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The Distribution of Envy in Matching Markets

Josué Ortega, Gabriel Ziegler, R. Pablo Arribillaga, Geng Zhao

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.

The Distribution of Envy in Matching Markets

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 proposing agents unenvied in expectation. Our results show that these clearly unimprovable proposing agents constitute a vanishing fraction of the market.
Paper Structure (7 sections, 4 theorems, 11 equations, 1 figure, 1 table)

This paper contains 7 sections, 4 theorems, 11 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Model and Results
  3. Students whom Nobody Envies
  4. Students who Envy Nobody
  5. Simulations
  6. Comparison with Random Serial Dictatorship
  7. Discussion

Key Result

Proposition 1

In a random school choice problem of size $n$,

Figures (1)

  • Figure 1: Average number of students whom nobody envies (blue) or who envy nobody (red) under DA, together with their theoretical predictions (averages over 2,000 i.i.d. random problems for each $n$).

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • proof