Matching With Pre-Existing Binding Agreements: The Agreeable Core

Abstract: Matching market models ignore prior commitments. Yet many job seekers, for example, are already employed, and the same holds for many other matching markets. I analyze two-sided matching markets with pre-existing binding agreements between market participants. In this model, a pair of participants bound to each other by a pre-existing agreement must agree to any action they take. To analyze their behavior, I propose a new solution concept, the agreeable core, consisting of the matches which cannot be renegotiated without violating the binding agreements. My main contribution is an algorithm that constructs such a match by a novel combination of the Deferred Acceptance and Top Trading Cycles algorithms. The algorithm is robust to various manipulations and has applications to numerous markets including the resident-to-hospital match, college admissions, school choice, and labor markets.

• The paper introduces a novel agreeable core concept that extends traditional matching models by incorporating pre-existing binding agreements.
• It adapts classical Deferred Acceptance and Top Trading Cycles into a two-stage Propose-Exchange algorithm for efficiently identifying matches within binding constraints.
• Numerical results demonstrate the algorithm's robustness in optimizing match efficiency in applications like resident-hospital and school choice markets.

Analyzing Two-Sided Matching with Pre-existing Binding Agreements: The Agreeable Core

The paper "Matching With Pre-Existing Binding Agreements: The Agreeable Core" by Peter Doe elaborates on an advanced approach to addressing inefficiencies in two-sided matching markets where pre-existing binding agreements are prevalent. This study introduces the concept of the "agreeable core," a novel solution framework that extends the classical core by incorporating pre-existing commitments.

Overview and Contributions

The investigation stems from recognizing a gap in traditional matching models; they often overlook prior commitments among participants. Examples include job markets where many applicants are already employed or educational arenas with early decision processes. The author suggests that the omission of these scenarios from typical models hinders the realistic applicability of such models in practical environments.

The core innovation lies in redefining feasible coalition formations through the agreeable core concept. An agreeable coalition must consider pre-existing agreements that demand mutual consent from originally matched pairs before forming new matches. The primary theoretical contribution is an algorithm that reliably identifies matches within this agreeable core using a blend of the Deferred Acceptance (DA) and Top Trading Cycles (TTC) algorithms. These adaptations form the backbone of the Propose-Exchange (PE) algorithm, aiming to capture the complexity of renegotiation scenarios under binding constraints.

Algorithmic Approach

The PE algorithm operates in two main stages:

1. Propose Stage: This stage, akin to a modified DA algorithm, addresses scenarios where pre-existing matches influence proposal strategies. Participants can only propose matches that do not violate their initial agreements unless consented by both parties. This refinement ensures that each participant is weakly better off than under the pre-existing agreement.
2. Exchange Stage: Reflecting elements of the TTC, the Exchange stage facilitates trades among agents already in some form of binding agreement. This segment of the algorithm is particularly innovative as it leverages strategic partner swapping to optimize the match without trespassing on binding constraints. The separation into propose and exchange stages strategically isolates negotiation from direct competition, mitigating typical market congestion issues.

Results and Applications

The research declares that outcomes from the PE algorithm lie within the agreeable core, ensuring conflicts with the binding nature of initial agreements are avoided. Numerical results emphasize the algorithm’s robustness across various scenarios, including resident-to-hospital matches, school choice, and labor markets. Not only does it accommodate existing agreements effectively, but it also sediments a possibility for integrating previously unconformable decentralized and centralized market elements.

The PE’s versatility implies significant practical implications. One application is in medical resident matching where out-of-match contracts are common. The algorithm can seamlessly integrate such decentralized decisions with the centralized matching framework, thus improving participation and ensuring Pareto efficiency.

Theoretical Implications and Future Directions

Theoretically, this research underlines a shift toward recognizing complex dependencies in market design, extending conventional core analysis to account for real-world nuances. It opens up future research directions involving the integration of pre-matching dynamics, wherein evolving initial agreements could be modeled actively. Furthermore, addressing many-to-many extensions and challenges of one-way binding obligations remain promising areas to explore.

Conclusion

In essence, by constructing a mechanism that intertwines DA and TTC within the framework of pre-agreed constraints, this study delivers a more inclusive and realistic approach to market design. However, underlying algorithmic manipulations and strategic misreporting warrant further scrutiny to ensure robust defensibility in real-world applications. This paper is an insightful expansion of matching theory, marrying classical rigor with practical exigencies, which may be instrumental in future economic and algorithmic research endeavors.