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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
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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 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.


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

  • 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 $f(\lambda (x)+g(y))=\mu (x)+h(x+y)$, 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

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