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

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.