Matching in Networks with Bilateral Contracts

Abstract

We introduce a model in which firms trade goods via bilateral contracts which specify a buyer, a seller, and the terms of the exchange. This setting subsumes (many-to-many) matching with contracts, as well as supply chain matching. When firms' relationships do not exhibit a supply chain structure, stable allocations need not exist. By contrast, in the presence of supply chain structure, a natural substitutability condition characterizes the maximal domain of firm preferences for which stable allocations always exist. Furthermore, the classical lattice structure, rural hospitals theorem, and one-sided strategy-proofness results all generalize to this setting.The theoretical literature on two-sided matching began with the simple one-to-one (marriage) model of Gale and Shapley (1962), in which agents on opposite sides of a market (men and women) seek to match into pairs. The central solution concept in this literature is stability, the requirement that, if two agents are not matched to each other, at least one of them prefers his or her assigned partner to the other agent. Gale and Shapley (1962) showed that stable one-to-one matches exist in general, and obtained conditions under which this existence result is preserved even if agents on one side of the market are allowed to match to multiple partners, that is, when the matching is many-to-one (as in college admissions and doctor-hospital matching). Following high-profile applications of matching in labor markets and school choice programs, 1 the foundational work on matching has been extensively generalized. 2 Kelso and Crawford (1982) extended many-to-one matching to a setting in which matches are supplemented by wage negotiations; Hatfield and Milgrom (2005) generalized this framework still further, by allowing agents to negotiate contracts which fully specify both a matching and the conditions of the match; the possibility of such a generalization was first noted by remarks of Crawford and Knoer (1981) and Kelso and Crawford (1982).Meanwhile, a host of work has studied the existence of stable matchings in many-to-many matching settings, two-sided markets in which all agents may match to multiple partners (as in the matching of consultants to firms). Many-to-many matching has been studied, for example, in the work of Sotomayor (1999Sotomayor ( , 2004, Echenique and Oviedo (2006), and Konishi andÜnver (2006). Recently, Walzl (2009) andKominers (2010) merged this line of research with that of Hatfield and Milgrom (2005), introducing a theory of many-to-many matching with contracts.

Trading networks with bilateral contracts

“…We can show graphically that our structure is more general than the setting described by Ostrovsky (2008) or Hatfield and Kominers (2012). Each firm f ∈ F is associated with a vertex of a directed multigraph (F, X) and each contract x ∈ X is a directed edge of this graph.…”

“…Hatfield et al (2015) showed that in general contract networks set-stable outcomes are equivalent to strongly trail-stable outcomes whenever choice functions satisfy full substitutability and Laws of Aggregate Demand and Supply. 9 However, Fleiner (2009) and Hatfield and Kominers (2012) showed that a set-stable outcomes may not exist in general contract networks (see Example 1 below). Moreover, our first result demonstrates that set stability is computationally intractable.…”

“…This creates a contract cycle. However, Hatfield and Kominers (2012) showed that if a contract network has a contract cycle then set-stable outcomes may fail to exist. Our first result shows that checking whether an outcome is in fact set-stable is computationally hard.…”

On Stable Matchings and Flows

“…They constructed a special stable marriage scheme with the help of a finite procedure, the so-called deferred acceptance algorithm. It also turned out that for the existence of a stable scheme, it is not necessary that the number of men is the same as the number of women or that After the original versions of our present work ( [7,8]), Hatfield et al generalized the Hatfield-Kominers result [6] in [9] to a "half-discrete" model (prices are arbitrary, but trading is done with integral amounts of goods) that allows cycles. The authors claim that under full substitutable preferences, there always exists a competitive equilibrium that corresponds to a stable outcome.…”

A many-to-many ‘rural hospital theorem’

“…Hatfield and Kojima (2010) introduced a weaker condition than substitutability called bilateral substitutability. 12 Hatfield and Kojima (2010, Theorem 6) extended R1 to bilaterally substitutable and cardinally monotonic preferences for many-to-one matching with contracts. Hatfield and Kominers (2012a) studied many-to-many matching with contracts where multiple contracts can be signed between any firm-worker pair.…”