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Multi-to -one dimensional and semi-discrete screening

Omar Abdul Halim, Brendan Pass

TL;DR

This work tackles multi‑dimensional screening by pairing a high‑dimensional consumer space with a one‑dimensional product space and introducing nestedness as a structural simplification that renders the problem tractable. It develops a semi‑discrete framework with a finite set of products and proves discrete nestedness under explicit convexity‑type conditions, enabling decoupled, one‑dimensional first‑order equations and often closed‑form solutions; it also extends the analysis to a continuum of products, showing existence of a nested solution and convergence of discrete approximations. The results connect monopolistic pricing to discrete and continuous optimal transport via $b$‑convex duality, yielding clear economic structure: when nested, agents’ choices partition neatly into consecutive product regimes with non‑intersecting indifference curves and a simple pricing rule. The approaches yield both analytical and efficient numerical methods, with explicit solutions in key cases and a rigorous continuum limit, offering a practical pathway to solving high‑dimensional screening problems in economics and related fields.

Abstract

We study the monopolist's screening problem with a multi-dimensional distribution of consumers and a one-dimensional space of goods. We establish general conditions under which solutions satisfy a structural condition known as nestedness, which greatly simplifies their analysis and characterization. Under these assumptions, we go on to develop a general method to solve the problem, either in closed form or with relatively simple numerical computations, and illustrate it with examples. These results are established both when the monopolist has access to only a discrete subset of the one-dimensional space of products, as well as when the entire continuum is available. In the former case, we also establish a uniqueness result.

Multi-to -one dimensional and semi-discrete screening

TL;DR

This work tackles multi‑dimensional screening by pairing a high‑dimensional consumer space with a one‑dimensional product space and introducing nestedness as a structural simplification that renders the problem tractable. It develops a semi‑discrete framework with a finite set of products and proves discrete nestedness under explicit convexity‑type conditions, enabling decoupled, one‑dimensional first‑order equations and often closed‑form solutions; it also extends the analysis to a continuum of products, showing existence of a nested solution and convergence of discrete approximations. The results connect monopolistic pricing to discrete and continuous optimal transport via ‑convex duality, yielding clear economic structure: when nested, agents’ choices partition neatly into consecutive product regimes with non‑intersecting indifference curves and a simple pricing rule. The approaches yield both analytical and efficient numerical methods, with explicit solutions in key cases and a rigorous continuum limit, offering a practical pathway to solving high‑dimensional screening problems in economics and related fields.

Abstract

We study the monopolist's screening problem with a multi-dimensional distribution of consumers and a one-dimensional space of goods. We establish general conditions under which solutions satisfy a structural condition known as nestedness, which greatly simplifies their analysis and characterization. Under these assumptions, we go on to develop a general method to solve the problem, either in closed form or with relatively simple numerical computations, and illustrate it with examples. These results are established both when the monopolist has access to only a discrete subset of the one-dimensional space of products, as well as when the entire continuum is available. In the former case, we also establish a uniqueness result.

Paper Structure

This paper contains 13 sections, 24 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formulation of the monopolist's problem
  3. Nestedness of solutions the semi-discrete monopolist's problem
  4. Significant intermediate results
  5. An alternate characterization of discretely nested solutions
  6. Explicit solutions
  7. Numerical computation
  8. Nestedness for a continuum of products
  9. Conclusion
  10. Connection to optimal transport
  11. Semi-discrete optimal transport
  12. Proofs
  13. Uniqueness of solutions

Key Result

Theorem 1

Under the hypotheses condition2--condition4, any solution $u\in \mathcal{U}$ of the monopolist's problem with data $(\mu, Y_N,c)$ is discretely nested.

Figures (2)

  • Figure 1: Comparison of regions $X_i$ for non-nested and nested $u\in\mathcal{U}$.
  • Figure 2: Comparison of regions $X_i$ and indifference curves behavior under varying model parameters.

Theorems & Definitions (56)

  • Definition 1
  • Example 1
  • Example 2
  • Theorem 1
  • Example 3
  • Theorem 2: Purchased goods are consecutive
  • Lemma 1
  • Theorem 3: Indifference curves cannot intersect at optimality
  • Lemma 2
  • Lemma 3
  • ...and 46 more