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Correlation of Rankings in Matching Markets

Rémi Castera, Patrick Loiseau, Bary S. R. Pradelski

TL;DR

The study develops a copula-based framework to model differential correlation of candidate rankings across multiple decision-makers in a continuum matching market. It shows that higher cross-group correlation generally boosts overall efficiency, but increasing a group’s own correlation raises its own odds of remaining unmatched, creating a systematic inequality across groups. The authors extend tie-breaking analysis to multiple priority classes and intermediate correlation levels, and provide theoretical guarantees for a decreasing-cutoffs regime while validating findings through extensive numerical experiments. The results illuminate how algorithmic monoculture and differential information can generate efficiency gains alongside group inequalities, informing policy design in school, university, and job admissions.

Abstract

We study the role of correlation in matching markets, where multiple decision-makers simultaneously face selection problems from the same pool of candidates. We propose a model in which a candidate's priority scores across different decision-makers exhibit varying levels of correlation dependent on the candidate's sociodemographic group. Such differential correlation can arise in school choice due to the varying prevalence of selection criteria, in college admissions due to test-optional policies, or due to algorithmic monoculture, that is, when decision-makers rely on the same algorithms and data sets to evaluate candidates. We show that higher correlation for one of the groups generally improves the outcome for all groups, leading to higher efficiency. However, students from a given group are more likely to remain unmatched as their own correlation level increases. This implies that it is advantageous to belong to a low-correlation group. Finally, we extend the tie-breaking literature to multiple priority classes and intermediate levels of correlation. Overall, our results point to differential correlation as a previously overlooked systemic source of group inequalities in school, university, and job admissions.

Correlation of Rankings in Matching Markets

TL;DR

The study develops a copula-based framework to model differential correlation of candidate rankings across multiple decision-makers in a continuum matching market. It shows that higher cross-group correlation generally boosts overall efficiency, but increasing a group’s own correlation raises its own odds of remaining unmatched, creating a systematic inequality across groups. The authors extend tie-breaking analysis to multiple priority classes and intermediate correlation levels, and provide theoretical guarantees for a decreasing-cutoffs regime while validating findings through extensive numerical experiments. The results illuminate how algorithmic monoculture and differential information can generate efficiency gains alongside group inequalities, informing policy design in school, university, and job admissions.

Abstract

We study the role of correlation in matching markets, where multiple decision-makers simultaneously face selection problems from the same pool of candidates. We propose a model in which a candidate's priority scores across different decision-makers exhibit varying levels of correlation dependent on the candidate's sociodemographic group. Such differential correlation can arise in school choice due to the varying prevalence of selection criteria, in college admissions due to test-optional policies, or due to algorithmic monoculture, that is, when decision-makers rely on the same algorithms and data sets to evaluate candidates. We show that higher correlation for one of the groups generally improves the outcome for all groups, leading to higher efficiency. However, students from a given group are more likely to remain unmatched as their own correlation level increases. This implies that it is advantageous to belong to a low-correlation group. Finally, we extend the tie-breaking literature to multiple priority classes and intermediate levels of correlation. Overall, our results point to differential correlation as a previously overlooked systemic source of group inequalities in school, university, and job admissions.

Paper Structure

This paper contains 51 sections, 18 theorems, 38 equations, 12 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Outline
  4. Setup
  5. Model
  6. Correlation and coherence
  7. Stable matching and the supply and demand framework
  8. Welfare metrics and preliminary results
  9. Main results
  10. Comparative statics
  11. Tie-breaking
  12. The decreasing cutoffs assumption
  13. Theoretical guarantees.
  14. Numerical Experiments.
  15. Cutoffs.
  16. ...and 36 more sections

Key Result

Proposition 1

If two groups, $G_j, G_\ell$, have the same marginal distribution at college $C^i$, then for students whose first choice is college $C^i$, the probability of obtaining this college is the same for both groups, i.e., $R^{1, \sigma}_j = R^{1, \sigma}_\ell$.

Figures (12)

  • Figure 1: Differential correlation between two groups at two colleges.
  • Figure 2: Illustration of the effect of correlation increase on cutoffs and efficiency.
  • Figure 3: Gaussian copula and joint distributions for two colleges.
  • Figure 4: Variations of efficiency ($E$) and inequality ($L$).
  • Figure 5: Cutoffs as functions of groups 2's correlation $\theta_2$.
  • ...and 7 more figures

Theorems & Definitions (27)

  • Definition 1: Matchings, cutoffs, and stability
  • Definition 2: Market-clearing
  • Definition 3: Rank functions
  • Proposition 1
  • Proposition 2
  • Proposition 2
  • Definition 4: Efficiency and Inequality
  • Proposition 3
  • Theorem 1: Efficiency is increasing in correlation
  • Theorem 2: Low-correlation groups are advantaged
  • ...and 17 more