Fuhito Kojima, Parag Pathak and I have a paper coming out that suggests a beginning of an answer to the empirical puzzle presented by the fact that the many annual labor matching markets with couples that use the Roth-Peranson algorithm overwhelmingly find stable matchings, even though in principle they might fail to exist. In large markets with short preference lists and without too many couples, the answer seems to be related to the fact that there remain sufficiently many unfilled positions so that vacancy chains are more likely to end than to cycle.
Here's the paper:
Kojima, Fuhito, Parag A. Pathak, and Alvin E. Roth, “Matching with Couples: Stability and Incentives in Large Markets,” April 2010, revised April 2013, Quarterly Journal of Economics, forthcoming.
Abstract: Accommodating couples has been a longstanding issue in the design of centralized labor market clearinghouses for doctors and psychologists, because couples view pairs of jobs as complements. A stable matching may not exist when couples are present. This paper's main result is that a stable matching exists when there are relatively few couples and preference lists are sufficiently short relative to market size. We also discuss incentives in markets with couples. We relate these theoretical results to the job market for psychologists, in which stable matchings exist for all years of the data, despite the presence of couples.
Manlove, D. (2013) Algorithmics Of Matching Under Preferences. Series: Theoretical Computer Science . World Scientific Publishing. ISBN 9789814425247
Publisher's URL: http://www.worldscientific.com/doi/pdf/10.1142/9789814425254_fmatter
Abstract
A new book by Dr David Manlove of the School of Computing Science has recently been published by World Scientific as part of their Series on Theoretical Computer Science. This book, called “Algorithmics of Matching Under Preferences”, deals with algorithms and complexity issues surrounding the matching of agents to one another when preferences are involved. For example, in several countries, centralised matching schemes handle the annual allocation of intending junior doctors to hospitals based on their preferences over one another. Efficient algorithms required to solve the underlying theoretical matching problems. Similar examples arise in the allocation of pupils to schools, students to projects, kidney patients to donors, and so on. The book surveys algorithmic results for a range of matching problems involving preferences, with practical applications areas including those mentioned above. It covers the classical Stable Marriage, Hospitals/Residents and Stable Roommates problems, where so-called stable matchings are sought, thereby providing an update to “The Stable Marriage problem, Structure and Algorithms”, by Dan Gusfield and Rob Irving, published by MIT Press in 1989. It also extends the coverage to the House Allocation problem, where stability is no longer the key requirement for a matching, and other definitions of optimality hold. This book builds on the author’s prior research in this area, and also his practical experience of developing, with colleagues including Rob Irving and Gregg O’Malley, algorithms for matching kidney patients to donors in the UK (collaborating with NHS Blood and Transplant), for assigning medical students to hospitals in Scotland (in collaboration with NHS Education for Scotland), and for allocating students to elective courses and projects (within the Schools of Medicine and Computing Science at the University of Glasgow, respectively). The book is also timely, as the research area recently came to the forefront in 2012 following the award of the Nobel Prize in Economic Sciences to Alvin Roth and Lloyd Shapley, two leading contributors to the field of matching theory and its application in practical settings, whose work is described in detail throughout the book. A Foreword is contributed by Kurt Mehlhorn of Max-Planck Institut fur Informatik, Saarbrucken, who wrote: “This book covers the research area in its full breadth and beauty. Written by one of the foremost experts in the area, it is a timely update to “The Stable Marriage Problem: Structure and Algorithms” (D. Gusfield and R.W. Irving, 1989). This book will be required reading for anybody working on the subject; it has a good chance of becoming a classic.”