Advertisement

Israel Journal of Mathematics

, Volume 121, Issue 1, pp 221–246 | Cite as

An operator approach to zero-sum repeated games

  • Dinah RosenbergEmail author
  • Sylvain Sorin
Article

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.

Keywords

Incomplete Information Recursive Formula Stochastic Game Repeat Game Recursive Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. J. aumann and M. Maschler with the collaboration of R. E. Stearns,Repeated Games with Incomplete Information, MIT Press, 1995.Google Scholar
  2. [2]
    T. Bewley and E. Kohlberg,The asymptotic theory of stochastic games, Mathematics of Operations Research1 (1976), 197–208.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    T. Bewley and E. Kohlberg,The asymptotic solution of a recursion equation occurring in stochastic games, Mathematics of Operations Research1 (1976), 321–336.zbMATHMathSciNetGoogle Scholar
  4. [4]
    E. Kohlberg,Repeated games with absorbing states, Annals of Statistics2 (1974), 724–738.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    E. Kohlberg and A. Neyman,Asymptotic behavior of non expansive mappings in normed linear spaces, Israel Journal of Mathematics38 (1981), 269–275.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    R. Laraki,Repeated games with lack of information on one side: the dual differential approach, preprint, 1999; Mathematics of Operations Research, to appear.Google Scholar
  7. [7]
    E. Lehrer and S. Sorin,A uniform tauberian theorem in dynamic programming, Mathematics of Operations Research17 (1992), 303–307.MathSciNetGoogle Scholar
  8. [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. [9]
    J.-F. Mertens and A. Neyman,Stochastic games, International Journal of Game Theory10 (1981), 53–56.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J.-F. Mertens, S. Sorin and S. Zamir,Repeated Games, Parts A, B, and C, CORE D.P. 9420-9422, 1994.Google Scholar
  11. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [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.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [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.Google Scholar
  14. [14]
    D. Rosenberg,Sur les jeux répétés à somme nulle, Thèse, Université Paris X-Nanterre, 1998.Google Scholar
  15. [15]
    D. Rosenberg,Absorbing games with incomplete information on one side: asymptotic analysis, preprint, 1999; SIAM Journal on Control and Optimization, to appear.Google Scholar
  16. [16]
    L.S. Shapley,Stochastic games, Proceedings of the National Academy of Sciences of the United States of America39 (1953), 1095–1100.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    S. Sorin,Big match with lack of information on one side (part 1), International Journal of Game Theory13 (1984), 201–255.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    S. Sorin,Big match with lack of information on one side (part 2), International Journal of Game Theory14 (1985), 173–204.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2001

Authors and Affiliations

  1. 1.LAGA, Institut GaliléeUniversité Paris 13VilletaneuseFrance
  2. 2.Laboratoire d’EconométrieEcole PolytechniqueParisFrance
  3. 3.MODALX and THEMA, UFR SEGMIUniversité Paris XNanterreFrance

Personalised recommendations