Local non-bossiness

The student-optimal stable mechanism (DA), the most popular mechanism in school choice, is the only one that is stable and strategy-proof. However, when DA is implemented, a student can change the schools of others without changing her own. We show that this drawback is limited: a student cannot change her schoolmates while remaining at the same school. We refer to this new property as local non-bossiness and use it to provide a new characterization of DA that does not rely on stability. Furthermore, we show that local non-bossiness plays a crucial role in providing incentives to be truthful when students have preferences over their colleagues. As long as students first consider the school to which they are assigned and then their schoolmates, DA induces the only stable and strategy-proof mechanism. There is limited room to expand this preference domain without compromising the existence of a stable and strategy-proof mechanism.

💡 Research Summary

**
The paper investigates a subtle but important limitation of the well‑known student‑optimal stable mechanism (Deferred Acceptance, DA) used in school choice. While DA is famously the only mechanism that is both stable and strategy‑proof when students have strict preferences over schools, it suffers from “bossiness”: a student can alter the assignments of others without changing her own assignment. The authors introduce a new property, local non‑bossiness, which restricts this ability. Specifically, a student cannot change the set of her schoolmates without also changing the school to which she is assigned. This holds even when the student remains unassigned, preventing her from reshuffling the pool of unassigned students.

The first major contribution is to prove that DA satisfies local non‑bossiness (Theorem 1) and that this property is independent of both stability and strategy‑proofness. Moreover, any mechanism that is locally non‑bossy and strategy‑proof automatically satisfies a form of group‑strategy‑proofness restricted to schoolmates (local group strategy‑proofness). Consequently, DA emerges as the unique mechanism that is both stable and locally group‑strategy‑proof (Corollary 1).

The second contribution is an axiomatic characterization of DA that does not invoke stability. The authors consider a set of schools with capacities and define a mechanism as a function mapping any student population and their preference profiles to a matching. They show that a mechanism satisfies the following six axioms iff it is the DA mechanism for some priority profile:

1. Individual Rationality (IR) – no student is assigned to a school she finds unacceptable.
2. Weak Non‑Wastefulness (WNW) – if a seat is empty and some unassigned student prefers that school, the matching is inefficient.
3. Population‑Monotonicity (PM) – shrinking the set of students cannot make any remaining student worse off.
4. Strategy‑Proofness (SP) – truthful reporting is a weakly dominant strategy.
5. S‑WrARP – a restricted version of the revealed‑preference axiom applied to schools’ choice functions, ensuring consistency with priority orders and capacities.
6. Weak Local Non‑Bossiness (WLNB) – a relaxation of local non‑bossiness that only restricts unassigned students from altering the set of other unassigned students.

Theorem 2 proves the equivalence. The inclusion of S‑WrARP and WLNB is crucial for handling many‑to‑one settings (schools with multiple seats); they are trivially satisfied in the unit‑capacity case. Strengthening SP to group‑strategy‑proofness yields a corollary that characterizes DA when schools’ priority orders are acyclic, a setting where DA is also Pareto‑efficient.

The third major contribution extends the analysis to environments with externalities, where students care not only about their own school but also about the composition of their peers. In such settings, classic results often break down: stable matchings may not exist, and strategy‑proof mechanisms may be impossible. The authors focus on a restricted preference domain called school‑lexicographic preferences over colleagues. Under this domain, each student first ranks schools (as usual) and, conditional on being assigned to a particular school, is indifferent among all matchings that give her the same set of schoolmates. There is no restriction on how she orders matchings that assign her to the same school with different colleagues.

Theorem 3 shows that when students have school‑lexicographic preferences over colleagues, applying DA to the underlying school rankings yields a mechanism that is both stable and strategy‑proof on this richer domain. The intuition is that a student’s incentive to misreport can arise in two ways: (i) to obtain a better school (addressed by SP) and (ii) to keep the same school but alter her peers (addressed by local non‑bossiness). Because DA is locally non‑bossy, the second type of manipulation is impossible, and the first type is ruled out by SP. Hence DA uniquely satisfies stability and strategy‑proofness in this setting (Corollary 3).

Finally, the authors demonstrate that the preference domain cannot be significantly enlarged. Even a single student whose preferences are school‑lexicographic but not school‑lexicographic over colleagues suffices to destroy the existence of a stable, strategy‑proof mechanism. This highlights the delicate balance between modeling realistic peer effects and preserving desirable mechanism properties.

Practical implications: The results reassure policymakers that the widely used DA mechanism remains robust when students have realistic peer‑related concerns, provided those concerns are limited to the composition of classmates at the same school. Moreover, the new axiomatization offers a tool for evaluating alternative mechanisms without relying on stability, which can be hard to verify in practice. The paper also cautions that expanding preference reporting to capture richer externalities must be done carefully, as modest relaxations can render any stable, strategy‑proof design impossible.

In sum, the paper makes three intertwined contributions: (1) identifying and proving the local non‑bossiness property of DA; (2) delivering a stability‑free axiomatic characterization of DA using six natural axioms; and (3) showing that, under a modest extension of preferences to include classmates, DA uniquely delivers a stable and strategy‑proof outcome, while any broader extension quickly eliminates such mechanisms.