Existence of the uniform value in zero-sum repeated games with a more informed controller

Fabien Gensbittel; Miquel Oliu-Barton; Xavier Venel

doi:10.3934/jdg.2014.1.411

Article Contents

2014, Volume 1, Issue 3: 411-445. Doi: 10.3934/jdg.2014.1.411

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Existence of the uniform value in zero-sum repeated games with a more informed controller

1.
TSE (GREMAQ, Université Toulouse 1 Capitole), Manufacture des Tabacs, 21, Allée de Brienne, 31015 Toulouse Cedex 6
2.
Université de Neuchâtel, Institut de Mathématiques, Emilie-Argand 11, 2000 Neuchatel
3.
Department of Statistics and Operations Research, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv

Received: October 31, 2012

Revised: August 31, 2013

Published: June 2014

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Abstract

We prove that in a two-player zero-sum repeated game where one of the players, say player $1$, is more informed than his opponent and controls the evolution of information on the state, the uniform value exists. This result extends previous results on Markov decision processes with partial observation (Rosenberg, Solan, Vieille [15]), and repeated games with an informed controller (Renault [14]). Our formal definition of a more informed player is more general than the inclusion of signals, allowing therefore for imperfect monitoring of actions. We construct an auxiliary stochastic game whose state space is the set of second order beliefs of player $2$ (beliefs about beliefs of player $1$ on the state variable of the original game) with perfect monitoring and we prove it has a value by using a result of Renault [14]. A key element in this work is to prove that player $1$ can use strategies of the auxiliary game in the original game in our general framework, from which we deduce that the value of the auxiliary game is also the value of our original game by using classical arguments.

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Mathematics Subject Classification: Primary: 91A20; Secondary: 91A26, 91A15.

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