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Multi-District School Choice: Playing on Several Fields

Yannai A. Gonczarowski, Michael Yin, Shirley Zhang

TL;DR

This paper extends the Pathak–Sönmez framework to a two-district setting where students can be constrained to their home district or unconstrained to choose any district. It shows that core predictions from the single-district literature can fail: sophisticated students may prefer district-level DA over BM and may even prefer for sincere peers to become sophisticated, with these effects robust in large random markets. The authors introduce a uniform two-district random model to demonstrate abundance of these phenomena and examine robustness to alternative definitions, including sincere-unconstrained heuristics. They also analyze the role of constraint types and provide counterexamples showing that in fully DA environments, some of these effects disappear, highlighting the importance of inter-district structure for mechanism design and equity considerations in school choice.

Abstract

We extend the seminal model of Pathak and Sönmez (2008) to a setting with multiple school districts, each running its own separate centralized match, and focus on the case of two districts. In our setting, in addition to each student being either sincere or sophisticated, she is also either constrained - able to apply only to schools within her own district of residence - or unconstrained - able to choose any single district within which to apply. We show that several key results from Pathak and Sönmez (2008) qualitatively flip: A sophisticated student may prefer for a sincere student to become sophisticated, and a sophisticated student may prefer for her own district to use Deferred Acceptance over the Boston Mechanism, irrespective of the mechanism used by the other district. We furthermore investigate the preferences of students over the constraint levels of other students. Many of these phenomena appear abundantly in large random markets.

Multi-District School Choice: Playing on Several Fields

TL;DR

This paper extends the Pathak–Sönmez framework to a two-district setting where students can be constrained to their home district or unconstrained to choose any district. It shows that core predictions from the single-district literature can fail: sophisticated students may prefer district-level DA over BM and may even prefer for sincere peers to become sophisticated, with these effects robust in large random markets. The authors introduce a uniform two-district random model to demonstrate abundance of these phenomena and examine robustness to alternative definitions, including sincere-unconstrained heuristics. They also analyze the role of constraint types and provide counterexamples showing that in fully DA environments, some of these effects disappear, highlighting the importance of inter-district structure for mechanism design and equity considerations in school choice.

Abstract

We extend the seminal model of Pathak and Sönmez (2008) to a setting with multiple school districts, each running its own separate centralized match, and focus on the case of two districts. In our setting, in addition to each student being either sincere or sophisticated, she is also either constrained - able to apply only to schools within her own district of residence - or unconstrained - able to choose any single district within which to apply. We show that several key results from Pathak and Sönmez (2008) qualitatively flip: A sophisticated student may prefer for a sincere student to become sophisticated, and a sophisticated student may prefer for her own district to use Deferred Acceptance over the Boston Mechanism, irrespective of the mechanism used by the other district. We furthermore investigate the preferences of students over the constraint levels of other students. Many of these phenomena appear abundantly in large random markets.
Paper Structure (27 sections, 8 theorems, 23 equations)

This paper contains 27 sections, 8 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Model
  4. Standard Concepts
  5. Single-District School Choice
  6. Mechanisms
  7. Multi-District School Choice
  8. Student Sophistication Types and Constraint Types
  9. Uniform (n; k) model
  10. Sophistication Types
  11. Example: A sophisticated student may prefer for a sincere student to become sophisticated
  12. Large-Market Analysis
  13. Necessity of a BM District
  14. Mechanism Choice
  15. Example: A sophisticated student may strictly prefer DA
  16. ...and 12 more sections

Key Result

Theorem 3.1

For every $p \in (0, 1)$, there exists $\tau > 0$ such that for any large enough $n$, in the uniform $(n; 3)$ model with category-probability lower bound $p$ and with one district using BM and the other using an arbitrary mechanism, there exists a set of sophisticated students of expected size at le

Theorems & Definitions (17)

  • Definition 2.2
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • Proposition 6.1
  • proof
  • ...and 7 more