Robust Price Discrimination

We consider a model of third-degree price discrimination, in which the seller has a valuation for the product which is unknown to the market designer, who aims to maximize the buyers' surplus by revealing information regarding the buyer's valuation to the seller. Our main result shows that the regret is bounded by $U^*(0)/e$, where $U^*(0)$ is the optimal buyer surplus in the case where the seller has zero valuation for the product. This bound is attained by randomly drawing a seller valuation and applying the segmentation of Bergemann et al. (2015) with respect to the drawn valuation. We show that the $U^*(0)/e$ bound is tight in the case of binary buyer valuation.