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K-12 School Choice Mechanisms: Parallels to College Admissions Matching

Source: k12_school_choice.md

K-12 School Choice Mechanisms: Parallels to College Admissions Matching

This document surveys the empirical and theoretical literature on K-12 school choice mechanism design, focusing on the Boston mechanism failure, the NYC high school match, charter school lotteries, and lessons for college admissions simulation.

Boston Mechanism Failure and Switch to DA

How the Boston Mechanism Works

The Boston mechanism (also called the Immediate Acceptance mechanism) operates through sequential rounds where assignments are final:

  1. Round 1: Each student applies to their first-choice school. Each school ranks applicants by priority (walk zone, sibling, lottery number) and fills seats in priority order. Assignments are permanent.
  2. Round 2: Unassigned students apply to their second-choice school. Schools with remaining capacity fill seats by priority among new applicants only. Again, assignments are permanent.
  3. Round k: Unassigned students apply to their k-th choice. The process continues until all students are assigned or all schools are full.

The critical design feature is that assignments are irrevocable in each round. A student assigned in Round 1 can never be displaced, even if a higher-priority student lists that school second.

Why the Boston Mechanism Is Manipulable

The Boston mechanism is not strategy-proof: a student can be strictly better off by misrepresenting their true preferences. The core problem is that listing a popular school as your true first choice is risky. If you are rejected in Round 1, you lose priority at your second-choice school to students who listed it first -- even if you have higher baseline priority.

This creates a strategic dilemma: sophisticated families learn to avoid listing oversubscribed "reach" schools and instead rank less popular schools first to secure a guaranteed seat. The mechanism rewards strategic gaming over truthful preference revelation.

Abdulkadiroglu, Pathak, Roth, and Sonmez (2006) documented this empirically in Boston:

This fairness argument -- that the mechanism punishes honest, unsophisticated participants -- became the primary rationale for reform.

The Switch to Deferred Acceptance

In July 2005, the Boston School Committee voted to replace the Boston mechanism with a student-proposing Deferred Acceptance (DA) mechanism. The key differences:

Feature Boston Mechanism Deferred Acceptance
Assignments per round Final (irrevocable) Tentative (can be displaced)
Strategy-proof? No Yes (for students)
Favors Sophisticated families All families equally
Stability Not guaranteed Guaranteed

Under DA, truthful preference revelation is a dominant strategy -- families cannot gain by misrepresenting preferences, regardless of what others do. This property ("strategy-proofness") was framed as an equal access argument: all families, regardless of sophistication, receive the same quality of information from truthful reporting.

How Deferred Acceptance Works (Student-Proposing)

  1. Round 1: Each student "proposes" to their first-choice school. Each school tentatively holds the highest-priority applicants up to capacity and rejects the rest.
  2. Round 2+: Rejected students propose to their next-choice school. Schools consider new proposals together with currently held students, keep the top applicants by priority up to capacity, and reject the rest -- possibly displacing previously held students.
  3. Termination: The algorithm terminates when no student is rejected. All tentative holds become final assignments.

The student-proposing DA produces the student-optimal stable matching: the best outcome for students among all stable matchings.

Empirical Evidence from Boston's Reform

Research comparing outcomes before and after Boston's switch found:

The TTC Alternative

Abdulkadiroglu and Sonmez (2003) also proposed a Top Trading Cycles (TTC) mechanism for school choice:

NYC High School Matching

The Pre-2003 System: Uncoordinated Chaos

Before 2003, NYC's high school admissions for approximately 80,000 students per year was a deeply dysfunctional decentralized process:

The 2003 Reform: Introducing DA

In 2003, economists Atila Abdulkadiroglu, Parag Pathak, and Alvin Roth designed a centralized clearinghouse for NYC high school admissions based on the student-proposing Deferred Acceptance algorithm. Key features:

Results: Dramatic Improvement

The first year of the new mechanism (2003-04 admissions cycle):

Ongoing Challenges

Despite the dramatic improvements, the NYC system still faces complexities:

Empirical Welfare Gains

The Landmark NYC Welfare Study

Abdulkadiroglu, Agarwal, and Pathak (2017) conducted the most rigorous empirical welfare analysis of coordinated school assignment in their paper "The Welfare Effects of Coordinated Assignment: Evidence from the New York City High School Match" (American Economic Review).

Key findings:

  1. Magnitude of gains: The coordinated DA mechanism achieves 80% of the possible welfare gains when measured against the spectrum from a pure neighborhood-assignment baseline (no choice) to a utilitarian optimum
  2. Coordination dominates algorithm choice: The welfare gains from coordinating offers (eliminating the uncoordinated multi-offer problem) are far larger than the gains from choosing between specific algorithm variants (e.g., DA vs. TTC). Simply coordinating the market matters more than the fine details of the algorithm.
  3. Distributional effects: Students who were most likely to be administratively assigned under the old system experienced the largest gains in both welfare and subsequent academic achievement
  4. Attendance effects: Students matched to preferred schools showed improved attendance and reduced dropout rates

Why Coordination Matters More Than Algorithm

This finding has profound implications for market design: the primary source of inefficiency in the pre-2003 system was not that schools used the "wrong" algorithm, but that the admissions process was uncoordinated. Multiple simultaneous offers, seat-hoarding, and administrative placements destroyed value. Centralizing the market and using any reasonable mechanism would have captured most of the gains.

Boston's Evidence

Structural estimation of Boston's pre-reform system (Agarwal and Somaini, 2018) found:

Broader Empirical Patterns

Across multiple jurisdictions that adopted DA-based centralized matching:

Common findings: reduced administrative placements, increased match rates, greater transparency, and disproportionate benefits for disadvantaged families.

Charter School Lotteries

How Charter Lotteries Work

When a charter school is oversubscribed (more applicants than seats), federal and state law generally requires a random lottery to allocate seats. The process is straightforward:

  1. Applications are submitted during an open enrollment period
  2. If applications exceed capacity, a random lottery number is assigned to each applicant
  3. Students are offered seats in lottery order until capacity is filled
  4. Remaining students are placed on a waitlist in lottery order

Some charter schools have priority categories (siblings, neighborhood, etc.) that operate before the general lottery.

Lotteries as Research Instruments

Charter school lotteries have become one of the most important quasi-experimental research instruments in education economics. Because lottery assignment is random, it mimics a randomized controlled trial:

This is methodologically powerful because students who apply to charter schools differ systematically from those who do not (motivation, family engagement, etc.). The lottery eliminates this selection problem.

Key Findings from Lottery-Based Charter Research

A comprehensive review by Cohodes and Roy (2024), "Thirty Years of Charter Schools: What Does Lottery-Based Research Tell Us?", summarizes the evidence:

Limitations of the Lottery Framework

Connection to Matching Theory

Charter lotteries exist within the broader school choice ecosystem. In cities with unified enrollment systems (e.g., New Orleans, Denver, DC), charter and district school lotteries are integrated into a single DA-based matching mechanism:

The integration of charter lotteries into centralized matching represents a major advance in market design, reducing the coordination failures that plagued decentralized enrollment.

Implications for College Admissions Market Design

Why College Admissions Remains Decentralized

Unlike K-12, U.S. college admissions operates as a decentralized market: students apply to multiple colleges independently, colleges make independent admission decisions, and students choose among acceptances. This structure resembles the pre-2003 NYC system in important ways:

Problem Pre-2003 NYC Current College Admissions
Multiple simultaneous offers Yes Yes (students hold multiple acceptances)
Seat hoarding / yield uncertainty Principals used back channels Colleges use yield management, waitlists
Strategic behavior Students gamed rankings Students game ED/EA timing, demonstrated interest
Uncoordinated timing Multiple offer rounds ED, EA, REA, RD, EDII across different dates
Information asymmetry Families varied in sophistication First-gen students lack admissions knowledge

What If Colleges Adopted Centralized DA?

The theoretical and empirical K-12 evidence suggests several potential effects:

Potential benefits:

Potential obstacles and drawbacks:

Early Decision as a Partial Matching Mechanism

The current ED system functions as a primitive, inefficient matching mechanism:

The NRMP Analogy

The closest real-world analog to centralized college matching is the National Resident Matching Program (NRMP) for medical residencies:

The key lesson from the NRMP: centralized matching works best when both sides of the market agree to participate, and when there is an institutional framework to enforce participation.

Relevance to College Simulator

Direct Modeling Implications

The K-12 school choice literature provides several concrete insights for our college admissions simulator:

  1. Round structure maps to mechanism design: Our simulation's rounds (ED -> EA/REA -> EDII -> RD -> Student decisions -> Waitlist) are a decentralized multi-round mechanism. The K-12 literature clarifies that this structure creates strategic incentives that differ from a single centralized round.

  2. ED as a partial commitment device: The ED round in our simulator (with its 1.5x multiplier) parallels the Boston mechanism's first-round advantage. Students who "commit" early (ED) gain a priority boost, just as first-choice listings gain priority in the Boston mechanism. This creates the same incentive for strategic behavior: sophisticated students should apply ED to a school where the boost maximizes their probability of acceptance, not necessarily their true first choice.

  3. Yield management is the college-side response to decentralization: Our simulation models colleges accepting more students than they have seats (overadmission) because yield is uncertain. In a centralized matching, this would be unnecessary -- the algorithm handles coordination. The current yield management mechanics in our simulator are thus a realistic response to the decentralized market structure.

  4. Waitlist mechanics model real coordination failures: The waitlist round in our simulator represents the cleanup phase where the decentralized market tries to resolve uncoordinated offers -- directly analogous to the administrative placement phase in pre-2003 NYC.

  5. Information asymmetry and hooks: The K-12 literature's finding that unsophisticated families are harmed most maps to our simulation's hook system: recruited athletes (3.5x), donors (4x), and legacy students (2.5x) represent applicants who have information and coordination advantages analogous to sophisticated families in the Boston mechanism.

Potential Simulator Extensions

The K-12 literature suggests several possible extensions:

Key Takeaway

The central lesson from K-12 school choice for college admissions simulation: the structure of the matching mechanism -- not just the evaluation criteria -- fundamentally shapes outcomes. Our simulator currently models the evaluation side in detail (GPA, SAT, hooks, essays) but treats the mechanism (round structure, timing, commitment) as fixed background. The K-12 literature shows that mechanism design choices can matter as much as or more than evaluation criteria for determining who ends up where.

References