American Institute of Mathematical Sciences

July  2021, 8(3): 233-266. doi: 10.3934/jdg.2021009

A zero sum differential game with correlated informations on the initial position. A case with a continuum of initial positions

 Univ Brest, Laboratoire de Mathématiques de Bretagne Atlantique, CNRS-UMR 6205, 6, avenue Victor Le Gorgeu, CS 93837, 29238 Brest cedex 3, France

Received  June 2020 Published  July 2021 Early access  March 2021

We study a two player zero sum game where the initial position $z_0$ is not communicated to any player. The initial position is a function of a couple $(x_0,y_0)$ where $x_0$ is communicated to player Ⅰ while $y_0$ is communicated to player Ⅱ. The couple $(x_0,y_0)$ is chosen according to a probability measure $dm(x,y) = h(x,y) d\mu(x) d\nu(y)$. We show that the game has a value and, under additional regularity assumptions, that the value is a solution of Hamilton Jacobi Isaacs equation in a dual sense.

Citation: Chloé Jimenez. A zero sum differential game with correlated informations on the initial position. A case with a continuum of initial positions. Journal of Dynamics and Games, 2021, 8 (3) : 233-266. doi: 10.3934/jdg.2021009
References:
 [1] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, MPS/SIAM Series on Optimization, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006. [2] R. Buckdahn, M. Quincampoix, C. Rainer and Y. Xu, Differential games with asymmetric information and without Isaacs' condition, Internat. J. Game Theory, 45 (2016), 795-816.  doi: 10.1007/s00182-015-0482-x. [3] P. Cardaliaguet, Introduction to Differential Games, lecture notes, 2010. Available from: https://www.ceremade.dauphine.fr/ cardaliaguet/CoursJeuxDiff.pdf. [4] 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. [5] P. Cardaliaguet, Differential games with asymmetric information, SIAM J. Control Optim., 46 (2007), 816-838.  doi: 10.1137/060654396. [6] P. Cardaliaguet, A double obstacle problem arising in differential game theory, J. Math. Anal. Appl., 360 (2009), 95-107.  doi: 10.1016/j.jmaa.2009.06.041. [7] P. Cardaliaguet, C. Jimenez and M. Quincampoix, Pure and random strategies in differential game with incomplete informations, Journal of Dynamics and Games, 1 (2014), 363-375.  doi: 10.3934/jdg.2014.1.363. [8] I. Ekeland, On the variational principle, Journal of Math. Anal. and Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0. [9] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. [10] F. Gensbittel and C. Rainer, A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides, SIAM J. Control Optim., 55, (2017), 693–723. doi: 10.1137/16M106217X. [11] F. Gensbittel and C. Rainer, A two player zerosum game where only one player observes a Brownian motion, Dynamic Games and Applications, 8, (2018), 280–314. doi: 10.1007/s13235-017-0219-5. [12] L. G. Hanin, An extension of the Kantorovich norm. Monge Ampère equation: applications to geometry and optimization, Contemp. Math., 226 (1999), 113-130.  doi: 10.1090/conm/226/03238. [13] C. Jimenez and M. Quincampoix, Hamilton Jacobi Isaacs equations for differential games with asymmetric information on probabilistic initial condition, J. Math. Anal. Appl., 457 (2018), 1422-1451.  doi: 10.1016/j.jmaa.2017.08.012. [14] C. Jimenez, M. Quincampoix and Y. Xu, Differential games with incomplete information on a continuum of initial positions and without Isaacs condition, Dyn. Games Appl., 6 (2016), 82-96.  doi: 10.1007/s13235-014-0134-y. [15] M. Oliu-Barton, Differential games with asymmetric and correlated information, Dyn. Games Appl., 5 (2015), 378-396.  doi: 10.1007/s13235-014-0131-1. [16] 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. [17] F. Santambrogio, Optimal Transport for Applied Mathematicians, Calculus of variations, PDEs, and modeling. Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2. [18] C. Villani, Topics in Optimal Transportation, Graduate studies in Mathematics, Vol.58, AMS, 2003. doi: 10.1090/gsm/058. [19] X. Wu, Existence of value for differential games with incomplete information and signals on initial states and payoffs, J. Math. Anal. Appl., 446 (2017), 1196-1218.  doi: 10.1016/j.jmaa.2016.09.035.

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References:
 [1] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, MPS/SIAM Series on Optimization, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006. [2] R. Buckdahn, M. Quincampoix, C. Rainer and Y. Xu, Differential games with asymmetric information and without Isaacs' condition, Internat. J. Game Theory, 45 (2016), 795-816.  doi: 10.1007/s00182-015-0482-x. [3] P. Cardaliaguet, Introduction to Differential Games, lecture notes, 2010. Available from: https://www.ceremade.dauphine.fr/ cardaliaguet/CoursJeuxDiff.pdf. [4] 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. [5] P. Cardaliaguet, Differential games with asymmetric information, SIAM J. Control Optim., 46 (2007), 816-838.  doi: 10.1137/060654396. [6] P. Cardaliaguet, A double obstacle problem arising in differential game theory, J. Math. Anal. Appl., 360 (2009), 95-107.  doi: 10.1016/j.jmaa.2009.06.041. [7] P. Cardaliaguet, C. Jimenez and M. Quincampoix, Pure and random strategies in differential game with incomplete informations, Journal of Dynamics and Games, 1 (2014), 363-375.  doi: 10.3934/jdg.2014.1.363. [8] I. Ekeland, On the variational principle, Journal of Math. Anal. and Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0. [9] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. [10] F. Gensbittel and C. Rainer, A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides, SIAM J. Control Optim., 55, (2017), 693–723. doi: 10.1137/16M106217X. [11] F. Gensbittel and C. Rainer, A two player zerosum game where only one player observes a Brownian motion, Dynamic Games and Applications, 8, (2018), 280–314. doi: 10.1007/s13235-017-0219-5. [12] L. G. Hanin, An extension of the Kantorovich norm. Monge Ampère equation: applications to geometry and optimization, Contemp. Math., 226 (1999), 113-130.  doi: 10.1090/conm/226/03238. [13] C. Jimenez and M. Quincampoix, Hamilton Jacobi Isaacs equations for differential games with asymmetric information on probabilistic initial condition, J. Math. Anal. Appl., 457 (2018), 1422-1451.  doi: 10.1016/j.jmaa.2017.08.012. [14] C. Jimenez, M. Quincampoix and Y. Xu, Differential games with incomplete information on a continuum of initial positions and without Isaacs condition, Dyn. Games Appl., 6 (2016), 82-96.  doi: 10.1007/s13235-014-0134-y. [15] M. Oliu-Barton, Differential games with asymmetric and correlated information, Dyn. Games Appl., 5 (2015), 378-396.  doi: 10.1007/s13235-014-0131-1. [16] 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. [17] F. Santambrogio, Optimal Transport for Applied Mathematicians, Calculus of variations, PDEs, and modeling. Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2. [18] C. Villani, Topics in Optimal Transportation, Graduate studies in Mathematics, Vol.58, AMS, 2003. doi: 10.1090/gsm/058. [19] X. Wu, Existence of value for differential games with incomplete information and signals on initial states and payoffs, J. Math. Anal. Appl., 446 (2017), 1196-1218.  doi: 10.1016/j.jmaa.2016.09.035.
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