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

The problem of determining all utility measures over binary gambles that ar e both separable and additive leads to the functional equation f(v) = f(vw) + f [vQ(w)], v,vQ(w) is an element of [0; k); w is an element of [0; 1]. The following conditions are more or less natural to the problem: f strictl y 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 w ithout any of these conditions (except f : [0; k) --> R+:= [0; infinity), Q : [0,1] --> R+, both into). If we exclude two trivial solutions, then we g et as general solution f(v) = alpha v(beta) (beta >0, alpha >0; alpha = 1 f or f(1) = 1), which satisfies all the above conditions. The paper concludes with a remark on the case where the equation is satisfi ed only almost everywhere.