# American Institute of Mathematical Sciences

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 and Games, 2014, 1 (3) : 411-445. doi: 10.3934/jdg.2014.1.411
##### References:
 [1] R. J. Aumann and M. Maschler, Repeated Games with Incomplete Information, with the collaboration of R. Stearns, MIT Press, 1995. [2] D. Blackwell and T. S. Fergusson, The big match, The Annals of Mathematical Statistics, 39 (1968), 159-163. doi: 10.1214/aoms/1177698513. [3] D. Blackwell and L. E. Dubins, An extension of Skorohod's almost sure representation theorem, Proceedings of the American Mathematical Society, 89 (1983), 691-692. doi: 10.2307/2044607. [4] K. Fan, Minimax theorems, Proceedings of the National Academy of Sciences of the United States of America, 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42. [5] L. Kantorovich and G. S. Rubinstein, On a space of totally additive functions, Vestnik Leningrad. Univ, 13 (1958), 52-59. [6] J. F. Mertens, Repeated games, in Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1986, 1528-1577. [7] J. F. Mertens and A. Neyman, Stochastic games, International Journal of Game Theory, 10 (1981), 53-66. doi: 10.1007/BF01769259. [8] J. F. Mertens, S. Sorin and S. Zamir, Repeated Games, CORE Discussion Papers 9420, 9421 and 9422, Université Catholique De Louvain, Belgium, 1994. [9] J. F. Mertens and S. Zamir, The value of two-person zero-sum repeated games with lack of information on both sides, International Journal of Game Theory, 1 (1971), 39-64. doi: 10.1007/BF01753433. [10] A. Neyman and S. Sorin, Repeated games with public uncertain duration process, International Journal of Game Theory 39 (2010), 29-52. doi: 10.1007/s00182-009-0197-y. [11] A. Neyman, Existence of optimal strategies in Markov games with incomplete information, International Journal of Game Theory, 37 (2008), 581-596. doi: 10.1007/s00182-008-0134-5. [12] J. Renault, The value of Markov chain games with lack of information on one side, Mathematics of Operations Research, 31 (2006), 490-512. doi: 10.1287/moor.1060.0199. [13] J. Renault, Uniform value in Dynamic Programming, Journal of the European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254. [14] J. Renault, The value of repeated games with an informed controller, Mathematics of Operations Research, 37 (2012), 154-179. doi: 10.1287/moor.1110.0518. [15] D. Rosenberg, E. Solan and N. Vieille, Blackwell optimality in Markov decision processes with partial observation, Annals of Statistics, 30 (2002), 1178-1193. doi: 10.1214/aos/1031689022. [16] D. Rosenberg, E. Solan and N. Vieille, Stochastic games with a single controller and incomplete information, SIAM Journal on Control and Optimization, 43 (2004), 86-110. doi: 10.1137/S0363012902407107. [17] S. Sorin, "Big Match'' with lack of information on one side (part I), International Journal of Game Theory, 13 (1984), 201-255. doi: 10.1007/BF01769463. [18] S. Sorin, A First Course on Zero-Sum Repeated games, Mathématiques & Applications, Springer, 2002.

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##### References:
 [1] R. J. Aumann and M. Maschler, Repeated Games with Incomplete Information, with the collaboration of R. Stearns, MIT Press, 1995. [2] D. Blackwell and T. S. Fergusson, The big match, The Annals of Mathematical Statistics, 39 (1968), 159-163. doi: 10.1214/aoms/1177698513. [3] D. Blackwell and L. E. Dubins, An extension of Skorohod's almost sure representation theorem, Proceedings of the American Mathematical Society, 89 (1983), 691-692. doi: 10.2307/2044607. [4] K. Fan, Minimax theorems, Proceedings of the National Academy of Sciences of the United States of America, 39 (1953), 42-47. doi: 10.1073/pnas.39.1.42. [5] L. Kantorovich and G. S. Rubinstein, On a space of totally additive functions, Vestnik Leningrad. Univ, 13 (1958), 52-59. [6] J. F. Mertens, Repeated games, in Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1986, 1528-1577. [7] J. F. Mertens and A. Neyman, Stochastic games, International Journal of Game Theory, 10 (1981), 53-66. doi: 10.1007/BF01769259. [8] J. F. Mertens, S. Sorin and S. Zamir, Repeated Games, CORE Discussion Papers 9420, 9421 and 9422, Université Catholique De Louvain, Belgium, 1994. [9] J. F. Mertens and S. Zamir, The value of two-person zero-sum repeated games with lack of information on both sides, International Journal of Game Theory, 1 (1971), 39-64. doi: 10.1007/BF01753433. [10] A. Neyman and S. Sorin, Repeated games with public uncertain duration process, International Journal of Game Theory 39 (2010), 29-52. doi: 10.1007/s00182-009-0197-y. [11] A. Neyman, Existence of optimal strategies in Markov games with incomplete information, International Journal of Game Theory, 37 (2008), 581-596. doi: 10.1007/s00182-008-0134-5. [12] J. Renault, The value of Markov chain games with lack of information on one side, Mathematics of Operations Research, 31 (2006), 490-512. doi: 10.1287/moor.1060.0199. [13] J. Renault, Uniform value in Dynamic Programming, Journal of the European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254. [14] J. Renault, The value of repeated games with an informed controller, Mathematics of Operations Research, 37 (2012), 154-179. doi: 10.1287/moor.1110.0518. [15] D. Rosenberg, E. Solan and N. Vieille, Blackwell optimality in Markov decision processes with partial observation, Annals of Statistics, 30 (2002), 1178-1193. doi: 10.1214/aos/1031689022. [16] D. Rosenberg, E. Solan and N. Vieille, Stochastic games with a single controller and incomplete information, SIAM Journal on Control and Optimization, 43 (2004), 86-110. doi: 10.1137/S0363012902407107. [17] S. Sorin, "Big Match'' with lack of information on one side (part I), International Journal of Game Theory, 13 (1984), 201-255. doi: 10.1007/BF01769463. [18] S. Sorin, A First Course on Zero-Sum Repeated games, Mathématiques & Applications, Springer, 2002.
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