A functional equation arising from simultaneous utility representations (Q1411440)

This paper works out consequences of assuming that two families of utility preferences, one additive and the other of increasing increments, hold simultaneously for a class of binary gambles. The assumption leads to the functional equation \[ f[h(x-y)+y]=f[h(x)]-f[h(y)]+f(y) \] for \(0\leq y\leq x<\kappa \), and the inequality \(h(x)\leq x\) for \(0\leq x\leq \kappa \), for strictly increasing functions \(f,h\) on a real interval \([0,\kappa [ \). The authors find all differentiable solutions. An open problem is to find all solutions without assuming differentiability.