Consistency and monotonicity in matching markets with contracts

Acta Scientiarum Mathematicarum, 2013

In matching theory of contracts the substitutes condition plays an essential role to ensure the existence of stable matchings. We study manyto-many matchings where groups of individuals, of size possibly greater than two, are matched to a set of institutions. Real-world examples include orphan brothers accepting an adoptive family conditional on all of them being included; hiring contracts that may only be chosen together; or a situation where a …rm accepts to hire several workers only if they accept to work on di¤erent days (part-time jobs). We demonstrate by several examples that such extra conditions may alter the natural choice maps so that stable matchings cannot be obtained by applying the standard theorems. We overcome this di¢ culty by introducing a new construction of choice maps. We prove that they yield stable matchings if the construction respects an "anti-trust" rule on the supply side of the market.

2000

Some properties of the set of many-to-one stable matchings for firms that have responsive preferences and quotas are not necessarily true when firms' preferences are substitutable. In particular, we provide examples in which firms have substitutable preferences but firms and workers may be``single'' in one stable matching and matched in another one. We identify a set of axioms on firms' preferences guaranteeing that the set of unmatched agents is the same under every stable matching. We also propose a weaker condition than responsiveness, called separability with quotas or q-separability, that together with substitutability implies this set of axioms.

ACM SIGecom Exchanges, 2011

This note surveys recent work in generalized matching theory, focusing on trading networks with transferable utility. In trading networks with a finite set of contractual opportunities, the substitutability of agents' preferences is essential for the guaranteed existence of stable outcomes and the correspondence of stable outcomes with competitive equilibria. Closely analogous results hold when venture participation is continuously adjustable, but under a concavity condition on agents' preferences which allows for some types of complementarity.

Working Papers, 2006

Games and Economic Behavior, 2007

We study two-sided matching markets with couples and show that for a natural preference domain for couples, the domain of weakly responsive preferences, stable outcomes can always be reached by means of decentralized decision making. Starting from an arbitrary matching, we construct a path of matchings obtained from 'satisfying' blocking coalitions that yields a stable matching. Hence, we establish a generalization of Roth and Vande Vate's (1990) result on path convergence to stability for decentralized singles markets. Furthermore, we show that when stable matchings exist, but preferences are not weakly responsive, for some initial matchings there may not exist any path obtained from 'satisfying' blocking coalitions that yields a stable matching. JEL classification: C62, C78, D70.

2014

Abstract. This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties. JEL classification: C78

2022

In a many-to-one matching model with responsive preferences in which indifferences are allowed, we study three notions of core, three notions of stability, and their relationships. We show that (i) the core contains the stable set, (ii) the strong core coincides with the strongly stable set, and (iii) the super core coincides with the super stable set. We also show how the core and the strong core in markets with indifferences relate to the stable matchings of their associated tie-breaking strict markets.

Review of Economic Design, 2001

This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties.

2021

We explore the possibility of designing matching mechanisms that can accommodate non-standard choice behavior. We pin down the necessary and sufficient conditions on participants’ choice behavior for the existence of stable and incentive compatible mechanisms. Our results imply that well-functioning matching markets can be designed to adequately accommodate a plethora of choice behaviors, including the standard behavior that is consistent with preference maximization. To illustrate the significance of our results in practice, we show that a simple modification in a commonly used matching mechanism enables it to accommodate non-standard choice behavior.

International Journal of Game Theory, 2008

For the many-to-one matching model we give a procedure to partition the set of substitutable preference pro…les into equivalence classes with the property that all pro…les in the same class have the same set of stable matchings. This partition allows to reduce the amount of information required by centralized stable mechanisms.