Random fixed points of multifunctions in games and dynamic programming

Andrzej Nowak

Published: 1986-09-30

Vol. 2 (1986)

Random fixed points of multifunctions in games and dynamic programming

Andrzej Nowak

Abstract

Recently several authors demonstrated random fixed point theorems for various classes of multifunctions ([7], [8], [2], [3], [12], [10]). On the other hand we do not know any work on applications of these theorems. In this paper we apply to games and dynamic programming a random analogue of the Fan-Kakutani fixed point theorem. We consider a zero-sum two-person game depending on a random parameter, and present sufficient conditions for the existence of a measurable solution. Then we study the existence of measurable stationary optimal programs in discounted dynamic programming with a random parameter.

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Nowak, A. (1986). Random fixed points of multifunctions in games and dynamic programming. Annales Mathematicae Silesianae, 2, 53–59. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14321

References

1. C. Berge, Espaces Topologiques (Fonctions multivoques), Dunod, Paris 1959.
2. H.W. Engl, Random fixed point theorems for multivalued mappings, Pacific J. Math. 76 (1978), 351-360.
3. H.W. Engl, Random fixed point theorems, in: Nonlinear Equations in Abstract Spaces, Academic Press, New York 1978.
4. Ky Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. USA 38 (1952), 121-126.
5. O. Hanš, Random operator equations, in: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part I, Berkeley 1961, 185-202.
6. C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72.
7. S. Itoh, A random fixed point theorem for a multivalued contraction mapping, Pacific J. Math. 68 (1977), 85-90.
8. S. Itoh, Measurable or condensing multivalued mappings and random fixed point theorems, Kodai Math. J. 2 (1979), 293-299.
9. A. Nowak, Stationary optimal process in discounted dynamic programming, Zastos. Mat. 25 (1977), 475-487.
10. A. Nowak, Random fixed points of multifunctions, Prace Nauk. Uniw. Śląsk., Prace Matematyczne 11 (1981), 36-41.
11. A. Nowak, Sequences of contractions and random fixed point theorems in dynamic programming, Demonstratio Math. 14 (1981), 343-353.
12. S. Reich, A random fixed point theorem for set-valued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 64 (1978), 65-66.
13. W. Sutherland, On optimal development in multi-sectoral economy: The discounted case, Rev. Econom. Stud. 37 (1970), 585-596.
14. D.H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), 859-903. Google Scholar

Vol. 2 (1986)
Published: 1986-09-30

ISSN: 0860-2107

eISSN: 2391-4238

10.2478/amsil

Publisher

University of Silesia Press

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