Differentiability properties of rank-linear utilities

We study the differentiability properties of concave functionals defined as integrals of the quantile. These functionals generalize the rank dependent expected utility and are called rank-linear utilities in decision theory. Their superdifferential is described as well as the set of random variables where they are Gâteaux-differentiable. Our results generalize those obtained for the rank dependent expected utility in Ref. [Carlier, G., Dana, R.-A., 2003. Core of a convex distortion of a probability. Journal of Economic Theory 113, 199-222.].