The core of Shapley-Scarf markets with full preferences
Jun Zhang
TL;DR
The paper probes core nonemptiness and efficiency in the Shapley–Scarf housing market with full (including weak) preferences. It shows that the exclusion core can be empty under weak preferences and introduces two rectified concepts—rectified exclusion core (RExC) and rectified strong core (RSC)—which are always nonempty, Pareto efficient, and equivalence-closed, and relate to the strong and weak cores via $SC \subseteq ExC \subseteq RExC \subseteq RSC \subseteq WC$, with RExC and RSC coinciding with SC when SC is nonempty. The authors provide a Generalized Top Trading Cycles (GTTC) algorithm to locate elements of RExC and discuss a special multi-unit-object setting where ExC equals TTC outcomes. They further compare these rectified cores to other solution concepts like the vNM stable set, MSS, and the bargaining set, highlighting trade-offs between stability, efficiency, and informational assumptions under weak preferences and offering guidance for mechanism design in environments with indifferences.
Abstract
We examine core concepts in the classical model of Shapley and Scarf (1974) under full preferences. Among the standard concepts, the strong core may be empty, whereas the nonempty weak core may be overly large and contain inefficient elements. Our main findings are: (1) The exclusion core of Balbuzanov and Kotowski (2019) -- a recent concept outperforming standard concepts in complex environments under strict preferences -- can also be empty, yet it is more often nonempty than the strong core. (2) We introduce two new core concepts, respectively built on the exclusion core and the strong core. Both are nonempty and Pareto efficient, and coincide with the strong core whenever it 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.