Beyond Expected Profit: How Risk Aversion Alters Learning in the Data-Driven Newsvendor

Abstract

We study how risk aversion changes learning and the value of data in the classical newsvendor problem. We evaluate data-driven policies through the Conditional Value-at-Risk (CVaR) of realized loss on the combined draw of an i.i.d.\ historical sample and an independent future demand from the same distribution. We measure worst-case regret over demand distributions on a bounded interval relative to an oracle that knows the distribution. We show that, for every fixed positive risk level, Sample Average Approximation (SAA) can retain regret bounded away from zero as the sample size \(n\) grows, even at a fixed demand distribution, and characterize its asymptotic worst-case regret. We trace this failure to persistent variability in the learned order, which contributes to tail risk under this combined-risk evaluation. We establish that risk aversion also slows the minimax learning rate over all data-driven policies from \(Θ(n^{-1/2})\) under expectation to \(Θ(n^{-1/4})\). We construct policies that average nearby order statistics and attain this optimal rate. For finite samples, we reduce worst-case analysis for mixtures of convex combinations of order statistics to Bernoulli demand on the endpoints of the interval. We use this reduction to formulate a linear program for computing minimax policies within this class to any accuracy. In numerical experiments, we find that minimizing this objective alone can leave substantial average CVaR regret when each learned order is evaluated over future demand alone. We therefore develop a two-stage linear program that minimizes the worst-case value of this average regret while allowing a prescribed increase in the original objective. This yields improvements across a broad range of scenarios. Our results show that risk aversion changes both the rate at which data improve decisions and the policies needed to realize that improvement.