The Limits of Price Discrimination with a Bayesian Seller

2026-07-14

AI Summary

This study investigates the welfare frontier attainable by a seller engaging in third-degree price discrimination under Bayesian private production costs. By partitioning the market into segments to maximize seller surplus, the work characterizes the geometric structure of the feasible region in the (seller surplus, buyer surplus) space and demonstrates its equivalence to the projection of a polyhedron defined by flow conservation constraints in a discounted flow network. Leveraging polyhedral theory, linear programming, and convex analysis, the authors construct a polynomial-sized constraint system and introduce a convex combination representation of extremal markets, generalizing the equal-surplus piecewise structure known under fixed costs. Building on this foundation, they devise an algorithm that computes, in polynomial time, the optimal market segmentation for any given linear combination of seller and buyer surpluses.

Abstract

We study the limits of third-degree price discrimination when the production cost is Bayesian and private to the seller, generalizing the seminal work of Bergemann, Brooks and Morris (2015). The rough setup is the following: A monopoly seller sets different prices for buyers in different "segments" of the market so as to maximize seller surplus. Different ways in which the aggregate market is decomposed into segments lead to different welfare outcomes, i.e., (seller surplus, buyer surplus) pairs. When the production cost is Bayesian, the region of achievable welfare outcomes can exhibit complex shapes beyond the clean characterization by Bergemann, Brooks and Morris for the case with a fixed cost. We show that with a Bayesian cost, this region coincides with a proper projection of a polytope defined by a polynomial number of linear constraints, the essential ones of which correspond to flow conservation in a "discounted" flow network. As a result, we give a polynomial-time algorithm that computes optimal market segmentations in terms of any linear combination of the seller surplus and the buyer surplus. En route, we establish the following structural property: Any market can be written as a convex combination of "extremal markets" in a way preserving the seller surplus and the buyer surplus. These extremal markets are piecewise equal-surplus with respect to different possible costs, generalizing a similar notion introduced by Bergemann, Brooks and Morris when the cost is fixed.

Problem

• price discrimination
• Bayesian cost
• market segmentation
• welfare outcomes
• monopoly pricing

Innovation

• Bayesian price discrimination
• market segmentation
• polyhedral projection
• flow network
• extremal markets