Stable many-to-many matchings with contracts

Abstract

We consider several notions of setwise stability for many-to-many matching markets with contracts and provide an analysis of the relations between the resulting stable sets and pairwise stable sets for general, substitutable, and strongly substitutable preferences. Apart from obtaining "set inclusion results" on all three domains, we prove that for substitutable preferences the set of pairwise stable matchings is nonempty and coincides with the set of weakly setwise stable matchings. For strongly substitutable preferences the set of pairwise stable matchings coincides with the set of setwise stable matchings. We also show that Roth's (1984b) stability coincides with pairwise stability for substitutable preferences. JEL classification: C62, C78, D78, J41.

On the correspondence of contracts to salaries in (many-to-many) matching

“…6 Intuitively, this condition means that there are no two contracts x, z ∈ X which are sometimes "complements" in the sense that the availability of z makes x more attractive. 7 A number of alternative stability concepts are available for many-to-many matching settings (Blair, 1988;Echenique and Oviedo, 2006;Klaus and Walzl, 2009). In general, the choice of solution concept is somewhat immaterial for the present exercise: Embedding results analogous to Theorem 2 hold so long as the stability concept under consideration for the case of matching with salaries corresponds to that considered for the case of matching with contracts.…”

“…12 In modeling many-to-many matching with contracts, unitarity is an added assumption; the model of Klaus and Walzl (2009) imposes it, while that of Hatfield and Kominers (2010) does not. Hatfield and Kojima (2010) show (in their Theorem 3) that in unitary models of matching with contracts, substitutability implies the following Pareto separability condition similar to that of Roth (1984).…”

“…This latter condition, which I call unitarity, may be a natural assumption for (many-to-many) firm-worker matching markets in which each worker can serve in at most one role at each firm. 2 Unitarity is automatic in the many-to-one matching settings considered by Hatfield and Milgrom (2005) and Echenique (2012), as in those settings workers have unit demand for contracts; it is also assumed in the many-to-many matching with contracts model of Klaus and Walzl (2009). 3 Hatfield andMilgrom (2005) introduced the matching with contracts model as a generalization of the Kelso and Crawford (1982) model of many-to-one matching with salaries; the possibility of such a generalization was first noted in remarks of Crawford and Knoer (1981) and Kelso and Crawford (1982).…”

Stable schedule matchings

“…Our notion differs from the strong substitutes condition of Echenique and Ovideo [6], [7] and of Klaus and Walzl [14] because our definition is purely set-theoretical: we do not use any preference relations. In fact, we could introduce preference relations leading to our choice map.…”

“…This, together with a special preference relation yields a choice map for which the theory applies. Many papers have followed by adapting the theory to more realistic and more complex problems; we cite Hatfield and Milgrom [13], Echenique and Ovideo [6], [7], Klaus and Walzl [14] among the most recent ones: many others figure in their references.…”