July  2014, 1(3): 411-445. doi: 10.3934/jdg.2014.1.411

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, France

2. 

Université de Neuchâtel, Institut de Mathématiques, Emilie-Argand 11, 2000 Neuchatel, Switzerland

3. 

Department of Statistics and Operations Research, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv, Israel

Received  November 2012 Revised  September 2013 Published  July 2014

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.
Citation: Fabien Gensbittel, Miquel Oliu-Barton, Xavier Venel. Existence of the uniform value in zero-sum repeated games with a more informed controller. Journal of Dynamics & Games, 2014, 1 (3) : 411-445. doi: 10.3934/jdg.2014.1.411
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show all references

References:
[1]

MIT Press, 1995.  Google Scholar

[2]

The Annals of Mathematical Statistics, 39 (1968), 159-163. doi: 10.1214/aoms/1177698513.  Google Scholar

[3]

Proceedings of the American Mathematical Society, 89 (1983), 691-692. doi: 10.2307/2044607.  Google Scholar

[4]

Proceedings of the National Academy of Sciences of the United States of America, 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42.  Google Scholar

[5]

Vestnik Leningrad. Univ, 13 (1958), 52-59.  Google Scholar

[6]

in Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1986, 1528-1577.  Google Scholar

[7]

International Journal of Game Theory, 10 (1981), 53-66. doi: 10.1007/BF01769259.  Google Scholar

[8]

CORE Discussion Papers 9420, 9421 and 9422, Université Catholique De Louvain, Belgium, 1994. Google Scholar

[9]

International Journal of Game Theory, 1 (1971), 39-64. doi: 10.1007/BF01753433.  Google Scholar

[10]

International Journal of Game Theory 39 (2010), 29-52. doi: 10.1007/s00182-009-0197-y.  Google Scholar

[11]

International Journal of Game Theory, 37 (2008), 581-596. doi: 10.1007/s00182-008-0134-5.  Google Scholar

[12]

Mathematics of Operations Research, 31 (2006), 490-512. doi: 10.1287/moor.1060.0199.  Google Scholar

[13]

Journal of the European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254.  Google Scholar

[14]

Mathematics of Operations Research, 37 (2012), 154-179. doi: 10.1287/moor.1110.0518.  Google Scholar

[15]

Annals of Statistics, 30 (2002), 1178-1193. doi: 10.1214/aos/1031689022.  Google Scholar

[16]

SIAM Journal on Control and Optimization, 43 (2004), 86-110. doi: 10.1137/S0363012902407107.  Google Scholar

[17]

International Journal of Game Theory, 13 (1984), 201-255. doi: 10.1007/BF01769463.  Google Scholar

[18]

Mathématiques & Applications, Springer, 2002.  Google Scholar

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