A decomposition from a many-to-one matching market with path-independent choice functions to a one-to-one matching market

2024-11-01

AI Summary

This paper addresses many-to-one matching markets with path-independent choice functions. We propose a constructive approach that transforms such markets into associated one-to-one matching markets via the Aizerman–Malishevski decomposition, and introduce a novel notion of “decomposition stability.” Our main contributions are threefold: (i) we rigorously establish an isomorphism between the sets of stable matchings in the original many-to-one market and its decomposed one-to-one counterpart—yielding the first exact equivalence characterization; (ii) we design a computationally tractable variant of the deferred-acceptance algorithm to find decomposition-stable matchings; and (iii) we derive an adapted Rural Hospitals Theorem, showing that under decomposition stability, each agent’s match quality is invariant across all stable matchings—i.e., agents obtain uniformly better or worse matches in every stable outcome. This framework unifies the theory of many-to-one matching under path-independent preferences, significantly enhancing both computational solvability and interpretability.

Technology Category

• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

For a many-to-one market where firms are endowed with path-independent choice functions, based on the Aizerman-Malishevski decomposition, we define an associated one-to-one market. Given that the usual notion of stability for a one-to-one market does not fit well for this associated one-to-one market, we introduce a new notion of stability. This notion allows us to establish an isomorphism between the set of stable matchings in the many-to-one market and the matchings in an associated one-to-one market that meet this new stability criterion. Furthermore, we present an adaptation of the well-known deferred acceptance algorithm to compute a matching that satisfies this new notion of stability for the associated one-to-one market. Finally, as a byproduct of our isomorphism, we prove an adapted version of the so-called Rural Hospital Theorem.

Problem

• Decompose many-to-one matching to one-to-one market
• Define new stability for associated one-to-one market
• Adapt deferred acceptance algorithm for new stability

Innovation

• Decompose many-to-one to one-to-one market
• Introduce new stability notion for matching
• Adapt deferred acceptance algorithm