Conditional Stable Matchings

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. Address of V. Komornik (corresponding author): Département de mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg CEDEX, France, e-mail: [email protected] Address of Christelle K. Viauroux: University of Maryland Baltimore County, Department of Economics, 1000 Hilltop Circle, Baltimore, MD 21250, USA, e-mail: [email protected] 2000 Mathematics Subject Classi…cation. Primary: 91B68; Secondary: 90C27.