Law-invariant functionals that collapse to the mean: Beyond convexity

TL;DR

This paper explores conditions under which law-invariant functionals simplify to expectations, extending known results beyond convexity to quasiconvex and certain non-quasiconvex cases, with applications in finance, insurance, and economics.

Contribution

It provides a comprehensive framework for the collapse to the mean phenomenon for a wide class of law-invariant functionals beyond convexity, including quasiconvex and specific non-quasiconvex functionals.

Findings

Established collapse to the mean principles for quasiconvex functionals.

Extended collapse results to non-quasiconvex cases like risk measures and Choquet integrals.

Highlighted limitations of existing quantile formulations in optimization problems.

Abstract

We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the "collapse to the mean" for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the "collapse to the mean" to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis.