Pure and Random strategies in differential game with incomplete informations

Previous Article
Policy improvement for perfect information additive reward and additive transition stochastic games with discounted and average payoffs
JDG Home
This Issue
Next Article
Competing for customers in a social network

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
Related Papers

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 & 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, (1812). 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, (1964), 627.
[3]	R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information,, MIT Press, (1995).
[4]	R. Buckdahn, P. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory,, Dynamic Games Applications, 1 (2011), 74. 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. doi: 10.1007/s00182-012-0351-9.
[6]	P. Cardaliaguet, Differential games with asymmetric information,, SIAM J. Control Optim., 46 (2007), 816. 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. doi: 10.1142/S021919890800173X.
[8]	P. Cardaliaguet and C. Rainer, Stochastic differential games with assymetric information,, Appl. Math. Optim., 59 (2009), 1. 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. doi: 10.1007/s13235-012-0043-x.
[10]	C. Dellacherie and P. A. Meyer, Probabilities and Potential,, North-Holland Mathematics Studies, (1978).
[11]	J. F. Mertens, S. Sorin and S. Zamir, Repeated Games,, CORE Discussion Papers 9420, (9420). 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. doi: 10.1016/j.anihpb.2005.12.001.
[13]	D. Schmeidler, Equilibrium points of nonatomic games,, Journal of Statistical Physics, 7 (1973), 295. doi: 10.1007/BF01014905.
[14]	C. Villani, Topics in Optimal Transportation,, Graduate studies in Mathematics, (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, (1812). 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, (1964), 627.
[3]	R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information,, MIT Press, (1995).
[4]	R. Buckdahn, P. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory,, Dynamic Games Applications, 1 (2011), 74. 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. doi: 10.1007/s00182-012-0351-9.
[6]	P. Cardaliaguet, Differential games with asymmetric information,, SIAM J. Control Optim., 46 (2007), 816. 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. doi: 10.1142/S021919890800173X.
[8]	P. Cardaliaguet and C. Rainer, Stochastic differential games with assymetric information,, Appl. Math. Optim., 59 (2009), 1. 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. doi: 10.1007/s13235-012-0043-x.
[10]	C. Dellacherie and P. A. Meyer, Probabilities and Potential,, North-Holland Mathematics Studies, (1978).
[11]	J. F. Mertens, S. Sorin and S. Zamir, Repeated Games,, CORE Discussion Papers 9420, (9420). 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. doi: 10.1016/j.anihpb.2005.12.001.
[13]	D. Schmeidler, Equilibrium points of nonatomic games,, Journal of Statistical Physics, 7 (1973), 295. doi: 10.1007/BF01014905.
[14]	C. Villani, Topics in Optimal Transportation,, Graduate studies in Mathematics, (2003). doi: 10.1007/b12016.

[1]	Beatris A. Escobedo-Trujillo. Discount-sensitive equilibria in zero-sum stochastic differential games. Journal of Dynamics & Games, 2016, 3 (1) : 25-50. doi: 10.3934/jdg.2016002
[2]	Qingmeng Wei, Zhiyong Yu. Time-inconsistent recursive zero-sum stochastic differential games. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1051-1079. doi: 10.3934/mcrf.2018045
[3]	Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics & Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020
[4]	Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zero-sum games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3901-3931. doi: 10.3934/dcds.2015.35.3901
[5]	Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95
[6]	Sylvain Sorin, Guillaume Vigeral. Reversibility and oscillations in zero-sum discounted stochastic games. Journal of Dynamics & Games, 2015, 2 (1) : 103-115. doi: 10.3934/jdg.2015.2.103
[7]	Alexander J. Zaslavski. Structure of approximate solutions of dynamic continuous time zero-sum games. Journal of Dynamics & Games, 2014, 1 (1) : 153-179. doi: 10.3934/jdg.2014.1.153
[8]	Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa. Average optimal strategies for zero-sum Markov games with poorly known payoff function on one side. Journal of Dynamics & Games, 2014, 1 (1) : 105-119. doi: 10.3934/jdg.2014.1.105
[9]	Alexander J. Zaslavski. Turnpike properties of approximate solutions of dynamic discrete time zero-sum games. Journal of Dynamics & Games, 2014, 1 (2) : 299-330. doi: 10.3934/jdg.2014.1.299
[10]	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
[11]	Feimin Zhong, Jinxing Xie, Jing Jiao. Solutions for bargaining games with incomplete information: General type space and action space. Journal of Industrial & Management Optimization, 2018, 14 (3) : 953-966. doi: 10.3934/jimo.2017084
[12]	Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial & Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27
[13]	Josef Hofbauer, Sylvain Sorin. Best response dynamics for continuous zero--sum games. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 215-224. doi: 10.3934/dcdsb.2006.6.215
[14]	Beatris Adriana Escobedo-Trujillo, José Daniel López-Barrientos. Nonzero-sum stochastic differential games with additive structure and average payoffs. Journal of Dynamics & Games, 2014, 1 (4) : 555-578. doi: 10.3934/jdg.2014.1.555
[15]	Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
[16]	Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control & Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010
[17]	Zhi-Wei Sun. Unification of zero-sum problems, subset sums and covers of Z. Electronic Research Announcements, 2003, 9: 51-60.
[18]	John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics & Games, 2016, 3 (4) : 335-354. doi: 10.3934/jdg.2016018
[19]	Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889
[20]	Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719

Impact Factor:

American Institute of Mathematical Sciences