The Distribution of Envy in Matching Markets

2026-03-24

AI Summary

This study investigates the distribution of envy in random matching markets under the Deferred Acceptance (DA) algorithm, focusing on the number of proposers who are either unenvied by anyone or do not envy others. By establishing the first connection between the number of unenvied agents in DA and the classical coupon collector problem, and leveraging tools from applied probability, combinatorial analysis, and asymptotic methods, the authors derive an exact expression for the expected number of unenvied proposers in finite markets and provide asymptotic bounds for those who envy no one. A key finding is that both DA and Random Serial Dictatorship (RSD) yield an expected $H_n$ unenvied proposers, yet this fraction vanishes as market size grows. In contrast, RSD guarantees a constant fraction of agents their top choice, revealing a fundamental disparity in fairness between the two mechanisms.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the distribution of envy in random matching markets under the Deferred Acceptance (DA) algorithm. Using tools from applied probability, we compute the expected number of proposing agents whom nobody envies and those who envy nobody. We obtain an exact finite-market expression for the former, based on a connection with the coupon collector problem, and asymptotic bounds for the latter. To put these quantities into perspective, we compare them to their counterparts under Random Serial Dictatorship (RSD): while RSD assigns a constant fraction of agents to their top choice, both DA and RSD leave exactly $H_n$ proposing agents unenvied in expectation. Our results show that these clearly unimprovable proposing agents constitute a vanishing fraction of the market.

Problem

• envy
• matching markets
• Deferred Acceptance
• Random Serial Dictatorship
• coupon collector problem

Innovation

• Deferred Acceptance
• envy distribution
• matching markets
• coupon collector problem
• Random Serial Dictatorship