Working Paper
When is less choice enough? Centralization in multi-item pricing.
with Omar El Housni.
Abstract
We study the revenue loss from replacing a multi-item price menu by a single item-price offer. Under decentralized pricing, the seller posts item-specific prices and customers choose utility-maximizing items; under centralized pricing, the seller chooses one item and one price. We ask when this restriction preserves revenue, and when it is costly.
We show that the answer is governed by the geometry of preference heterogeneity. In finite-support markets, centralized revenue can be a factor equal to the number of customer segments below decentralized revenue, and this is tight. The loss is driven by horizontal heterogeneity: different segments prefer different items. By contrast, if all valuation vectors lie on a common nonnegative ray, customers may differ in willingness to pay but agree on relative item values, and centralized and decentralized pricing have the same optimal revenue. With k aligned preference blocks, a centralized offer obtains at least a 1/k fraction of decentralized revenue.
We prove that this no-gap result is robust to approximate alignment. If a valuation matrix is within entrywise distance ε of the nonnegative rank-one cone, the decentralized revenue advantage is at most O(nε), and this dependence is tight. We also extend the analysis to stochastic environments around a common aligned core and to additive item-level shocks. Overall, centralization can be costly under horizontal heterogeneity, but is revenue-safe when preferences are mostly aligned or vertical.
Draft available upon request.