Multi-to one-dimensional screening and semi-discrete optimal transport

2025-06-26

AI Summary

This paper studies the monopolist’s optimal screening problem under multidimensional consumer types and a one-dimensional product space, focusing on the structural condition of nestedness—its general validity mechanism. Methodologically, it develops a unified analytical framework integrating optimal transport theory, convex analysis, and variational inequalities. It establishes, for the first time, general sufficient conditions for nestedness in multi-to-one-dimensional screening. In the semi-discrete setting (finite product set), the paper rigorously proves solution uniqueness and proposes a hybrid solution method combining piecewise analytical derivation with low-dimensional numerical computation. The contributions significantly reduce computational complexity: closed-form solutions are obtained in the continuous case; structured, provably optimal pricing policies are derived for both discrete and continuous settings. These results provide an actionable theoretical foundation and algorithmic support for high-dimensional information design in mechanism design and pricing.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Study monopolist's screening with multi-dimensional consumers and one-dimensional goods
• Establish conditions for nestedness to simplify solution analysis
• Develop method for solving problem analytically or numerically

Innovation

• Multi-dimensional consumer distribution screening
• Nestedness structural condition analysis
• Semi-discrete optimal transport solution