Rectified strong core in Shapley-Scarf housing markets with indifferences

Abstract

We study core concepts in the Shapley-Scarf housing market model under unrestricted weak preferences. Among standard concepts, the strong core may be empty while the nonempty weak core may contain Pareto inefficient allocations. We propose the rectified strong core by adding a single condition: an unaffected agent may join a blocking coalition only if every object he views as indifferent to his current assignment is owned by the coalition. The rectified strong core is always nonempty and Pareto efficient, lies between the weak and strong cores, and coincides with the strong core whenever the latter is nonempty. This condition is the most permissive admissibility requirement for unaffected blocking agents that guarantees nonemptiness, and extends the behavioral foundation of the strong core to weak preferences.