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

Published electronically:
October 4, 2000

MathSciNet review:
1814138

Full-text PDF Free Access

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*, 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****2.**J. Aczél and J. K. Chung,*Integrable solutions of functional equations of a general type*, Studia Sci. Math. Hungar.**17**(1982), no. 1-4, 51–67. MR**761524****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.**Edwin Hewitt and Karl Stromberg,*Real and abstract analysis*, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing; Graduate Texts in Mathematics, No. 25. MR**0367121****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. 17 (1982), no. 1-4, 51–67; MR0761524 (85i:39008)]*, Studia Sci. Math. Hungar.**19**(1984), no. 2-4, 273–274. MR**874494****7.**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****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 𝑓(𝜆(𝑥)+𝑔(𝑦))=𝜇(𝑥)+ℎ(𝑥+𝑦)*, Aequationes Math.**16**(1977), no. 1-2, 21–30. MR**0611541****11.**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****12.**A. Wayne Roberts and Dale E. Varberg,*Convex functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Pure and Applied Mathematics, Vol. 57. MR**0442824**

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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:
http://dx.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