Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable Matching with Contingent Priorities

Ignacio Rios, Federico Bobbio, Margarida Carvalho, Alfredo Torrico

TL;DR

The paper studies stable matching in school choice when priorities can depend on the current matching through contingent priorities that favor siblings. It introduces two explicit implementations—absolute and partial—and a soft-priority variant, analyzes existence, complexity, and incentives, and provides mathematical programming formulations to compute or certify contingent-stable matchings. It also compares against Chilean school-choice benchmarks and demonstrates that incorporating contingent priorities can substantially increase the share of students assigned to their top choices and keep siblings together, with a practical hybrid soft approach balancing existence and gains. Overall, the work offers a rigorous framework and computational tools for designing family- and sibling-friendly school-choice mechanisms, with broad applicability beyond education to domains with joint preferences and complementarities.

Abstract

Using school choice as a motivating example, we introduce a stylized model of a many-to-one matching market where the clearinghouse aims to implement contingent priorities, i.e., priorities that depend on the current assignment, to prioritize students with siblings and match them together. We provide a series of guidelines and introduce two natural approaches to implement them: (i) absolute, whereby a prioritized student can displace any student without siblings assigned to the school, and (ii) partial, whereby prioritized students can only displace students that have a less favorable lottery than their priority provider. We study several properties of the corresponding mechanisms, including the existence of a stable assignment under contingent priorities, the complexity of deciding whether there exists one, and its incentive properties. Furthermore, we introduce a soft version of these priorities to guarantee existence, and we provide mathematical programming formulations to find such stable matching or certify that one does not exist. Finally, using data from the Chilean school choice system, we show that our framework can significantly increase the number of students assigned to their top preference and the number of siblings assigned together relative to current practice.

Stable Matching with Contingent Priorities

TL;DR

The paper studies stable matching in school choice when priorities can depend on the current matching through contingent priorities that favor siblings. It introduces two explicit implementations—absolute and partial—and a soft-priority variant, analyzes existence, complexity, and incentives, and provides mathematical programming formulations to compute or certify contingent-stable matchings. It also compares against Chilean school-choice benchmarks and demonstrates that incorporating contingent priorities can substantially increase the share of students assigned to their top choices and keep siblings together, with a practical hybrid soft approach balancing existence and gains. Overall, the work offers a rigorous framework and computational tools for designing family- and sibling-friendly school-choice mechanisms, with broad applicability beyond education to domains with joint preferences and complementarities.

Abstract

Using school choice as a motivating example, we introduce a stylized model of a many-to-one matching market where the clearinghouse aims to implement contingent priorities, i.e., priorities that depend on the current assignment, to prioritize students with siblings and match them together. We provide a series of guidelines and introduce two natural approaches to implement them: (i) absolute, whereby a prioritized student can displace any student without siblings assigned to the school, and (ii) partial, whereby prioritized students can only displace students that have a less favorable lottery than their priority provider. We study several properties of the corresponding mechanisms, including the existence of a stable assignment under contingent priorities, the complexity of deciding whether there exists one, and its incentive properties. Furthermore, we introduce a soft version of these priorities to guarantee existence, and we provide mathematical programming formulations to find such stable matching or certify that one does not exist. Finally, using data from the Chilean school choice system, we show that our framework can significantly increase the number of students assigned to their top preference and the number of siblings assigned together relative to current practice.
Paper Structure (67 sections, 16 theorems, 46 equations, 4 figures, 8 tables)

This paper contains 67 sections, 16 theorems, 46 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Framework.
  4. Properties.
  5. Formulations.
  6. Impact.
  7. Organization
  8. Literature
  9. Matching with families.
  10. Matching with couples.
  11. Matching with complementarities.
  12. Model
  13. Students.
  14. Levels.
  15. Schools.
  16. ...and 52 more sections

Key Result

Proposition 1

A contingent stable matching under absolute priorities may not exist regardless of the tie-breaking rule, even if families are of size at most two and the initial tie-breakers are at the family level.

Figures (4)

  • Figure 1: Instance for Evaluation
  • Figure 2: Preference of Assignment by Group
  • Figure 3: Sensitivity to Tie-Breaking Rule
  • Figure 4: Capacities are next to each school in parenthesis where each component indicates the level's capacity.

Theorems & Definitions (48)

  • Definition 1: Contingent Siblings Priority
  • Example 1
  • Definition 2: Contingent Sibling Priority, Refined
  • Definition 3: Absolute Contingent Priority
  • Definition 4: Partial Contingent Priority
  • Definition 5: Contingent Justified-Envy and Stability
  • Remark 1
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • ...and 38 more