Dynamic Games and Applications

, Volume 8, Issue 2, pp 280–314 | Cite as

A Two-Player Zero-sum Game Where Only One Player Observes a Brownian Motion

  • Fabien Gensbittel
  • Catherine Rainer


We study a two-player zero-sum game in continuous time, where the payoff—a running cost—depends on a Brownian motion. This Brownian motion is observed in real time by one of the players. The other one observes only the actions of his/her opponent. We prove that the game has a value and characterize it as the largest convex subsolution of a Hamilton–Jacobi equation on the space of probability measures.


Zero-sum continuous-time game Incomplete information Hamilton–Jacobi equations Brownian motion Measure-valued process 

Mathematics Subject Classification

91A05 91A23 49N70 



We thank Pierre Cardaliaguet for very fruitful discussions. We thank the referees for their careful reading and their pertinent remarks. This work has been partially supported by the French National Research Agency ANR-16-CE40-0015-01 MFG.


  1. 1.
    Aumann RJ, Maschler MB (1995) Repeated games with incomplete information. With the collaboration of Richard E. Stearns. MIT Press, CambridgeGoogle Scholar
  2. 2.
    Bertsekas DP, Shreve SE (1978) Stochastic optimal control: the discrete time case. Academic Press, New YorkzbMATHGoogle Scholar
  3. 3.
    Blackwell D, Dubins LE (1983) An extension of Skorohod’s almost sure representation theorem. Proc Am Math Soc 89:691–692MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cardaliaguet P Notes on Mean Field Games, unpublishedGoogle Scholar
  5. 5.
    Cardaliaguet P (2006) Differential games with asymmetric information. SIAM J Control Optim 46(3):816–838MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cardaliaguet P, Delarue F, Lasry JF, Lions PL (2015) The master equation and the convergence problem in mean field games. arXiv:1509.02505
  7. 7.
    Cardaliaguet P, Laraki R, Sorin S (2012) A continuous time approach for the asymptotic value in two-person zero-sum repeated games. SIAM J Control Optim 50(3):1573–1596MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cardaliaguet P, Rainer C (2009) Stochastic differential games with asymmetric information. Appl Math Optim 59(1):1–36MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cardaliaguet P, Rainer C (2009) On a continuous-time game with incomplete information. Math Oper Res 34(4):769–794MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cardaliaguet P, Rainer C (2012) Games with incomplete information in continuous time and for continuous types. Dyn Games Appl 2:206–227MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cardaliaguet P, Rainer C, Rosenberg D, Vieille N (2016) Markov games with frequent actions and incomplete information. Math Oper Res 41(1):49–71MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Crandall MG, Ishii H, Lions PL (1992) User’s guide to viscosity solutions of second order partial differential equations. Am Math Soc Bull New Ser 27(1):1–67MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dudley RM (2002) Real analysis and probability. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  14. 14.
    Gensbittel F (2016) Continuous-time limits of dynamic games with incomplete information and a more informed player. Int J of Game Theory 45(1):321–352MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gensbittel F, Rainer C (2016) A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides. arXiv:1602.06140
  16. 16.
    Gruen C (2012) A BSDE approach to stochastic differential games with incomplete information. Stoch Process Appl 122(4):1917–1946MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jacod J, Shiryaev AN (2003) Limit theorems for stochastic processes, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  18. 18.
    Kurtz TG (1991) Random time changes and convergence in distribution under the Meyer- Zheng conditions. Ann Probab 19:1010–1034MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lions P-L (2005–2006) Cours au Collège de France.
  20. 20.
    Meyer P-A, Zheng WA (1984) Tightness criteria for laws of semimartingales. Ann Inst H Poincar Probab Statist 20(4):353–372MathSciNetzbMATHGoogle Scholar
  21. 21.
    Neyman A (2013) Stochastic games with short-stage duration. Dyn Games Appl 3:236–278MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Oliu Barton M (2015) Differential games with asymmetric and correlated information. Dyn Games Appl 5(3):378–396MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sorin S (2002) A first course on zero-sum repeated games. Springer, BerlinzbMATHGoogle Scholar
  24. 24.
    Sorin S (2016) Limit value of dynamic zero-sum games with vanishing stage duration. arXiv:1603.09089
  25. 25.
    Villani C (2003) Topics in optimal transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, ProvidenceGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Manufacture des Tabacs Allée de BrienneToulouse Cedex 6France
  2. 2.Université de Bretagne OccidentaleBrest CedexFrance

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