Stability and Competitive Equilibrium in Matching Markets with Transfers JOHN WILLIAM HATFIELD Graduate School of Business, Stanford University and SCOTT DUKE KOMINERS Becker Friedman Institute, University of Chicago 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 substi- tutability 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. Categories and Subject Descriptors: J.4 [Computer Applications]: Social and Behavioral Sci- ences—Economics; K.4.4 [Computers and Society]: Electronic Commerce General Terms: Economics, Theory Additional Key Words and Phrases: Matching, Networks, Joint Ventures, Stability, Competitive Equilibrium, Core, Efficiency 1. INTRODUCTION In the half-century since Gale and Shapley [1962] introduced the stable marriage model, matching theory has been extended to encompass successively more general economic settings with relationship-specific utilities. The fundamental solution concept in this literature is stability, the condition that no group of agents can block the match outcome by recontracting. Stable outcomes have been shown to exist in two-sided matching markets—including those for which the matching process determines contractual terms in addition to partnerships—even when agents on both sides of the market may match to multiple agents on the other side. Crucial for these results, however, is a substitutability condition on agents’ preferences, which requires that when an agent is presented with new matching opportunities, that agent never desires a previously-rejected opportunity. 1 The existence results 1 Substituable preferences are sufficient and necessary for the existence of stable outcomes in set- tings of many-to-one matching (Roth [1984] proved the sufficiency result; Hatfield and Kojima [2008] proved the necessity result), and in settings of many-to-many matching with and without contracts (Roth [1984], Echenique and Oviedo [2006], Klaus and Walzl [2009], and Hatfield and Kominers [2011a] proved sufficiency results; Hatfield and Kojima [2008] and Hatfield and Komin- Authors’ addresses: hatfield john@gsb.stanford.edu, skominers@uchicago.edu This article describes the results of [Hatfield et al. 2011] and [Hatfield and Kominers 2011b]. The authors appreciate the helpful comments of Fuhito Kojima, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp. ACM SIGecom Exchanges, Vol. 10, No. 3, December 2011, Pages 29–34