Matching with Ownership

Abstract We consider a hybrid model at the intersection of the standard two-sided matching market as proposed by Gale and Shapley (1962) and a housing market as proposed by Shapley and Scarf (1974). Two sets of agents have to be matched in pairs to a common set of objects. Agents of one type have preferences that depend on not only the object they are matched to but also the agent of the other type matched to this object. The crucial difference lies in the fact that the common side is interpreted as an object and has no intrinsic preferences over the agents matched to it. We introduce a natural definition of the ownership of the objects that determines which agent owns the object he is matched to. Ownership restricts the objections of agents who are not owners and defines a notion of stability. We consider two natural ownership structures and show that stable matchings exist in both structures. The first ownership structure, i.e., one-side ownership, always gives ownership to agents of the same side. Even if this structure shares similarities with the classical two-sided matching framework, we show the following important difference: stable matchings and Pareto-efficient matchings can be disjoints, implying that the core can be empty. We also propose two subdomains of preferences, i.e., lexicographic and couple preferences, where core matchings exist in one-sided ownership structures. The second notion is joint ownership, where any reallocation of objects must be jointly agreed upon by the two agents initially assigned to them. As discussed in Morrill (2010), this notion is equivalent to Pareto-efficient matchings, and we discuss possible algorithms that can be used to check Pareto efficiency. Finally, we propose a general definition of ownership structures and show that one-sided ownerships are not the only ones that can guarantee the existence of stable matchings. To further investigate the link with the housing market literature, we also introduce an initial allocation to objects and define a core notion with respect to this initial allocation. We also show that in contrast to the standard setting, this housing market core can be empty. However, we show that in this housing market framework, there always exists a Pareto-efficient matching that is not blocked by any coalition of size two. In both settings, pairwise stability is the only minimal requirement that one can ensure.