Robust Stable Matchings: Dealing with Changes in Preferences

TL;DR

This work studies stability under perturbations by requiring matchings to be stable across two instances on the same agents. It develops a unified framework around stableMatching lattices, rotations, and Birkhoff representations to analyze robustness across increasingly general perturbations, identifying precise thresholds where lattice structure and polyhedral integrality persist. It provides both structural characterizations (sublattice and semi-sublattice properties) and algorithmic tools (Deferred Acceptance variants, bouquet computations, and LP formulations) that yield tractable results for several regimes, notably (0,1) and (1,n), including polynomial-time computation of worker- and firm-optimal robust matchings and XP-time algorithms for the general case. A key contribution is the sharp boundary: integrality of the robust polytope holds for (n,1) or (1,n) but can fail at (2,2), aligning with the breakdown of lattice structure, which informs the computational limits of LP-based approaches. The results advance understanding of robustness in stable matchings with uncertainty and offer practical algorithms for markets where preferences may change or be misreported.

Abstract

We study stable matchings that are robust to preference changes in the two-sided stable matching setting of Gale and Shapley [GS62]. Given two instances $A$ and $B$ on the same set of agents, a matching is said to be robust if it is stable under both instances. This notion captures desirable robustness properties in matching markets where preferences may evolve, be misreported, or be subject to uncertainty. While the classical theory of stable matchings reveals rich lattice, algorithmic, and polyhedral structure for a single instance, it is unclear which of these properties persist when stability is required across multiple instances. Our work initiates a systematic study of the structural and computational behavior of robust stable matchings under increasingly general models of preference changes. We analyze robustness under a hierarchy of perturbation models: 1. a single upward shift in one agent's preference list, 2. an arbitrary permutation change by a single agent, and 3. arbitrary preference changes by multiple agents on both sides. For each regime, we characterize when: 1. the set of robust stable matchings forms a sublattice, 2. the lattice of robust stable matchings admits a succinct Birkhoff partial order enabling efficient enumeration, 3. worker-optimal and firm-optimal robust stable matchings can be computed efficiently, and 4. the robust stable matching polytope is integral (by studying its LP formulation). We provide explicit counterexamples demonstrating where these structural and geometric properties break down, and complement these results with XP-time algorithms running in $O(n^k)$ time, parameterized by $k$, the number of agents whose preferences change. Our results precisely delineate the boundary between tractable and intractable cases for robust stable matchings.

Figures (11)

• Figure 1: Finding an integral stable matching from a fractional solution (Image from MM.Book-Online)
• Figure 2: Two examples of compressions. Lattice $\mathcal{L} = L(P)$. $P_1$ and $P_2$ are compressions of $P$, and they generate the sublattices in $\mathcal{L}$, of red and blue elements, respectively.
• Figure 3: $E_1$ (red edges) and $E_2$ (blue edges) define the sublattices in \ref{['ex:compression']}, of red and blue elements, respectively. $E_2$ and $E_3$ define the same compression and represent the same sublattice. All black edges in $E_1, E_2$ and $E_3$ are directed from top to bottom (not shown in the figure).
• Figure 4: Examples of: (a) canonical path, and (b) bouquet.
• Figure 5: An example in which $\mathcal{M}_A \setminus \mathcal{M}_B$ is not a sublattice of $\mathcal{L}_A$.

Theorems & Definitions (170)

• Definition 1
• Theorem 2: Birkhoff Birkhoff
• Lemma 1: GusfieldI, Theorem 2.5.4
• Lemma 2: GusfieldI, Lemma 3.2.1
• Lemma 3: GusfieldI, Lemma 3.3.2
• Lemma 4: Teo-Sethuraman-Lp
• Lemma 5: Teo-Sethuraman-Lp
• Theorem 3: VANDEVATE
• Theorem 4
• proof