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

Pure and Random strategies in differential game with incomplete informations

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

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).
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.  Google Scholar

[2]

R. J. Aumann, Mixed and behavior strategies in infinite extensive games,, in Advances in Game Theory, (1964), 627.   Google Scholar

[3]

R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information,, MIT Press, (1995).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. Cardaliaguet, Differential games with asymmetric information,, SIAM J. Control Optim., 46 (2007), 816.  doi: 10.1137/060654396.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Dellacherie and P. A. Meyer, Probabilities and Potential,, North-Holland Mathematics Studies, (1978).   Google Scholar

[11]

J. F. Mertens, S. Sorin and S. Zamir, Repeated Games,, CORE Discussion Papers 9420, (9420).  doi: 10.1057/9780230226203.3424.  Google Scholar

[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.  Google Scholar

[13]

D. Schmeidler, Equilibrium points of nonatomic games,, Journal of Statistical Physics, 7 (1973), 295.  doi: 10.1007/BF01014905.  Google Scholar

[14]

C. Villani, Topics in Optimal Transportation,, Graduate studies in Mathematics, (2003).  doi: 10.1007/b12016.  Google Scholar

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.  Google Scholar

[2]

R. J. Aumann, Mixed and behavior strategies in infinite extensive games,, in Advances in Game Theory, (1964), 627.   Google Scholar

[3]

R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information,, MIT Press, (1995).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. Cardaliaguet, Differential games with asymmetric information,, SIAM J. Control Optim., 46 (2007), 816.  doi: 10.1137/060654396.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Dellacherie and P. A. Meyer, Probabilities and Potential,, North-Holland Mathematics Studies, (1978).   Google Scholar

[11]

J. F. Mertens, S. Sorin and S. Zamir, Repeated Games,, CORE Discussion Papers 9420, (9420).  doi: 10.1057/9780230226203.3424.  Google Scholar

[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.  Google Scholar

[13]

D. Schmeidler, Equilibrium points of nonatomic games,, Journal of Statistical Physics, 7 (1973), 295.  doi: 10.1007/BF01014905.  Google Scholar

[14]

C. Villani, Topics in Optimal Transportation,, Graduate studies in Mathematics, (2003).  doi: 10.1007/b12016.  Google Scholar

[1]

Zhongbao Zhou, Yanfei Bai, Helu Xiao, Xu Chen. A non-zero-sum reinsurance-investment game with delay and asymmetric information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 909-936. doi: 10.3934/jimo.2020004

[2]

Wai-Ki Ching, Jia-Wen Gu, Harry Zheng. On correlated defaults and incomplete information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 889-908. doi: 10.3934/jimo.2020003

[3]

Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020033

[4]

Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028

[5]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[6]

Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300

[7]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, 2021, 14 (1) : 77-88. doi: 10.3934/krm.2020049

[8]

San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038

[9]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

[10]

Tinghua Hu, Yang Yang, Zhengchun Zhou. Golay complementary sets with large zero odd-periodic correlation zones. Advances in Mathematics of Communications, 2021, 15 (1) : 23-33. doi: 10.3934/amc.2020040

[11]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[12]

Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017

[13]

Honglin Yang, Jiawu Peng. Coordinating a supply chain with demand information updating. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020181

[14]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145

[15]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[16]

Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007

[17]

Chuan Ding, Da-Hai Li. Angel capitalists exit decisions under information asymmetry: IPO or acquisitions. Journal of Industrial & Management Optimization, 2021, 17 (1) : 369-392. doi: 10.3934/jimo.2019116

[18]

Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135

[19]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

[20]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386

 Impact Factor: 

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (13)

[Back to Top]