Essentially Stable Matchings

We propose a solution to the trade-off between Pareto efficiency and stability in matching markets. We define a matching to be essentially stable if any claim initiates a chain of reassignments that ultimately results in the initial claimant losing the object she claimed to a third agent. Our definition is not only compatible with Pareto efficiency but also is practical, since explaining why claims are not valid is straightforward. We study the essentially stable set and show that it shares some, though not all, of the structure possessed by the stable set. We classify popular Pareto efficient mechanisms using our definition: those based on Shapley and Scarf's TTC mechanism are not essentially stable while Kesten's EADA mechanism is. Our analysis points to a trade-off among essential stability, Pareto efficiency, and strategy proofness: mechanisms exist that satisfy any two of these properties but no mechanism achieves all three