For simplicity, we describe in detail the case of two-person repeated games with symmetric uncertainty about the number of stages. (The extension to general n-person games and/or asymmetric uncertainty about the number of stages is straightforward.) The model consists of two basic components:(a) First, a repeated game is given and described as follows. M is a state space on which a family of normal form two-person games is dened by move spaces I and J for Player 1 and Player 2 respectively, and a payoff function g = (g1, g2) from
M IJ...