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Solution of a functional equation arising from utility that is both separable and additive
Authors:Já  nos Aczé  l   Roman Ger   Antal Já  rai
Affiliation:Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada ~N2L 3G1 ; Institute of Mathematics, Silesian University, Bankowa 14, PL-40-007 Katowice, Poland ; Universität GH Paderborn, FB 17, D-33095 Paderborn, Germany
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.

Keywords:Utility measures   binary gambles   bounded   monotonic   continuous   differentiable functions   elevating boundedness to differentiability   functional equation with two unknown functions
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