We develop a model of matching with contracts which incorporates, as special cases, the college admissions problem, the Kelso-Crawford labor market matching model, and ascending package auctions. We introduce a new "law of aggregate demand" for the case of discrete heterogeneous workers and show that, when workers are substitutes, this law is satisfied by profit-maximizing firms. When workers are substitutes and the law is satisfied, truthful reporting is a dominant strategy for workers in a worker-offering auction/matching algorithm. We also parameterize a large class of preferences satisfying the two conditions.

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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.

Extended polymers are relevant in a variety of situations ranging from the classic coil-stretch problem to recent single molecule polymer experiments with DNA. We present theoretical calculations and computer simulations of the dynamic properties of extended single polymers. We discuss the effects of tension and hydrodynamics on t, the fundamental relaxation time of the polymer, and find that tension dominates the behavior of t. Furthermore, the symmetry breaking caused by extending the polymer "splits" t, leading to distinct longitudinal and transverse relaxation times. Our results are in agreement with recent experiments, and we discuss implications for the coil-stretch transition.[ S0031-9007(99) PACS numbers: 83.10. Nn, Extended polymers play an important role in many rheological problems: Under shear or strain forces, polymers will extend. A classic example is the coilstretch transition, in which polymers in a strain flow will undergo a phase transition from a coiled state to a highly stretched state when the strain rate exceeds a critical value [1]. The physics of this transition depend crucially on the relationship between hydrodynamic and entropic forces as a polymer is extended. Recent experiments with single molecules of DNA [2,3] have revealed its importance in a context outside biology-as a model system for studying polymer dynamics. When DNA is partially extended, one can use optical microscopy to image the internal modes of the polymer and study the normal mode structure. However, to analyze the results completely requires a detailed theory of the effects of extension.Here we calculate the fundamental relaxation time of a polymer as a function of its extension. The effects of nonlinear force curves and changing hydrodynamic interactions are incorporated in the theory. We have performed computer simulations of an extended polymer, the results of which are in agreement with the theory. The theoretical predictions are compared with the single molecule DNA data of Ref.[2], and their implications for the coil-stretch transition are discussed.Earlier theoretical studies of the dynamics of extended polymers [4,5] have made use of the blob model [6]. However, the blob model gives an incorrect prediction of polymer forces, which limits its applicability. For example, experiments with the synthetic polymer dextran have demonstrated that the force is best described by a modified freely jointed chain model [7]. We show below that a proper understanding of the forces of an extended polymer is crucial in predicting the dynamics. In the case of DNA, its stiffness precludes the application of blobs since the assumption fb͞k B T ø 1 is not generally met, where f is the force extending the polymer, b is the Kuhn length, k B is the Boltzmann constant, and T is the temperature. Other authors have considered the subtleties of extending polyelectrolyte polymers with flows and electric fields, but their primary results do not concern the dynamics of the polymer [8,9].Polymer dynamics are usually considered within the co...