A Minimax Perspective on Almost-Stable Matchings

Abstract

Stability is crucial in matching markets, yet in many real-world settings - from hospital residency allocations to roommate assignments - full stability is either unachievable or comes at the cost of leaving agents unmatched. In these cases, algorithmicists and market designers face a critical question: how should instability be measured and distributed among participants? Existing approaches to "almost-stable" matchings focus on aggregate measures, minimising the number of blocking pairs or the count of agents involved in blocking pairs. However, these objectives can result in concentrated instability on a few agents, raising concerns about fairness and incentives to deviate. We introduce a fairness-oriented approach to approximate stability based on the minimax principle: we seek matchings that minimise the maximum number of blocking pairs any agent is in. Equivalently, we minimise the maximum number of agents that anyone has justified envy towards. This distributional objective protects the worst-off agents from bearing a disproportionate amount of instability. We characterise the computational complexity of this notion across fundamental matching settings. Surprisingly, even very modest guarantees with respect to the distribution of instability prove computationally intractable: we show that it is NP-complete to decide whether a matching exists in which no agent is in more than one blocking pair, even when preference lists are bounded. This result applies to both Stable Roommates and maximum cardinality Stable Marriage. On the positive side, we provide polynomial-time algorithms when agents rank at most two others, and present approximation algorithms and integer programs for general settings. Our results map the algorithmic landscape and reveal fundamental trade-offs between distributional guarantees on justified envy and computational feasibility in matching market design.