Random fixed points of multifunctions in games and dynamic programming
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.
References
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.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
