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Post-Match Error Mitigation for Deferred Acceptance

Abraham Gale, Amélie Marian, David M. Pennock

TL;DR

This work proposes models for this new problem of deferred-acceptance matching algorithms that exhibit errors or changes to the matching inputs that are discovered only after the algorithm has been run and the results are announced to participants.

Abstract

Real-life applications of deferred-acceptance (DA) matching algorithms sometimes exhibit errors or changes to the matching inputs that are discovered only after the algorithm has been run and the results are announced to participants. Mitigating the effects of these errors is a different problem than the original match since the decision makers are often constrained by the offers they already sent out. We propose models for this new problem, along with mitigation strategies to go with these models. We explore three different error scenarios: resource reduction, additive errors, and subtractive errors. For each error type, we compute the expected number of students directly harmed, or helped, by the error, the number indirectly harmed or helped, and the number of students with justified envy due to the errors. Error mitigation strategies need to be selected based on the goals of the administrator, which include restoring stability, avoiding direct harm to any participant, and focusing the extra burden on the schools that made the error. We provide empirical simulations of the errors and the mitigation strategies.

Post-Match Error Mitigation for Deferred Acceptance

TL;DR

This work proposes models for this new problem of deferred-acceptance matching algorithms that exhibit errors or changes to the matching inputs that are discovered only after the algorithm has been run and the results are announced to participants.

Abstract

Real-life applications of deferred-acceptance (DA) matching algorithms sometimes exhibit errors or changes to the matching inputs that are discovered only after the algorithm has been run and the results are announced to participants. Mitigating the effects of these errors is a different problem than the original match since the decision makers are often constrained by the offers they already sent out. We propose models for this new problem, along with mitigation strategies to go with these models. We explore three different error scenarios: resource reduction, additive errors, and subtractive errors. For each error type, we compute the expected number of students directly harmed, or helped, by the error, the number indirectly harmed or helped, and the number of students with justified envy due to the errors. Error mitigation strategies need to be selected based on the goals of the administrator, which include restoring stability, avoiding direct harm to any participant, and focusing the extra burden on the schools that made the error. We provide empirical simulations of the errors and the mitigation strategies.
Paper Structure (30 sections, 3 theorems, 31 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 30 sections, 3 theorems, 31 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Motivations
  5. Related Work
  6. Problem Statement
  7. Traditional Matching Goals
  8. Error Mitigation Goals
  9. Affected Students
  10. Types of Errors
  11. Resource Reduction
  12. Size of Each Affected Students Group
  13. Mitigation Model
  14. Stable Expansion
  15. Subtractive Error
  16. ...and 15 more sections

Key Result

lemma 1

Stable Expansion produces a stable match.

Figures (5)

  • Figure 1: Affected and unaffected students change in outcome when the best school is removed, under the original condition (no removal), removal, and Stable expansion
  • Figure 2: The actual vs expected group size as we vary the proportion of lost applications
  • Figure 3: The effect of mitigation strategies when a subtractive error with $p=0.5$
  • Figure 4: The actual vs expected group size as we vary the proportion of erroneously added students
  • Figure 5: The effect of mitigation strategies with an Additive Error $N=10$

Theorems & Definitions (10)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • lemma 1
  • theorem 1
  • theorem 2