Law-invariant functionals that collapse to the mean

We discuss when law-invariant convex functionals "collapse to the mean''. More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine transformation, the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation. This extends results obtained in the literature in a bounded setting and under additional assumptions on the functionals. We illustrate the implications of our general results for pricing rules and risk measures. (C) 2021 The Author(s). Published by Elsevier B.V.