On the core of the Shapley-Scarf housing market model with full preferences

We examine core concepts in the classical model of \cite{shapley1974cores} under full preferences. Among the standard notions, the strong core may be empty, whereas the weak core, though always nonempty, can be overly large and include unreasonable allocations. Our main findings are: (1) The exclusion core of Balbuzanov and Kotowski (2019) -- a recent concept shown to outperform standard cores in complex environments under strict preferences -- can also be empty. We establish a necessary and sufficient condition for its nonemptiness, showing that it is more often nonempty than the strong core. (2) We introduce two new core concepts, built on the exclusion core and the strong core respectively, by refining the assumptions on how indifferent agents may block. Both are nonempty and Pareto efficient, and coincide with the strong core whenever the latter is nonempty. (3) These core concepts are ordered by set inclusion, with the strong core as the smallest and the weak core as the largest.