On the Heavy-Tail Behavior of the Distributionally Robust Newsvendor

Since the seminal work of Scarf (A min-max solution of an inventory problem) in 1958 on the newsvendor problem with ambiguity in the demand distribution, there has been a growing interest in the study of the distributionally robust newsvendor problem. The model is criticized at times for being conservative because the worst-case distribution is discrete with a few support points. However, it is the order quantity prescribed by the model that is of practical relevance. Interestingly, the order quantity from Scarf's model is optimal for a heavy-tailed distribution. In this paper, we generalize this observation by showing a heavy-tail optimality property of the distributionally robust order quantity for an ambiguity set where information on the first and the ath moment is known, for any real alpha > 1. We show that the optimal order quantity for the distributionally robust newsvendor is also optimal for a regularly varying distribution with parameter alpha. In the high service level regime, when the original demand distribution is given by an exponential or a lognormal distribution and contaminated with a regularly varying distribution, the distributionally robust order quantity is shown to outperform the optimal order quantity of the original distribution, even with a small amount of contamination.