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Solution of a functional equation arising in an axiomatization of the utility of binary gambles


Authors: János Aczél, Gyula Maksa and Zsolt Páles
Journal: Proc. Amer. Math. Soc. 129 (2001), 483-493
MSC (2000): Primary 39B22, 39B72, 39B12; Secondary 26A51, 91B16
Published electronically: August 29, 2000
MathSciNet review: 1707502
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Abstract | References | Similar Articles | Additional Information

Abstract:

For a new axiomatization, with fewer and weaker assumptions, of binary rank-dependent expected utility of gambles the solution of the functional equation

\begin{displaymath}(z/p)\gamma^{-1}[z\gamma(p)] = \varphi^{-1}[\varphi(z)\psi(p)] \qquad (z,p\in]0,1[) \end{displaymath}

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

\begin{displaymath}h(u)[g(u+v)-g(v)]=f(v)g(u+v)\qquad (u\in\mathbb{R} _+, v\in\mathbb{R} ). \end{displaymath}


References [Enhancements On Off] (What's this?)

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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

DOI: https://doi.org/10.1090/S0002-9939-00-05545-3
Keywords: Functional equation, binary gamble, convexity
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
Article copyright: © Copyright 2000 American Mathematical Society