When Marilda Sotomayor and I wrote our 1990 book on matching, we divided the book into models without money (like the marriage model) and models with money (like auction models), and showed how similar results obtained for both kinds of models. There's been lots of recent progress in unifying those models and understanding why the results were so similar. The most recent paper in this literature is by John Hatfield and Scott Kominers, Many-to-Many Matching with Contracts, and I asked them to write a blog post about it, which appears below.
"A few years ago, Hatfield and Milgrom (2005) introduced "(many-to-one) matching with contracts," a generalization of classical two-sided matching in which contracts between agents specify both (1) a matching and (2) conditions of the match (such as wages, hours, or specific responsibilities). The results of Hatfield and Milgrom (2005) are surprisingly general--although presented using matching-theoretic language, the Hatfield and Milgrom (2005) model encompasses not only two-sided matching but also several package auction models.Hatfield and Milgrom (2005) assumed a many-to-one matching market, that is, a market in which agents on one side of the market (the "doctors") were never allowed to sign more than one contract. Although reasonable for some applications of matching (such as school choice), some matching applications (such as the United Kingdom Medical Intern match) allow "many-to-many matching," in which every agent can be assigned to multiple agents on the other side of the market.In "Many-to-Many Matching with Contracts," we develop a theory of "many-to-many” version of matching with contracts which extends the model of Hatfield and Milgrom (2005) to allow all agents to accept multiple contracts.This framework extends contract matching to a host of applications not previously covered by generalized matching theory, including the United Kingdom Medical Intern match (discussed above), the United States National Resident Matching Program (which allows agents to pair together and match as "couples" who can receive two jobs), and the matching of consultants to firms. Additionally, many-to-many matching with contracts generalizes the theory of bilateral "buyer-seller markets".
We prove that substitutability of preferences (for agents on both sides of the market) is both sufficient and necessary to guarantee the existence of stable many-to-many contract allocations; in many-to-one applications, in contrast, weaker conditions than substitutability guarantee existence. This result shows that, in general, a stable match is not guaranteed to exist in the matching with couples problem, since couples' preferences are generally assumed to be non-substitutable by nature. These results also provide insight into why the necessity result does not hold in the many-to-one matching case, and allows us to identify a new class of non-substitutable many-to-one preferences which are in some sense projections of substitutable many-to-many preferences, and for which the existence of a stable match is guaranteed."