🤖 AI Summary
In Shapley–Scarf housing markets, the strong core may be empty under strict preferences, the weak core—though always nonempty—often contains Pareto-inefficient allocations, and the exclusion core may also be empty. This paper systematically analyzes the existence and rationality of these three core concepts. First, we establish a necessary and sufficient condition for the nonemptiness of the exclusion core. Building upon this, we propose two novel core notions: the *robust exclusion core* and the *enhanced exclusion core*, both of which are guaranteed to be nonempty and Pareto optimal, and coincide with the strong core whenever the latter is nonempty. Using set-inclusion modeling and formal blocking analysis, we rigorously characterize the hierarchical relationships among all core variants. Our results demonstrate that the new concepts overcome the fundamental limitations of traditional cores—namely, emptiness and inefficiency—thereby significantly enhancing allocation stability and rationality.
📝 Abstract
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.