Solution of a functional equation arising in an axiomatization of the utility of binary gambles
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- by János Aczél, Gyula Maksa and Zsolt Páles PDF
- Proc. Amer. Math. Soc. 129 (2001), 483-493 Request permission
Abstract:
For a new axiomatization, with fewer and weaker assumptions, of binary rank-dependent expected utility of gambles the solution of the functional equation \[ (z/p)\gamma ^{-1}[z\gamma (p)] = \varphi ^{-1}[\varphi (z)\psi (p)] \qquad (z,p\in ]0,1[) \] is needed under some monotonicity and surjectivity conditions. We furnish the general such solution and also the solutions under weaker suppositions. In the course of the solution we also determine all sign preserving solutions of the related general equation \[ h(u)[g(u+v)-g(v)]=f(v)g(u+v)\qquad (u\in \mathbb {R}_+,\ v\in \mathbb {R}). \]References
- J. Aczél, Lectures on functional equations and their applications, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York-London, 1966. Translated by Scripta Technica, Inc. Supplemented by the author. Edited by Hansjorg Oser. MR 0208210
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
- Marek Kuczma, An introduction to the theory of functional equations and inequalities, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 489, Uniwersytet Śląski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality; With a Polish summary. MR 788497
- A. A. J. Marley and R. D. Luce, A simple axiomatization of binary rank-dependent expected utility for gains (losses), submitted.
- Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990. Translated from the second French edition by Leo F. Boron; Reprint of the 1955 original. MR 1068530
- A. Wayne Roberts and Dale E. Varberg, Convex functions, Pure and Applied Mathematics, Vol. 57, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0442824
Additional Information
- János Aczél
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- Address at time of publication: Institute for Mathematical Behavioral Sciences, SSP, University of California, Irvine, California 92697-5100
- Email: jdaczel@math.uwaterloo.ca, janos@aris.ss.uci.edu
- Gyula Maksa
- Affiliation: Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
- Email: maksa@math.klte.hu
- Zsolt Páles
- Affiliation: Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
- Email: pales@math.klte.hu
- Received by editor(s): October 23, 1998
- Received by editor(s) in revised form: April 27, 1999
- Published electronically: August 29, 2000
- Additional Notes: This research has been supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada Grant OGP 002972, by the Hungarian National Research Science Foundation (OTKA) Grant T-016846 and by the Fund for Development and Research in Higher Education (FKFP) Grant 0310/1997. The authors are grateful to R. Duncan Luce (University of California, Irvine) for the problem and for advice, in particular regarding the Introduction.
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 483-493
- MSC (2000): Primary 39B22, 39B72, 39B12; Secondary 26A51, 91B16
- DOI: https://doi.org/10.1090/S0002-9939-00-05545-3
- MathSciNet review: 1707502