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The Limits of Interval-Regulated Price Discrimination

Kamesh Munagala, Yiheng Shen, Renzhe Xu

TL;DR

This paper analyzes third-degree price discrimination under interval-based regulation, introducing passive and active intermediary models where a regulator constrains prices to a contiguous set $F$ of valuations. It proves that, for any feasible $F$, there exists market-segmentation schemes that preserve the unregulated uniform-price revenue for the seller while transferring the remaining surplus to buyers, and it fully characterizes the achievable CS–PS region as a right triangle with three algebraically defined extreme points. The authors develop three distinct segmentation constructions (one each for maximizing producer surplus, maximizing consumer surplus, and minimizing social welfare) and show how convex combinations yield any point within the region; they also extend the results to the active intermediary setting, where the regulator directly enforces prices within $F$. Collectively, the results provide a principled framework for the fairness-efficiency trade-offs inherent in interval-regulated price discrimination and offer constructive methods for regulator-friendly implementations in platforms and markets. The work advances the understanding of how pricing regulation interacts with optimal signaling and market segmentation to shape welfare outcomes.

Abstract

In this paper, we study third-degree price discrimination in a model first presented by Bergemann, Brooks, and Morris [2015]. Since such price discrimination might create market segments with vastly different posted prices, we consider regulating these prices, specifically, by restricting them to lie within an interval. Given a price interval, we consider segmentations of the market where a seller, who is oblivious to the existence of such regulation, still posts prices within the price interval. We show the following surprising result: For any market and price interval where such segmentation is feasible, there is always a different segmentation that optimally transfers all excess surplus to the consumers. In addition, we characterize the entire space of buyer and seller surplus that is achievable by such segmentation, including maximizing seller surplus, and simultaneously minimizing buyer and seller surplus. A key technical challenge is that the classical segmentation method of Bergemann, Brooks, and Morris [2015] fails under price constraints. To address this, we develop three intuitive but fundamentally distinct segmentation constructions, each tailored to a different surplus objective. These constructions maintain different invariants, reflect different economic intuitions, and collectively form the core of our regulated surplus characterization.

The Limits of Interval-Regulated Price Discrimination

TL;DR

This paper analyzes third-degree price discrimination under interval-based regulation, introducing passive and active intermediary models where a regulator constrains prices to a contiguous set of valuations. It proves that, for any feasible , there exists market-segmentation schemes that preserve the unregulated uniform-price revenue for the seller while transferring the remaining surplus to buyers, and it fully characterizes the achievable CS–PS region as a right triangle with three algebraically defined extreme points. The authors develop three distinct segmentation constructions (one each for maximizing producer surplus, maximizing consumer surplus, and minimizing social welfare) and show how convex combinations yield any point within the region; they also extend the results to the active intermediary setting, where the regulator directly enforces prices within . Collectively, the results provide a principled framework for the fairness-efficiency trade-offs inherent in interval-regulated price discrimination and offer constructive methods for regulator-friendly implementations in platforms and markets. The work advances the understanding of how pricing regulation interacts with optimal signaling and market segmentation to shape welfare outcomes.

Abstract

In this paper, we study third-degree price discrimination in a model first presented by Bergemann, Brooks, and Morris [2015]. Since such price discrimination might create market segments with vastly different posted prices, we consider regulating these prices, specifically, by restricting them to lie within an interval. Given a price interval, we consider segmentations of the market where a seller, who is oblivious to the existence of such regulation, still posts prices within the price interval. We show the following surprising result: For any market and price interval where such segmentation is feasible, there is always a different segmentation that optimally transfers all excess surplus to the consumers. In addition, we characterize the entire space of buyer and seller surplus that is achievable by such segmentation, including maximizing seller surplus, and simultaneously minimizing buyer and seller surplus. A key technical challenge is that the classical segmentation method of Bergemann, Brooks, and Morris [2015] fails under price constraints. To address this, we develop three intuitive but fundamentally distinct segmentation constructions, each tailored to a different surplus objective. These constructions maintain different invariants, reflect different economic intuitions, and collectively form the core of our regulated surplus characterization.
Paper Structure (61 sections, 35 theorems, 117 equations, 3 figures, 4 tables, 1 algorithm)

This paper contains 61 sections, 35 theorems, 117 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Information Intermediary.
  3. Regulation via Price Intervals.
  4. Overview of Results
  5. Passive Intermediary Model
  6. Feasibility with the Optimal Uniform Price Excluded.
  7. Feasibility Implies Consumer Optimality.
  8. Producer-Consumer Surplus Region.
  9. Insights.
  10. Technical Contribution: Segmentation Algorithms.
  11. Active Intermediary Model
  12. Additional Related Work
  13. Information Design and Persuasion.
  14. Price Discrimination.
  15. Price Regulation.
  16. ...and 46 more sections

Key Result

Theorem 1.1

In the passive intermediary model, for any feasible $F$, there is a market segmentation where the seller revenue equals that of uniform pricing without regulation The seller revenue in this model is always at least that of uniform pricing, since the seller can choose to ignore the segmentation., whi

Figures (3)

  • Figure 1: The illustration of the CS-PS feasible area in three different types of regulation.
  • Figure 2: Illustration of the idea behind constructing SW-minimizing market scheme.
  • Figure 3: Proportion of feasible sets among all regulated sets that exclude the optimal price. Details are provided in \ref{['sect:justification-feasibility']}.

Theorems & Definitions (77)

  • Example 1.1
  • Example 1.2: Continuation of \ref{['ex:intro_four']}
  • Theorem 1.1: Informal; see \ref{['thrm:passive-overview']}
  • Example 1.3: Continuation of \ref{['ex:intro_four']}
  • Theorem 1.2: Informal, see \ref{['thm:RSeller_base_bounds']}
  • Definition 2.1: Market Scheme
  • Definition 2.2: $F$-valid Market Scheme
  • Definition 2.3: $\mathbf{x}^*$-Feasible Price Set
  • Theorem 3.1: Main Theorem
  • Definition 3.1: Equal-Revenue Market
  • ...and 67 more