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The core is a traditional and useful solution concept in economic theory. Butin discrete exchange economies without transfers, when endowments are complex,the core may be empty. This motivates Balbuzanov and Kotowski (2019) tointerpret endowments as exclusion rights and propose a new concept calledexclusion core. Our contribution is twofold. First, we propose a rectificationof the core to solve its problem under complex endowments. Second, we propose arefinement of Balbuzanov and Kotowski's exclusion core to improve itsperformance. Our two core concepts share a common idea of correcting themisused altruism of unaffected agents in blocking coalitions. We propose amechanism to find allocations in the two cores.

Cores in discrete exchange economies with complex endowments