We propose a solution to the conflict between fairness and efficiency in one-sided matching markets. A matching is essentially stable if any priority-based claim initiates a chain of reassignments that results in the initial claimant losing the object. We show that an essentially stable and Pareto efficient matching always exists and that Kesten's (2010) EADA mechanism always selects one while other common Pareto efficient mechanisms do not. Additionally, we show that there exists a student-pessimal essentially stable matching and that the Rural Hospital Theorem extends to essential stability. Finally, we analyze the incentive properties of essentially stable mechanisms.

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