Learning from Historical Transactions: Robust Supplier Pricing and Stocking with Sparse Data

Abstract

Upstream suppliers often set wholesale prices and reserve capacity without direct access to the detailed demand information held by downstream retailers. We study how a supplier can learn from a short history of wholesale prices, the retail prices subsequently chosen by a better-informed retailer, and the associated purchase probabilities. A quantile-only method (Q) uses each retail price and purchase probability as a demand observation. Our decision-informed method (DI) also uses the wholesale price under which the retailer chose that price. This additional context rules out demand curves that fit the observed sales outcomes but cannot explain the retailer's past choices. We characterize the worst-case retailer response to a new wholesale price under a broad, nonparametric class of demand curves. When the retailer has a unique best price, the relevant uncertainty is summarized by the lowest demand that remains possible. When several prices are tied, the supplier instead evaluates a finite collection of induced-demand cases. We also distinguish two forms of imperfect behavior. Under condition error (CE), the observed price is retained but its link to the wholesale price may be imperfect. Under price error (PE), the observed price may lie near an unobserved exact optimum. Both extensions remain computationally finite. The resulting downstream-demand summary leads directly to robust wholesale-pricing and stocking decisions. Numerical experiments across six demand environments show that DI provides substantial value with only a few transactions and remains useful under moderate error; with three observations and $\alpha=0$, DI improves the profit-to-oracle ratio by 9.1--11.8 percentage points.