Characterization of Priority-Neutral Matching Lattices

TL;DR

The paper investigates the structure of priority-neutral matchings, a lattice-generalization of stable matchings that allows controlled priority violations to enable Pareto improvements. It proves that the $\mathsf{PN}$ lattice is not distributive and introduces a tight, constructive movement-lattice representation that captures all $\mathsf{PN}$ lattices via a good-chain property. This representation not only clarifies the richer order structure of $\mathsf{PN}$ but also yields a polynomial-time procedure to decide whether a given matching is priority-neutral. The results bridge PN with movement graphs and rotations, providing both a deeper theoretical understanding and practical algorithmic implications for testing priority-neutrality in one-to-one matching markets.

Abstract

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.

Figures (21)

• Figure 1: An example instance where the stable matching lattice is $\{ \mu_1, \mu_2, \mu_3\}$, with $\mu_1 < \mu_2 < \mu_3$. This illustrates the simple ordering constraints on stable matchings: we must ensure that $d_A$ is improved "before" $h_D$ is made worse. In contrast, we show that the ordering constraints on priority-neutral matchings are more involved: they may require only that some change, among many possibilities, is made "before" another change.
• Figure 2: Preferences and Priorities.
• Figure 3: Priority-neutral lattice.
• Figure 4: Matching and rotations used to denote the matchings in \ref{['fig:non-distributive-b']}.
• Figure 6: Illustration of matching lattices.

Theorems & Definitions (49)

• definition 2.1: Stable matchings and priority violations
• theorem 2.2: Distributive Lattices Characterize Stable Matchings
• definition 2.3: $\mathsf{EADAM}$ and the Kesten-Tang-Yu sequence
• definition 2.4: Priority-Neutrality
• theorem 2.5: Reny22
• proposition 3.1
• proof
• theorem 3.2: $\mathsf{PN}$ Is Not Distributive
• proof
• definition 4.1: Rotation DAG of distributive lattices