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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Solution of a functional equation
arising from utility that is both
separable and additive


Authors: János Aczél, Roman Ger and Antal Járai
Journal: Proc. Amer. Math. Soc. 127 (1999), 2923-2929
MSC (1991): Primary 39B22, 90D05
Published electronically: April 23, 1999
MathSciNet review: 1605911
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Abstract: The problem of determining all utility measures over binary gambles that are both separable and additive leads to the functional equation

\begin{equation*}f(v) = f(vw) + f[vQ(w)], \qquad v, vQ(w) \in [0,k),\quad w \in [0,1]\,. \end{equation*}

The following conditions are more or less natural to the problem: $f$ strictly increasing, $Q$ strictly decreasing; both map their domains onto intervals ($f$ onto a $[0,K)$, $Q$ onto $[0,1]$); thus both are continuous, $k>1$, $f(0)=0$, $f(1)=1$, $Q(1)=0$, $Q(0)=1$. We determine, however, the general solution without any of these conditions (except $f: [0,k) \to \mathbb{R}_{+}\!:= [0, \infty )$, $Q:[0,1]\to \mathbb{R}_{+}$, both into). If we exclude two trivial solutions, then we get as general solution $f(v) = \alpha v^{\beta }$ ($ \beta > 0$, $\alpha > 0$; $\alpha = 1$ for $f(1) = 1$), which satisfies all the above conditions. The paper concludes with a remark on the case where the equation is satisfied only almost everywhere.


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Additional Information

János Aczél
Affiliation: Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: jdaczel@math.uwaterloo.ca

Roman Ger
Affiliation: Institute of Mathematics, Silesian University, Bankowa 14, PL-40-007 Katowice, Poland
Email: romanger@us.edu.pl

Antal Járai
Affiliation: Universität GH Paderborn, FB 17, D-33095 Paderborn, Germany
Address at time of publication: Department of Numerical Analysis, Eötvös Lóránd University, Pázmány Péter Sétány 1/D, H-1117 Budapest, Hungary
Email: jarai@uni-paderborn.de, ajarai@moon.inf.elte.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04863-7
PII: S 0002-9939(99)04863-7
Keywords: Utility measures, binary gambles, bounded, monotonic, continuous, differentiable functions, elevating boundedness to differentiability, functional equation with two unknown functions
Received by editor(s): June 17, 1997
Received by editor(s) in revised form: December 23, 1997
Published electronically: April 23, 1999
Additional Notes: The research of the first author has been supported in part by the NSERC (Canada), grant no. OGP002972. He is also grateful for the hospitality of the Institute for Mathematical Behavioral Sciences, University of California, Irvine.
The research of the second author has been supported by the KBN (Poland) grant no. 2 P03A 049 09.
The research of the third author has been supported by the OTKA (Hungary) grant no. T-016846.
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society