A functional equation arising from ranked additive and separable utility

Authors:
János Aczél, Gyula Maksa, Che Tat Ng and Zsolt Páles

Journal:
Proc. Amer. Math. Soc. **129** (2001), 989-998

MSC (2000):
Primary 39B12, 39B22, 39B72; Secondary 26A48, 26A51, 91A30, 91C05

DOI:
https://doi.org/10.1090/S0002-9939-00-05686-0

Published electronically:
October 4, 2000

MathSciNet review:
1814138

Full-text PDF

Abstract | References | Similar Articles | Additional Information

All strictly monotonic solutions of a general functional equation are determined. In a particular case, which plays an essential role in the axiomatization of rank-dependent expected utility, all nonnegative solutions are obtained without any regularity conditions. An unexpected possibility of reduction to convexity makes the present proof possible.

**1.**J. Aczél,*Lectures on Functional Equations and Their Applications*, Academic Press, New York/London, 1966. MR**34:8020****2.**J. Aczél and J. K. Chung,*Integrable solutions of functional equations of a general type*, Studia Sci. Math. Hungar.**17**(1982), 51-67. MR**85i:39008****3.**J. Aczél, R. Ger and A. Járai,*Solution of a functional equation arising from utility that is both separable and additive*, Proc. Amer. Math. Soc.**127**(1999), 2923-2929. CMP**99:15****4.**J. Aczél, Gy. Maksa, and Zs. Páles,*Solution of a functional equation arising in an axiomatization of the utility of binary gambles*, Proc. Amer. Math. Soc. (to appear). CMP**99:17****5.**E. Hewitt and K. Stromberg,*Real and Abstract Analysis*, Springer, New York/Heidelberg, 1975. MR**51:3363****6.**A. Járai,*A remark to a paper of J. Aczél and J. K. Chung: ``Integrable solutions of functional equations of a general type''*, Studia Sci. Math. Hungar.**19**(1984), 273-274. MR**87m:39006****7.**M. Kuczma,*An Introduction to the Theory of Functional Equations and Inequalities*, Panstwowe Wydawnictwo Naukowe, Warszawa/Kraków/Katowice, 1985. MR**86i:39008****8.**R. D. Luce,*Coalescing, event commutativity, and theories of utility*, J. Risk Uncertainty,**16**(1998), 87-114.**9.**R. D. Luce and A. A. J. Marley,*Separable and additive utility of binary gambles of gains*, Math. Social Sci., in press.**10.**A. Lundberg,*On the functional equation*, Aequationes Math.**16**(1977), 21-30. MR**58:29529****11.**F. Riesz and B. Szokefalvi-Nagy,*Functional Analysis*, Dover, New York, 1990. MR**91g:00002****12.**A. W. Roberts and D. E. Varberg,*Convex Functions*, Academic Press, New York and London, 1973. MR**56:1201**

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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, 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

**Che Tat Ng**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Email:
ctng@math.uwaterloo.ca

**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/S0002-9939-00-05686-0

Keywords:
Functional equation,
binary gamble,
rank-dependent expected utility,
convexity

Received by editor(s):
June 7, 1999

Published electronically:
October 4, 2000

Additional Notes:
This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada Grants OGP 0002972 and OGP 0008212, by the Hungarian National Science Foundation (OTKA) Grant T-030082 and by the Higher Education Research Council (FKFP) Grant 0310/1997.

The authors are grateful to R. Duncan Luce for communicating the problem and explanations and to the referee for pointing out a confusing misprint in the first version of the manuscript.

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2000
American Mathematical Society