Core stability and core-like solutions for three-sided assignment games

In this paper, we study different notions of stability for three-sided assignment games. Since the core may be empty in this case, we first focus on other notions of stability such as the notions of subsolution and von Neumann-Morgenstern stable sets. The dominant diagonal property is necessary for the core to be a stable set, and also sufficient in the case where each sector of the market has two agents. Furthermore, for any three-sided assignment market, we prove that the union of the extended cores of all μ-compatible subgames, for a given optimal matching μ, is the core with respect to those allocations that are compatible with that matching, and this union is always non-empty.