We describe a fixed-point based approach to the theory of bipartite stable matchings. By this, we provide a common framework that links together seemingly distant results, like the stable marriage theorem of Gale and Shapley [11], the Menelsohn-Dulmage theorem [21], the Kundu-Lawler theorem [19], Tarski's fixed point theorem [32], the Cantor-Bernstein theorem, Pym's linking theorem [22, 23] or the monochromatic path theorem of Sands et al. [29]. In this framework, we formulate a matroid-generalization of the stable marriage theorem and study the lattice structure of generalized stable matchings. Based on the theory of lattice polyhedra and blocking polyhedra, we extend results of Vande Vate [33] and Rothblum [28] on the bipartite stable matching polytope.

We study two generalised stable matching problems motivated by the current matching scheme used in the higher education sector in Hungary. The first problem is an extension of the College Admissions problem in which the colleges have lower quotas as well as the normal upper quotas. Here, we show that a stable matching may not exist and we prove that the problem of determining whether one does is NP-complete in general. The second problem is a different extension in which, as usual, individual colleges have upper quotas, but in addition, certain bounded subsets of colleges have common quotas smaller than the sum of their individual quotas. Again, we show that a stable matching may not exist and the related decision problem is NP-complete. On the other hand we prove that, when the bounded sets form a nested set system, a stable matching can be found by generalising, in non-trivial ways, both the applicant-oriented and college-oriented versions of the classical Gale-Shapley algorithm. Finally, we present an alternative view of this nested case using the concept of choice functions, and with the aid of a matroid model we establish some interesting structural results for this case.

We show how frictions and continuous transfers jointly affect equilibria in a model of matching in trading networks. Our model incorporates distortionary frictions such as transaction taxes and commissions. When contracts are fully substitutable for firms, competitive equilibria exist and coincide with outcomes that satisfy a cooperative solution concept called trail stability. However, competitive equilibria are generally neither stable nor Pareto‐efficient.

In SODA'10, Huang introduced the laminar classified stable matching problem (LCSM for short) that is motivated by academic hiring. This problem is an extension of the wellknown hospitals/residents problem in which a hospital has laminar classes of residents and it sets lower and upper bounds on the number of residents that it would hire in that class. Against the intuition that stable matching problems with lower quotas are difficult in general, Huang proved that this problem can be solved in polynomial time. In this paper, we propose a matroid-based approach to this problem and we obtain the following results. (i) We solve a generalization of the LCSM problem. (ii) We exhibit a polyhedral description for stable assignments of the LCSM problem, which gives a positive answer to Huang's question. (iii) We prove that the set of stable assignments of the LCSM problem has a lattice structure similarly to the ordinary stable matching model.• We solve a generalization of the LCSM problem.• By exhibiting a polyhedral description for stable assignments of the LCSM problem, we give a positive answer to Huang's question in [9].• We prove that similarly to the ordinary stable matchings, the set of stable assignments of the LCSM problem has a natural lattice structure.for any class C of C Ci . Hence, I i , J i ∈ I Ci follows from I, J ∈ I C . Moreover, by the condition in the lemma,for any class C of C Ci on which I i ∩ C is deficient. So, by the induction hypothesis,

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We consider general networks of bilateral contracts that include supply chains. We define a new stability concept, called trail stability, and show that any network of bilateral contracts has a trailstable outcome whenever agents' choice functions satisfy full substitutability. Trail stability is a natural extension of chain stability, but is a stronger solution concept in general contract networks. Trail-stable outcomes are not immune to deviations of arbitrary sets of firms. In fact, we show that outcomes satisfying an even more demanding stability property -full trail stability -always exist. For fully trailstable outcomes, we prove results on the lattice structure, the rural hospitals theorem, strategy-proofness and comparative statics of firm entry and exit. We pin down a condition under which trail-stable and fully trail-stable outcomes coincide. We then completely describe the relationships between various other concepts. When contracts specify trades and prices, we also show that competitive equilibrium exists in networked markets even in the absence of fully transferrable utility. The competitive equilibrium outcome is (fully) trail-stable.Keywords: trail stability, chain stability, set stability, matching markets, supply chains, networks, contracts, competitive equilibrium.JEL Classification: C78, L14 * We would like to thank Samson Alva, Scott Kominers and Michael Ostrovsky for their valuable comments on the recent versions of the paper. Vincent Crawford, Umut Dur, Jens Gudmundsson, Claudia Herrestahl, Paul Klemperer, Collin Raymond, and Zaifu Yang also gave great comments on much earlier drafts. We had enlightening conversations with Alex Nichifor, Alex Westkamp and M. Bumin Yenmez about the project. Moreover, we are also grateful to seminar participants at the Southern

Abstract:We describe a flow model related to ordinary network flows the same way as stable matchings are related to maximum matchings in bipartite graphs. We prove that there always exists a stable flow and generalize the lattice structure of stable marriages to stable flows. Our main tool is a straightforward reduction of the stable flow problem to stable allocations. For the sake of completeness, we prove the results we need on stable allocations as an application of Tarski's fixed point theorem.

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