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

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.