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), 989998
MSC (2000):
Primary 39B12, 39B22, 39B72; Secondary 26A48, 26A51, 91A30, 91C05
Published electronically:
October 4, 2000
MathSciNet review:
1814138
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: All strictly monotonic solutions of a general functional equation are determined. In a particular case, which plays an essential role in the axiomatization of rankdependent 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 YorkLondon, 1966. Translated by Scripta Technica, Inc.
Supplemented by the author. Edited by Hansjorg Oser. MR 0208210
(34 #8020)
 2.
J.
Aczél and J.
K. Chung, Integrable solutions of functional equations of a general
type, Studia Sci. Math. Hungar. 17 (1982),
no. 14, 51–67. MR 761524
(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), 29232929. 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, SpringerVerlag, New
YorkHeidelberg, 1975. A modern treatment of the theory of functions of a
real variable; Third printing; Graduate Texts in Mathematics, No. 25. MR 0367121
(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.\ {17} (1982), no.\ 14,
51–67; MR0761524 (85i:39008)], Studia Sci. Math. Hungar.
19 (1984), no. 24, 273–274. MR 874494
(87m:39006)
 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
(86i:39008)
 8.
R. D. Luce, Coalescing, event commutativity, and theories of utility, J. Risk Uncertainty, 16 (1998), 87114.
 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. 12, 21–30. MR 0611541
(58 #29529)
 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
(91g:00002)
 12.
A.
Wayne Roberts and Dale
E. Varberg, Convex functions, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1973. Pure and
Applied Mathematics, Vol. 57. MR 0442824
(56 #1201)
 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), 5167. 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), 29232929. 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), 273274. 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), 87114.
 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), 2130. MR 58:29529
 11.
 F. Riesz and B. SzokefalviNagy, 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
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
39B12,
39B22,
39B72,
26A48,
26A51,
91A30,
91C05
Retrieve articles in all journals
with MSC (2000):
39B12,
39B22,
39B72,
26A48,
26A51,
91A30,
91C05
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 926975100
Email:
jdaczel@math.uwaterloo.ca, janos@aris.ss.uci.edu
Gyula Maksa
Affiliation:
Institute of Mathematics and Informatics, University of Debrecen, H4010 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, H4010 Debrecen, Pf. 12, Hungary
Email:
pales@math.klte.hu
DOI:
http://dx.doi.org/10.1090/S0002993900056860
PII:
S 00029939(00)056860
Keywords:
Functional equation,
binary gamble,
rankdependent 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 T030082 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
