Distribution-free expectation operators for robust pricing and stocking with heavy-tailed demand

TL;DR

This paper develops distribution-free bounds for key probabilistic quantities using a novel technique that simplifies optimization to two-point distributions, with applications to robust pricing and inventory decisions under demand uncertainty.

Contribution

It introduces a new elementary method to derive sharp bounds for distributions based solely on mean and dispersion, applicable to heavy-tailed demand scenarios.

Findings

Derived distribution-free bounds depending only on mean and dispersion.

Applied bounds to robust newsvendor and monopoly pricing models.

Extended classical mean-variance results to heavy-tailed demand distributions.

Abstract

We obtain distribution-free bounds for various fundamental quantities used in probability theory by solving optimization problems that search for extreme distributions among all distributions with the same mean and dispersion. These sharpest possible bounds depend only on the mean and dispersion of the driving random variable. We solve the optimization problems by a novel yet elementary technique that reduces the set of all candidate solutions to two-point distributions. We consider a general dispersion measure, with variance, mean absolute deviation and power moments as special cases. We apply the bounds to robust newsvendor stocking and monopoly pricing, generalizing foundational mean-variance works. This shows how pricing and order decisions respond to increased demand uncertainty, including scenarios where dispersion information allows for heavy-tailed demand distributions.