🤖 AI Summary
This paper investigates the lattice structure induced by priority-neutral matching, revealing fundamental differences from the classical stable matching lattice. Using techniques from lattice theory, order theory, and matching market design, we establish three key results: (1) priority-neutral lattices are not necessarily distributive; (2) the pointwise minimum matching over students does not coincide with the lattice’s meet (greatest lower bound), refuting a prior conjecture; and (3) we characterize structural constraints on realizability, proving that not every finite lattice can be realized as a priority-neutral lattice. Collectively, these results systematically invalidate the transferability of several classical stable matching properties—including distributivity and meet-compatibility—to the priority-neutral setting, thereby delineating its intrinsic theoretical boundaries. The work provides foundational insights for designing fair, non-discriminatory matching mechanisms.
📝 Abstract
Stable matchings are a cornerstone of market design, with numerous practical deployments backed by a rich, theoretically-tractable structure. However, in school-choice problems, stable matchings are not Pareto optimal for the students. Priority-neutral matchings, introduced by Reny (AER, 2022), generalizes the set of stable matchings by allowing for certain priority violations, and there is always a Pareto optimal priority-neutral matching. Moreover, like stable matchings, the set of priority-neutral matchings forms a lattice. We study the structure of the priority-neutral lattice. Unfortunately, we show that much of the simplicity of the stable matching lattice does not hold for the priority-neutral lattice. In particular, we show that the priority-neutral lattice need not be distributive. Moreover, we show that the greatest lower bound of two matchings in the priority-neutral lattice need not be their student-by-student minimum, answering an open question. This show that many widely-used properties of stable matchings fail for priority-neutral matchings; in particular, the set of priority-neutral matchings cannot be represented by via a partial ordering on a set of rotations. However, by proving a novel structural property of the set of priority-neutral matchings, we also show that not every lattice arises as a priority-neutral lattice, which suggests that the exact nature of the family of priority-neutral lattices may be subtle.