Pure and Random strategies in differential game with incomplete informations

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July 2014, 1(3): 363-375. doi: 10.3934/jdg.2014.1.363

Pure and Random strategies in differential game with incomplete informations

Pierre Cardaliaguet ^1, , Chloé Jimenez ^2, and Marc Quincampoix ^2,

1.	CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France
2.	Laboratoire de Mathématiques de Bretagne Atlantique, CNRS-UMR 6205, Université de Brest, 6, avenue Victor Le Gorgeu, CS 93837, 29238 Brest cedex 3, France, France

Received November 2012 Revised May 2013 Published July 2014

Abstract
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We investigate a two players zero sum differential game with incomplete information on the initial state: The first player has a private information on the initial state while the second player knows only a probability distribution on the initial state. This could be view as a generalization to differential games of the famous Aumann-Maschler framework for repeated games. In an article of the first author, the existence of the value in random strategies was obtained for a finite number of initial conditions (the probability distribution is a finite combination of Dirac measures). The main novelty of the present work consists in : first extending the result on the existence of a value in random strategies for infinite number of initial conditions and second - and mainly - proving the existence of a value in pure strategies when the initial probability distribution is regular enough (without atoms).

Keywords: Wasserstein distance., zero-sum games, incomplete information, Differential games, purification.

Mathematics Subject Classification: Primary: 49N70; Secondary: 91A0.

Citation: Pierre Cardaliaguet, Chloé Jimenez, Marc Quincampoix. Pure and Random strategies in differential game with incomplete informations. Journal of Dynamics and Games, 2014, 1 (3) : 363-375. doi: 10.3934/jdg.2014.1.363

References:

[1]	L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical Aspects of Evolving Interfaces, CIME Summer School in Madeira, Vol. 1812, Springer, 2003. doi: 10.1007/978-3-540-39189-0_1.
[2]	R. J. Aumann, Mixed and behavior strategies in infinite extensive games, in Advances in Game Theory, Princeton Univ. Press, Princeton, N.J., 1964, 627-650.
[3]	R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, MA, 1995.
[4]	R. Buckdahn, P. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory, Dynamic Games Applications, 1 (2011), 74-114. doi: 10.1007/s13235-010-0005-0.
[5]	R. Buckdahn, J. Li and M. Quincampoix, Value function of differential games without isaacs conditions. An approach with non-anticipative mixed strategies, Internat. J. of Game Theory, 42 (2013), 989-1020. doi: 10.1007/s00182-012-0351-9.
[6]	P. Cardaliaguet, Differential games with asymmetric information, SIAM J. Control Optim., 46 (2007), 816-838. doi: 10.1137/060654396.
[7]	P. Cardaliaguet and M. Quincampoix, Deterministic differential games under probability knowledge of initial condition, Int. Game Theory Rev., 10 (2008), 1-16. doi: 10.1142/S021919890800173X.
[8]	P. Cardaliaguet and C. Rainer, Stochastic differential games with assymetric information, Appl. Math. Optim., 59 (2009), 1-36. doi: 10.1007/s00245-008-9042-0.
[9]	P. Cardaliaguet and C. Rainer, Games with incomplete information in continuous time and for continuous types, Dyn. Games Appl., 2 (2012), 206-227. doi: 10.1007/s13235-012-0043-x.
[10]	C. Dellacherie and P. A. Meyer, Probabilities and Potential, North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1978.
[11]	J. F. Mertens, S. Sorin and S. Zamir, Repeated Games, CORE Discussion Papers 9420, 9421, 9422, 1994. doi: 10.1057/9780230226203.3424.
[12]	A. Pratelli, On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation, Ann. Inst. H. Poincaré Probab. Statist., 43 (2007), 1-13. doi: 10.1016/j.anihpb.2005.12.001.
[13]	D. Schmeidler, Equilibrium points of nonatomic games, Journal of Statistical Physics, 7 (1973), 295-300. doi: 10.1007/BF01014905.
[14]	C. Villani, Topics in Optimal Transportation, Graduate studies in Mathematics, Vol. 58, AMS, 2003. doi: 10.1007/b12016.

show all references

References:

[1]	L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical Aspects of Evolving Interfaces, CIME Summer School in Madeira, Vol. 1812, Springer, 2003. doi: 10.1007/978-3-540-39189-0_1.
[2]	R. J. Aumann, Mixed and behavior strategies in infinite extensive games, in Advances in Game Theory, Princeton Univ. Press, Princeton, N.J., 1964, 627-650.
[3]	R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information, MIT Press, Cambridge, MA, 1995.
[4]	R. Buckdahn, P. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory, Dynamic Games Applications, 1 (2011), 74-114. doi: 10.1007/s13235-010-0005-0.
[5]	R. Buckdahn, J. Li and M. Quincampoix, Value function of differential games without isaacs conditions. An approach with non-anticipative mixed strategies, Internat. J. of Game Theory, 42 (2013), 989-1020. doi: 10.1007/s00182-012-0351-9.
[6]	P. Cardaliaguet, Differential games with asymmetric information, SIAM J. Control Optim., 46 (2007), 816-838. doi: 10.1137/060654396.
[7]	P. Cardaliaguet and M. Quincampoix, Deterministic differential games under probability knowledge of initial condition, Int. Game Theory Rev., 10 (2008), 1-16. doi: 10.1142/S021919890800173X.
[8]	P. Cardaliaguet and C. Rainer, Stochastic differential games with assymetric information, Appl. Math. Optim., 59 (2009), 1-36. doi: 10.1007/s00245-008-9042-0.
[9]	P. Cardaliaguet and C. Rainer, Games with incomplete information in continuous time and for continuous types, Dyn. Games Appl., 2 (2012), 206-227. doi: 10.1007/s13235-012-0043-x.
[10]	C. Dellacherie and P. A. Meyer, Probabilities and Potential, North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1978.
[11]	J. F. Mertens, S. Sorin and S. Zamir, Repeated Games, CORE Discussion Papers 9420, 9421, 9422, 1994. doi: 10.1057/9780230226203.3424.
[12]	A. Pratelli, On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation, Ann. Inst. H. Poincaré Probab. Statist., 43 (2007), 1-13. doi: 10.1016/j.anihpb.2005.12.001.
[13]	D. Schmeidler, Equilibrium points of nonatomic games, Journal of Statistical Physics, 7 (1973), 295-300. doi: 10.1007/BF01014905.
[14]	C. Villani, Topics in Optimal Transportation, Graduate studies in Mathematics, Vol. 58, AMS, 2003. doi: 10.1007/b12016.

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