Newsvendor under Mean-Variance Ambiguity and Misspecification

Abstract: Consider a newsvendor problem with an unknown demand distribution. When addressing the issue of distributional uncertainty, we distinguish ambiguity under which the newsvendor does not differentiate demand distributions of common distributional characteristics (e.g., mean and variance) and misspecification under which such characteristics might be misspecified (due to, e.g., estimation error and/or distribution shift). The newsvendor hedges against ambiguity and misspecification by maximizing the worst-case expected profit regularized by a distribution's distance to an ambiguity set of distributions with some specified characteristics. Focusing on the popular mean-variance ambiguity set and optimal-transport cost for the misspecification, we show that the decision criterion of misspecification aversion possesses insightful interpretations as distributional transforms and convex risk measures. We derive the closed-form optimal order quantity that generalizes the solution of the seminal Scarf model under only ambiguity aversion. This highlights the impact of misspecification aversion: the optimal order quantity under misspecification aversion can decrease as the price or variance increases, reversing the monotonicity of that under only ambiguity aversion. Hence, ambiguity and misspecification, as different layers of distributional uncertainty, can result in distinct operational consequences. We quantify the finite-sample performance guarantee, which consists of two parts: the in-sample optimal value and the out-of-sample effect of misspecification that can be decoupled into estimation error and distribution shift. This theoretically justifies the necessity of incorporating misspecification aversion in a non-stationary environment, which is also well demonstrated in our experiments with real-world retailing data.