Before revision:  Our first notion lies between the strong core and the weak core and is independent of Balbuzanov and Kotowski's (2019) exclusion core . Our second notion refines the first and the exclusion core by combining their different merits. We generalize the You Request My House - I Get Your Turn mechanism to find our core allocations .
After revision:  Our first notion lies between the strong core and the weak core and is independent of Balbuzanov and Kotowski's exclusion core to improve its performance. Our two core concepts share a common idea of correcting the misused altruism of unaffected agents in blocking coalitions. We propose a mechanism to find our core allocations .