Characterization of Priority-Neutral Matching Lattices

We study the structure of the set of priority-neutral matchings. These matchings, introduced by Reny (AER, 2022), generalize stable matchings by allowing for priority violations in a principled way that enables Pareto-improvements to stable matchings. Known results show that the set of priority-neutral matchings is a lattice, suggesting that these matchings may enjoy the same tractable theoretical structure as stable matchings. In this paper, we characterize priority-neutral matching lattices, and show that their structure is considerably more intricate than that of stable matching lattices. To begin, we show priority-neutral lattices are not distributive, an important property that characterizes stable lattices and is satisfied by many other lattice structures considered in matching theory and algorithm design. Then, in our main result, we show that priority-neutral lattices are in fact characterized by a more-involved property which we term being a "movement lattice," which allows for significant departures from the order theoretic properties of distributive (and hence stable) lattices. While our results show that priority-neutrality is more intricate than stability, they also establish tractable properties. Indeed, as a corollary of our main result, we obtain the first known polynomial-time algorithm for checking whether a given matching is priority-neutral.