In the housing market model of the market design theory, when agents’ preferences are not strict, the strong core may be empty, while the nonempty weak core may be inefficient and larger than the set of intuitively reasonable allocations. We solve this problem by proposing two new core concepts through modifying behavioral assumptions on unaffected agents in blocking coalitions. The two cores are nonempty, efficient, and coincide with the strong core whenever the strong core is nonempty. We generalize the concept of competitive allocations and show that there exist competitive allocations in the two cores.