Near-Feasible Stable Matchings: Incentives and Optimality

TL;DR

This paper investigates near-feasible stable matchings, analyzing their stability, incentives, and optimal capacity modifications in complex matching models, and provides algorithms and experimental validation.

Contribution

It introduces a formal framework for incentive analysis in near-feasible matchings and develops efficient algorithms for optimal capacity modifications.

Findings

Capacity modifications can be optimal at individual and aggregate levels.

Minimal modifications align with minimal deviation incentives and are efficiently computable.

Algorithms and experiments demonstrate practical solutions for complex matching scenarios.

Abstract

Stable matching is a fundamental area with many practical applications, such as centralised clearinghouses for school choice or job markets. Recent work has introduced the paradigm of near-feasibility in capacitated matching settings, where agent capacities are slightly modified to ensure the existence of desirable outcomes. While useful when no stable matching exists, or some agents are left unmatched, it has not previously been investigated whether near-feasible stable matchings satisfy desirable properties with regard to their stability in the original instance. Furthermore, prior works often leave open deviation incentive issues that arise when the centralised authority modifies agents' capacities. We consider these issues in the Stable Fixtures problem model, which generalises many classical models through non-bipartite preferences and capacitated agents. We develop a formal framework to analyse and quantify agent incentives to adhere to computed matchings. Then, we embed near-feasible stable matchings in this framework and study the trade-offs between instability, capacity modifications, and computational complexity. We prove that capacity modifications can be simultaneously optimal at individual and aggregate levels, and provide efficient algorithms to compute them. We show that different modification strategies significantly affect stability, and establish that minimal modifications and minimal deviation incentives are compatible and efficiently computable under general conditions. Finally, we provide exact algorithms and experimental results for tractable and intractable versions of these problems.