Skip to main content
Log in

Optimal Strategies in Zero-Sum Repeated Games with Incomplete Information: The Dependent Case

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

Abstract

Using the duality techniques introduced by De Meyer (Math Oper Res 21:209–236, 1996a, Math Oper Res 21:237–251, 1996b), Rosenberg (Int J Game Theory 27:577–597, 1998) and De Meyer and Marino (Cahiers de la MSE 27, 2005) provided an explicit construction for optimal strategies in repeated games with incomplete information on both sides, in the independent case. In this note, we extend both the duality techniques and the construction of optimal strategies to the dependent case.

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

Access this article

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. One may apply Sion’s minmax theorem to the game in mixed strategies when pure strategies are endowed with the product topology, and then apply Kuhn’s theorem to deduce the result. See, e.g., chapter 3 and appendix A in [14] where the same method is applied in the discounted case.

References

  1. Aumann RJ, Maschler M (1995) Repeated games with incomplete information, with the collaboration of R. MIT Press, Stearns

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. De Meyer B (1996) Repeated games and partial differential equations. Math Oper Res 21:209–236

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. De Meyer B, Marino A (2005) Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides. Cahiers de la MSE, 27

  6. Heuer M (1992) Asymptotically optimal strategies in repeated games with incomplete information. Int J Game Theory 20:377–392

    Article  MathSciNet  Google Scholar 

  7. Mertens J-F, Zamir S (1971) The value of two-person zero-sum repeated games with lack of information on both sides. Int J Game Theory 1:39–64

    Article  MathSciNet  Google Scholar 

  8. Oliu-Barton M (2013) Discrete and continuous time games with incomplete information. Ph.D. thesis, Pierre and Marie Curie University, Paris 6

  9. Oliu-Barton M (2015) Differential games with asymmetric and correlated information. Dyn Games Appl 5:378–396

    Article  MathSciNet  Google Scholar 

  10. Oliu-Barton M (2018) The splitting game: value and optimal strategies. Dyn Games Appl 8(1):157–179

    Article  MathSciNet  Google Scholar 

  11. Oliu-Barton M (2019) Asymptotically optimal strategies in repeated games with incomplete information and vanishing weights. J Dyn Games 6(4):259–275

    Article  MathSciNet  Google Scholar 

  12. Rosenberg Dinah (1998) Duality and markovian strategies. Int J Game Theory 27:577–597

    Article  MathSciNet  Google Scholar 

  13. Sion M (1958) On general minimax theorems. Pac J Math 8(1):171–176

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are indebted to Sylvain Sorin and Bernard De Meyer for their insight and suggestions. The authors are also thankful to Mario Bravo for his comments. The first author gratefully acknowledges support from the Artificial and Natural Intelligence Toulouse Institute under Grant ANR-3IA, and funding from the French National Research Agency (ANR), under the Investments for the Future (Investissements d’Avenir) program under grant ANR-17-EURE-0010. The second author gratefully acknowledges funding from the French National Research Agency (ANR), under grant ANR CIGNE (ANR-15-CE38-0007-01).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fabien Gensbittel.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gensbittel, F., Oliu-Barton, M. Optimal Strategies in Zero-Sum Repeated Games with Incomplete Information: The Dependent Case. Dyn Games Appl 10, 819–835 (2020). https://doi.org/10.1007/s13235-020-00347-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-020-00347-y

Keywords

Navigation