Skip to main content
SpringerLink
Log in

An operator approach to zero-sum repeated games

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We consider two person zero-sum stochastic games. The recursive formula for the valuesvλ (resp.v n) of the discounted (resp. finitely repeated) version can be written in terms of a single basic operator Φ(α,f) where α is the weight on the present payoff andf the future payoff. We give sufficient conditions in terms of Φ(α,f) and its derivative at 0 for limv n and limvλ to exist and to be equal.

We apply these results to obtain such convergence properties for absorbing games with compact action spaces and incomplete information games.

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

References

  1. R. J. aumann and M. Maschler with the collaboration of R. E. Stearns,Repeated Games with Incomplete Information, MIT Press, 1995.

  2. T. Bewley and E. Kohlberg,The asymptotic theory of stochastic games, Mathematics of Operations Research1 (1976), 197–208.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Bewley and E. Kohlberg,The asymptotic solution of a recursion equation occurring in stochastic games, Mathematics of Operations Research1 (1976), 321–336.

    MATH  MathSciNet  Google Scholar 

  4. E. Kohlberg,Repeated games with absorbing states, Annals of Statistics2 (1974), 724–738.

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Kohlberg and A. Neyman,Asymptotic behavior of non expansive mappings in normed linear spaces, Israel Journal of Mathematics38 (1981), 269–275.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Laraki,Repeated games with lack of information on one side: the dual differential approach, preprint, 1999; Mathematics of Operations Research, to appear.

  7. E. Lehrer and S. Sorin,A uniform tauberian theorem in dynamic programming, Mathematics of Operations Research17 (1992), 303–307.

    MathSciNet  Google Scholar 

  8. J.-F. Mertens,Repeated games, inProceedings of the International Congress of Mathematicians (Berkeley), 1986, American Mathematical Society, Providence, 1987, pp. 1528–1577.

    Google Scholar 

  9. J.-F. Mertens and A. Neyman,Stochastic games, International Journal of Game Theory10 (1981), 53–56.

    Article  MATH  MathSciNet  Google Scholar 

  10. J.-F. Mertens, S. Sorin and S. Zamir,Repeated Games, Parts A, B, and C, CORE D.P. 9420-9422, 1994.

  11. J.-F. Mertens and S. Zamir,The value of two person zero sum repeated games with lack of information on both sides, International Journal of Game Theory,1 (1971-72), 39–64.

    Article  MATH  MathSciNet  Google Scholar 

  12. J.-F. Mertens and S. Zamir,A duality theorem on a pair of simultaneous functional equations, Journal of Mathematical Analysis and its Applications60 (1977), 550–558.

    Article  MATH  MathSciNet  Google Scholar 

  13. H.D. Mills,Marginal values of matrix games and linear programs, inLinear Inequalities and Related Systems (H. W. Kuhn and A. W. Tucker, eds.), Annals of Mathematics Studies 38, Princeton University Press, 1956, pp. 183–194.

  14. D. Rosenberg,Sur les jeux répétés à somme nulle, Thèse, Université Paris X-Nanterre, 1998.

  15. D. Rosenberg,Absorbing games with incomplete information on one side: asymptotic analysis, preprint, 1999; SIAM Journal on Control and Optimization, to appear.

  16. L.S. Shapley,Stochastic games, Proceedings of the National Academy of Sciences of the United States of America39 (1953), 1095–1100.

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Sorin,Big match with lack of information on one side (part 1), International Journal of Game Theory13 (1984), 201–255.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Sorin,Big match with lack of information on one side (part 2), International Journal of Game Theory14 (1985), 173–204.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LAGA, Institut Galilée, Université Paris 13, avenue Jean Baptiste Clément, 93430, Villetaneuse, France

    Dinah Rosenberg

  2. Laboratoire d’Econométrie, Ecole Polytechnique, 1 rue Descartes, 75005, Paris, France

    Sylvain Sorin

  3. MODALX and THEMA, UFR SEGMI, Université Paris X, 200 Avenue de la République, 92001, Nanterre, France

    Sylvain Sorin

Authors
  1. Dinah Rosenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sylvain Sorin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dinah Rosenberg.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rosenberg, D., Sorin, S. An operator approach to zero-sum repeated games. Isr. J. Math. 121, 221–246 (2001). https://doi.org/10.1007/BF02802505

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02802505

Keywords

Search

Navigation