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.
Similar content being viewed by others
References
R. J. aumann and M. Maschler with the collaboration of R. E. Stearns,Repeated Games with Incomplete Information, MIT Press, 1995.
T. Bewley and E. Kohlberg,The asymptotic theory of stochastic games, Mathematics of Operations Research1 (1976), 197–208.
T. Bewley and E. Kohlberg,The asymptotic solution of a recursion equation occurring in stochastic games, Mathematics of Operations Research1 (1976), 321–336.
E. Kohlberg,Repeated games with absorbing states, Annals of Statistics2 (1974), 724–738.
E. Kohlberg and A. Neyman,Asymptotic behavior of non expansive mappings in normed linear spaces, Israel Journal of Mathematics38 (1981), 269–275.
R. Laraki,Repeated games with lack of information on one side: the dual differential approach, preprint, 1999; Mathematics of Operations Research, to appear.
E. Lehrer and S. Sorin,A uniform tauberian theorem in dynamic programming, Mathematics of Operations Research17 (1992), 303–307.
J.-F. Mertens,Repeated games, inProceedings of the International Congress of Mathematicians (Berkeley), 1986, American Mathematical Society, Providence, 1987, pp. 1528–1577.
J.-F. Mertens and A. Neyman,Stochastic games, International Journal of Game Theory10 (1981), 53–56.
J.-F. Mertens, S. Sorin and S. Zamir,Repeated Games, Parts A, B, and C, CORE D.P. 9420-9422, 1994.
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.
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.
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.
D. Rosenberg,Sur les jeux répétés à somme nulle, Thèse, Université Paris X-Nanterre, 1998.
D. Rosenberg,Absorbing games with incomplete information on one side: asymptotic analysis, preprint, 1999; SIAM Journal on Control and Optimization, to appear.
L.S. Shapley,Stochastic games, Proceedings of the National Academy of Sciences of the United States of America39 (1953), 1095–1100.
S. Sorin,Big match with lack of information on one side (part 1), International Journal of Game Theory13 (1984), 201–255.
S. Sorin,Big match with lack of information on one side (part 2), International Journal of Game Theory14 (1985), 173–204.
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02802505