📈 economics

The Distribution of Envy in Matching Markets

This paper analyzes the distribution of envy in random matching markets under the Deferred Acceptance algorithm, establishing that while the expected number of unenvied proposing agents equals the harmonic number $H_n$ (matching Random Serial Dictatorship), this group constitutes a vanishing fraction of the total market.

Original authors: Josué Ortega, Gabriel Ziegler, R. Pablo Arribillaga, Geng Zhao

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Josué Ortega, Gabriel Ziegler, R. Pablo Arribillaga, Geng Zhao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive school fair where thousands of students are trying to get into their dream schools, and thousands of schools are trying to pick their favorite students. There's a rulebook for how this happens, called the Deferred Acceptance (DA) algorithm. It's the system used in many real-world places (like high school admissions) because it's "stable"—meaning no student and school would rather swap partners than stay where they are.

But here's the question the paper asks: How fair does this actually feel?

Specifically, the authors want to know about envy.

The "Jealous" Student: A student who looks at another student's school and thinks, "I wish I was there instead."
The "Unwanted" Student: A student who is so happy with their school that nobody wishes they were in their spot.

The researchers used some fancy math (probability and statistics) to count how many students fall into these two categories in a random, chaotic market. Here is the breakdown in plain English.

1. The "Nobody Envies" Group (The Lucky Few)

The Metaphor: Imagine a game of musical chairs, but the chairs are schools.
The paper asks: How many students end up in a chair that no one else wants to sit in?

The Finding: Surprisingly, very few. The number of students whom nobody envies is roughly equal to the Harmonic Number ( $H_n$ ).
What does that mean? If you have 10,000 students, the number of people nobody envies is only about 9 or 10.
The Analogy: Think of a lottery with 10,000 tickets. If you ask, "How many people won the jackpot?" the answer is usually just one or two. Here, the "jackpot" is being in a school so perfect that no one else wants to trade with you. The paper shows that in a stable market, this "perfect spot" is incredibly rare. As the market gets bigger, the percentage of these lucky people shrinks to almost zero.

2. The "Envy Nobody" Group (The Top Choice Winners)

The Metaphor: Imagine a line of people grabbing cookies.
The paper asks: How many students get their absolute #1 favorite cookie?

The Finding: This group is larger than the first one, but still small. For 10,000 students, about 1,100 get their top choice.
The Analogy: If you have a line of people picking from a buffet, the first few people get exactly what they want. But as the line gets longer, the good stuff runs out. The math shows that in this specific "stable" system, only about 1 in 10 students gets their dream school. The rest have to settle for their 2nd, 3rd, or 10th choice.

3. The Big Surprise: Comparing to a "Random Line"

The authors compared this complex "stable" system (DA) to a much simpler, chaotic system called Random Serial Dictatorship (RSD).

RSD: Imagine students are put in a random line. The first person picks their #1 choice. The second person picks their #1 choice from what's left, and so on.
The Comparison:
- Top Choices: In the Random Line (RSD), about 50% of students get their top choice. In the Stable System (DA), only 10% do. The Stable System is much worse at giving people their dream school.
- The "Unwanted" Spot: Here is the magic trick. Even though the two systems work totally differently, they both leave exactly the same tiny number of students (about 10 out of 10,000) in spots that nobody envies.

Why is this weird?
It's like two different chefs cooking completely different recipes. One chef makes a gourmet meal (DA) and the other makes a random sandwich (RSD). You'd expect the results to be totally different. But if you ask, "How many people in the room are sitting in a chair that no one else wants?", both chefs end up with the exact same answer.

The Takeaway

The paper concludes that in large, random markets:

Stability comes at a cost: The system that prevents "justified envy" (people fighting over spots) does a terrible job of giving people their top choices.
The "Perfect" is rare: Whether the system is complex or random, the number of people who are so happy with their match that no one wants to swap with them is vanishingly small.
Most people are "improvable": Because so few people are in these "perfect" or "unwanted" spots, it means that for the vast majority of students, there is some way to rearrange the matches to make them happier. The "stable" outcome isn't the most efficient one; it's just the one where no one has a legal reason to complain.

In short: The "stable" matching system is great at stopping fights, but it's not great at making everyone happy. And strangely, whether you use a complex algorithm or just a random line, the number of people who are "untouchable" (either because they are too happy or too lucky) is always the same tiny fraction.

1. Problem Statement

The paper investigates the distribution of envy in random two-sided matching markets, specifically focusing on the Deferred Acceptance (DA) algorithm. While DA is celebrated for producing stable matchings (eliminating justified envy), it does not guarantee Pareto efficiency or the absence of unjustified envy.

The authors aim to quantify two specific subsets of agents (students) in a random market of size $n$ (where $n$ students match with $n$ schools):

Students whom nobody envies: Agents assigned to schools that no other student prefers over their own assignment (under-demanded schools).
Students who envy nobody: Agents assigned to their top-choice school.

Understanding the size of these sets is critical because:

Agents in either category are unimprovable: No Pareto-improving exchange starting from the DA outcome can make them better off.
They serve as a lower bound for the number of agents who cannot be improved upon.
The "students who envy nobody" metric is a proxy for the welfare measure of "top-choice assignment," often used by policymakers.

2. Methodology

The authors analyze a random school choice problem where:

There are $n$ students and $n$ schools.
Preferences and priorities are drawn independently and uniformly at random (i.i.d.).
The analysis relies on tools from applied probability, specifically the Coupon Collector Problem.

Key Analytical Techniques:

Sequential DA Implementation: The authors utilize the McVitie and Wilson (1971) sequential implementation of DA, where unmatched students apply to their next preferred school one by one.
Principle of Deferred Decisions: Preferences are generated "on the fly" as students make proposals. This allows the mapping of the DA execution to a sequence of independent uniform draws from the set of schools.
Coupling with Coupon Collector:
- The process of schools receiving proposals is mapped to collecting coupons.
- A school is "under-demanded" (envied by no one) if and only if it receives a proposal from exactly one distinct student during the entire execution.
- This is mathematically equivalent to the number of singleton coupon types in a Coupon Collector problem at the stopping time when all $n$ coupon types have been collected.
Asymptotic Approximation: For students who envy nobody, the authors approximate the distribution of assignment ranks using a geometric distribution derived from the known asymptotic behavior of the total number of applications in DA ( $\approx n H_n$ ).

3. Key Contributions and Results

A. Students Whom Nobody Envies (Under-Demanded Schools)

The authors derive an exact finite-market expression for the expected number of students whom nobody envies ( $NE_n$ ).

Result (Proposition 1):
$NE_n = H_n$
Where $H_n = \sum_{k=1}^n \frac{1}{k}$ is the $n$ -th Harmonic number.
Asymptotic Behavior: Since $H_n \approx \ln n + \gamma$ , the number of unenvied students grows only logarithmically with market size.
Implication: The fraction of unenvied students vanishes as $n \to \infty$ ( $H_n/n \to 0$ ). For a market of 10,000, the expected number of unenvied students is fewer than 10.

B. Students Who Envy Nobody (Top-Choice Assignments)

The authors provide an asymptotic characterization for the expected number of students who envy nobody ( $EN_n$ ).

Result (Proposition 2 & Corollary 1):
The rank distribution of students under DA approximates a geometric distribution with parameter $1/H_n$ . Consequently:
$EN_n \approx \frac{n}{H_n}$
Implication: While larger than the set of unenvied students, this group is still a vanishing fraction of the market. For $n=10,000$ , roughly 1,100 students get their top choice (compared to 10,000 total).

C. Comparison with Random Serial Dictatorship (RSD)

The paper compares DA results against Random Serial Dictatorship (RSD), a mechanism known for high efficiency but lacking stability.

Students Who Envy Nobody:
- RSD: $\approx n/2$ (A constant fraction of students get their top choice).
- DA: $\approx n/\ln n$ (A vanishing fraction).
- Conclusion: DA performs significantly worse than RSD regarding top-choice assignments.
Students Whom Nobody Envies:
- RSD: $H_n$ (Exact).
- DA: $H_n$ (Exact).
- Conclusion: Despite the structural differences between the mechanisms, the expected number of unenvied students is identical ( $H_n$ ) for both.

4. Significance and Discussion

Fundamental Property of Random Markets: The fact that both DA and RSD yield exactly $H_n$ unenvied students suggests this is a fundamental property of random matching markets rather than a specific deficiency of the DA algorithm. This result also extends to the Top Trading Cycles (TTC) mechanism with random endowments.
Pareto Efficiency Limits: The results highlight that the set of "unimprovable" agents (those who envy no one or are envied by no one) constitutes a vanishing fraction of the market as $n$ grows. This implies that for large markets, the DA outcome leaves a vast majority of agents in a state where they could theoretically be made better off via Pareto improvements, even though DA is stable.
Welfare Implications: While DA is stable, it fails to deliver top-choice assignments to a significant portion of the market compared to RSD. However, the "unenvied" metric (stability from the perspective of others) remains robustly low across mechanisms.
Exactness vs. Asymptotics: The paper provides a rare exact result ( $H_n$ ) for finite markets regarding unenvied students, contrasting with the vast majority of matching literature which relies on asymptotic approximations.

Summary Table of Findings

Metric	Deferred Acceptance (DA)	Random Serial Dictatorship (RSD)
Students who envy nobody (Top Choice)	$\approx \frac{n}{H_n}$ (Vanishing fraction)	$\frac{n+1}{2}$ (Constant fraction)
Students whom nobody envies (Unenvied)	$H_n$ (Logarithmic growth)	$H_n$ (Logarithmic growth)
Fraction of Market	Both groups vanish as $n \to \infty$	Unenvied vanishes; Top-choice remains constant

The paper concludes that while DA is stable, the "unimprovable" core of the market is extremely small in large random settings, and the specific count of unenvied agents is a universal constant ( $H_n$ ) across different mechanism classes.