Strategy-Proofness and Essentially Single-Valued Cores Revisited

We consider general allocation problems with indivisibilities where agents' preferences possibly exhibit externalities. In such contexts many different core notions were proposed. One is the -core whereby blocking is only allowed via allocations where the non-blocking agents receive their endowment. We show that if there exists an allocation rule satisfying individual rationality, efficiency, and strategy-proofness, then for any problem for which the -core is non-empty, the allocation rule must choose a -core allocation and all agents are indifferent between all allocations in the -core. We apply our result to housing markets, coalition formation and networks.