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 individually-rational-core (IR-core) whereby blocking is only allowed via allocations where the non-blocking agents receive their endowments. We show that if there exists an allocation rule satisfying individual rationality, efficiency, and strategy-proofness, then for any problem for which the IR-core is non-empty, the allocation rule must choose an IR-core allocation and all agents are indifferent between all allocations in the IR-core. We further show that the result cannot be generalized to supersets of the IR-core. We apply our result to housing markets, coalition formation and networks.