Stable Matchings and the Core in a Matching Market with Ties and Matroid Constraints

Core Concepts

The set of weakly stable matchings is contained in the weak core, the set of strongly stable matchings coincides with the strong core, and the set of super-stable matchings coincides with the super core in a matching market with ties and matroid constraints.

Abstract

The paper considers a many-to-one matching market where ties in the preferences of agents are allowed. The agents' preferences are subject to matroid constraints, which generalize capacity constraints.
The key results are:

The set of weakly stable matchings is contained in the weak core.
The set of strongly stable matchings coincides with the strong core.
The set of super-stable matchings coincides with the super core.

These results generalize the findings of Bonifacio, Juarez, Neme, and Oviedo (2024) from the setting with capacity constraints to the more general matroid constraints.
The proofs rely on properties of matroids, such as the existence of fundamental circuits and the ability to exchange elements between bases. The paper also provides algorithmic implications, noting that under the assumption of having independence oracles for the matroids, the existence of strongly stable and super-stable matchings can be determined in polynomial time.

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Key Insights Distilled From

The Set of Stable Matchings and the Core in a Matching Market with Ties and Matroid Constraints

by Naoyuki Kami... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00342.pdf

Deeper Inquiries

How do the results extend to many-to-many matching markets with ties and matroid constraints

The results obtained in the paper can be extended to many-to-many matching markets with ties and matroid constraints by adapting the concepts and techniques discussed in the original context. In the case of many-to-many matching, the considerations would involve multiple agents being matched to multiple partners, introducing a more intricate network of preferences and constraints. By incorporating matroid constraints, which are more flexible and can capture a wider range of constraints beyond simple capacity limitations, the analysis can be expanded to encompass these complex scenarios. The key lies in modifying the definitions of stability, core, and blocking to suit the many-to-many setting while accounting for ties and matroid constraints.

What are the practical implications of the generalization from capacity constraints to matroid constraints in real-world matching problems

The generalization from capacity constraints to matroid constraints in real-world matching problems has significant practical implications. Matroid constraints offer a more nuanced and versatile way to model constraints in matching markets, allowing for the representation of various complex scenarios such as hierarchical capacity constraints or multi-dimensional constraints. This flexibility enables a more accurate reflection of real-world scenarios where matching is subject to diverse constraints beyond simple capacities. By applying matroid constraints, the matching process can be tailored to specific requirements, leading to more efficient and effective matching outcomes in practical applications such as school admissions, job placements, or resource allocations.

Can the techniques used in this paper be applied to other matching problems with more complex constraints, such as hierarchical constraints or multi-dimensional constraints

The techniques used in this paper can be applied to other matching problems with more complex constraints, such as hierarchical constraints or multi-dimensional constraints, by adapting the framework to accommodate these specific constraints. By incorporating the principles of stability, core, and blocking within the context of the given constraints, it is possible to analyze and derive optimal matching solutions that satisfy the constraints while ensuring stability and fairness. The key lies in understanding the underlying structure of the constraints, defining appropriate blocking criteria, and developing algorithms that can efficiently compute stable matchings within the given constraints. This approach can be extended to various domains where matching problems with intricate constraints arise, providing a systematic and effective way to address complex matching scenarios.