Skip to main content
SpringerLink
Log in

Continuous-Time Markov Games with Asymmetric Information

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

We study a two-player zero-sum stochastic differential game with asymmetric information where the payoff depends on a controlled continuous-time Markov chain X with finite state space which is only observed by player 1. This model was already studied in Cardaliaguet et al (Math Oper Res 41(1):49–71, 2016) through an approximating sequence of discrete-time games. Our first contribution is the proof of the existence of the value in the continuous-time model based on duality techniques. This value is shown to be the unique solution of the same Hamilton–Jacobi equation with convexity constraints which characterized the limit value obtained in Cardaliaguet et al. (2016). Our second main contribution is to provide a simpler equivalent formulation for this Hamilton–Jacobi equation using directional derivatives and exposed points, which we think is interesting for its own sake as the associated comparison principle has a very simple proof which avoids all the technical machinery of viscosity solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Recall that we may replace global maximum/minimum by local maximum/minimum in the definitions of viscosity solutions, see, e.g., [10].

References

  1. Alvarez O, Lasry JM, Lions PL (1997) Convex viscosity solutions and state constraints. Journal de mathématiques pures et appliquées 76(3):265–288

    Article  MathSciNet  MATH  Google Scholar 

  2. Bremaud P (1981) Point processes and queues: martingale dynamics. Springer series in statistics. Springer-Verlag, Berlin

    Book  MATH  Google Scholar 

  3. Buckdahn R, Quincampoix M, Rainer C, Xu Y (2016) Differential games with asymmetric information and without Isaacs’ condition. Int. J. Game Theory 45(4):795–816

    Article  MathSciNet  MATH  Google Scholar 

  4. Cardaliaguet P (2007) Differential games with asymmetric information. SIAM J Control Optim 46(3):816–838

    Article  MathSciNet  MATH  Google Scholar 

  5. Cardaliaguet P (2009) A double obstacle problem arising in differential game theory. J Math Anal Appl 360(1):95–107

    Article  MathSciNet  MATH  Google Scholar 

  6. Cardaliaguet P, Rainer C (2009) Stochastic differential games with asymmetric information. Appl Math Optim 59(1):1–36

    Article  MathSciNet  MATH  Google Scholar 

  7. Cardaliaguet P, Rainer C (2009) On a continuous-time game with incomplete information. Math Oper Res 34(4):769–794

    Article  MathSciNet  MATH  Google Scholar 

  8. Cardaliaguet P, Rainer C (2012) Games with incomplete information in continuous time and for continuous types. Dyn Games Appl 2(2):206–227

    Article  MathSciNet  MATH  Google Scholar 

  9. Cardaliaguet P, Rainer C, Rosenberg D, Vieille N (2016) Markov games with frequent actions and incomplete information. Math Oper Res 41(1):49–71

    Article  MathSciNet  MATH  Google Scholar 

  10. Crandall MG, Ishii H, Lions PL (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull Am Math Soc 27:1–67

    Article  MathSciNet  MATH  Google Scholar 

  11. De Meyer B (1996) Repeated games, duality and the central limit theorem. Math Oper Res 21(1):237–251

    Article  MathSciNet  MATH  Google Scholar 

  12. Fan K (1953) Minimax theorems. Proc Natl Acad Sci USA 39(1):42–47

    Article  MathSciNet  MATH  Google Scholar 

  13. Gensbittel F (2016) Continuous-time limit of dynamic games with incomplete information and a more informed player. Int J Game Theory 45(1–2):321–352

    Article  MathSciNet  MATH  Google Scholar 

  14. Gensbittel F, Grün C (2018) Zero-sum stopping games with asymmetric information. Math Oper Res. arXiv:1412.1412

  15. Gensbittel F, Rainer C (2018) A two player zero-sum game where only one player observes a Brownian motion. Dyn Games Appl 8(2):280–314

    Article  MathSciNet  MATH  Google Scholar 

  16. Gensbittel F, Renault J (2015) The value of Markov chain games with incomplete information on both sides. Math Oper Res 40(4):820–841

    Article  MathSciNet  MATH  Google Scholar 

  17. Grün C (2012) On Dynkin games with incomplete information. SIAM J Control Optim 51(5):4039–4065

    Article  MathSciNet  MATH  Google Scholar 

  18. Grün C (2012) A BSDE approach to stochastic differential games with incomplete information. Stoch Process Appl 122(4):1917–1946

    Article  MathSciNet  MATH  Google Scholar 

  19. Jimenez C, Quincampoix M, Xu Y (2016) Differential games with incomplete information on a continuum of initial positions and without Isaacs condition. Dyn Games Appl 6(1):82–96

    Article  MathSciNet  MATH  Google Scholar 

  20. Jimenez C, Quincampoix M (2018) Hamilton Jacobi Isaacs equations for differential games with asymmetric information on probabilistic initial condition. J Math Anal Appl 457(2):1422–1451

    Article  MathSciNet  MATH  Google Scholar 

  21. Mertens JF, Zamir S (1971) The value of two-person zero-sum repeated games with incomplete information. Int J Game Theory 1(1):39–64

    Article  MathSciNet  MATH  Google Scholar 

  22. Neyman A (2008) Existence of optimal strategies in Markov games with incomplete information. Int J Game Theory 37(4):581–596

    Article  MathSciNet  MATH  Google Scholar 

  23. Oliu-Barton M (2015) Differential games with asymmetric and correlated information. Dyn Games Appl 5(3):490–512

    Article  MathSciNet  MATH  Google Scholar 

  24. Parthasarathy KR (2005) Probability measures on metric spaces, vol 352. American Mathematical Society, Providence

    MATH  Google Scholar 

  25. Renault J (2006) The value of Markov chain games with lack of information on one side. Math Oper Res 31(3):490–512

    Article  MathSciNet  MATH  Google Scholar 

  26. Sorin S (2002) A first course on zero-sum repeated games. Springer, Berlin

    MATH  Google Scholar 

  27. Wu X (2017) Existence of value for differential games with incomplete information and signals on initial states and payoffs. J Math Anal Appl 446(2):1196–1218

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referees for their careful reading and their pertinent remarks.

Author information

Authors and Affiliations

  1. Toulouse School of Economics, University Toulouse 1 Capitole, Manufacture des Tabacs, MF213, 21, Allée de Brienne, 31015, Toulouse Cedex 6, France

    Fabien Gensbittel

Authors
  1. Fabien Gensbittel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabien Gensbittel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gensbittel, F. Continuous-Time Markov Games with Asymmetric Information. Dyn Games Appl 9, 671–699 (2019). https://doi.org/10.1007/s13235-018-0273-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-018-0273-7

Keywords

Mathematics Subject Classification

Search

Navigation