  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.
