The Distribution of Envy in Matching Markets

TL;DR

This paper analyzes how envy is distributed among agents in random matching markets using the Deferred Acceptance algorithm, providing exact and asymptotic results and comparing with Random Serial Dictatorship.

Contribution

It introduces a novel probabilistic analysis of envy distribution in matching markets, connecting it with the coupon collector problem and deriving finite-market and asymptotic results.

Findings

Expected number of proposing agents whom nobody envies is computed.

Exact finite-market expression for proposing agents with no envy, linked to coupon collector problem.

Both DA and RSD leave a vanishing fraction of proposing agents unenvied in expectation.

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.