Matching problems with preferences are all around us OCo they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their preference lists.In recent years there has been a sharp increase in the study of algorithmic aspects of matching problems with preferences, partly reflecting the growing number of applications of these problems worldwide. This book describes the most important results in this area, providing a timely update to The Stable Marriage Problem: Structure and Algorithms (D Gusfield and R W Irving, MIT Press, 1989) in connection with stable matching problems, whilst also broadening the scope to include matching problems with preferences under a range of alternative optimality criteria."

Matching under preferences involves matching agents to one another, subject to various optimality criteria such as stability, popularity, and Pareto-optimality, etc. Each agent expresses ordinal preferences over a subset of the others. Real-life applications include assigning graduating medical students to hospitals, high school students to colleges, public houses to applicants, and so on. We consider various matching problems with preferences. In this dissertation, we present efficient algorithms to solve them, prove hardness results, and develop linear programming theory around them. In the first part of this dissertation, we present two characterizations for the set of super-stable matchings. Super-stability is one of the optimality criteria when the preference lists contain ties. The first algorithm computes irreducible super-stable matchings in the super-stable matching lattice. The second algorithm takes O(mn) time, where m denotes the number of edges and n denotes the number of vertices and gives an explicit rotation poset that can be used to construct all super-stable matchings. In the second part, we present a polyhedral characterization of the set of all super-stable matchings, i.e. a linear system that is integral and describes the super-stable matching polytope. We also give alternative proof for the integrality of the strongly stable matching polytope. We also use linear programming techniques to solve an application of the stable matching problem. In the third part, we present NC algorithms for the popular matching problem. Popularity is another optimality criterion, where each agent gives a vote and the outcome matching has majority votes. In the last part, we consider envy-freeness, a relaxation of stability in the Hospitals/Residents setting, which allows blocking pairs involving a resident and an empty position of a hospital. Envy-free matching might not exist. We prove NP-hardness results of minimizing envy (if envy is inevitable) in terms of envy-pairs and envy-residents in the Hospitals/Residents Problem with Lower Quota

Two-sided matching provides a model of search processes such as those between firms and workers in labor markets or between buyers and sellers in auctions. This book gives a comprehensive account of recent results concerning the game-theoretic analysis of two-sided matching. The focus of the book is on the stability of outcomes, on the incentives that different rules of organization give to agents, and on the constraints that these incentives impose on the ways such markets can be organized. The results for this wide range of related models and matching situations help clarify which conclusions depend on particular modeling assumptions and market conditions, and which are robust over a wide range of conditions. 'This book chronicles one of the outstanding success stories of the theory of games, a story in which the authors have played a major role: the theory and practice of matching markets ... The authors are to be warmly congratulated for this fine piece of work, which is quite unique in the game-theoretic literature.' From the Foreword by Robert Aumann

Bachelor Thesis from the year 2009 in the subject Economics - Other, grade: 5.0, University of Zurich (Sozialökonomisches Institut (SOI)), language: English, abstract: Matching is the part of economics that deals with the question of who gets what, e.g. who gets which jobs, who goes to which university, who receives which organ or who marries whom. During the second part of the last century, many markets have been discovered to have failed in providing the necessary conditions for efficient matches. These market failures have partly evolved on ethical or institutional grounds, but are more generally attributed to congestion or coordination problems caused by the inability of the market to make it safe for participants to act on their private information. For this reason, a variety of allocation mechanisms have been developed and subsequently tested in field and laboratory experiments for possible implementation in real-world applications. This work attempts at giving a condensed review of different matching mechanisms and the performance of algorithms that have been implemented for solving the problems in their respective environments. The theoretical properties of these mechanisms as described in the increasingly vast literature on matching design will be used as a benchmark to compare their relative performance in terms of overall efficiency. The results yield some basic insights in the varying success of the competing algorithms in practice, indicating that both the quality of theoretical predictions and the actual performance of the algorithms decrease with the complexity of market environments. In particular, they show that imperfections of markets such as information asymmetry and incentive problems can have far-reaching consequences with respect to the effective working of matching procedures.

This book probes the stable marriage problem and its variants as a rich source of problems and ideas that illustrate both the design and analysis of efficient algorithms. It covers the most recent structural and algorithmic work on stable matching problems, simplifies and unifies many earlier proofs, strengthens several earlier results, and presents new results and more efficient algorithms.The authors develop the structure of the set of stable matchings in the stable marriage problem in a more general and algebraic context than has been done previously; they discuss the problem's structure in terms of rings of sets, which allows many of the most useful features to be seen as features of a more general set of problems. The relationship between the structure of the stable marriage problem and the more general stable roommates problem is demonstrated, revealing many commonalities.The results the authors obtain provide an algorithmic response to the practical, and political, problems created by the asymmetry inherent in the Gale Shapley solutions, leading to alternative methods and better compromises than are provided by the Gale Shapley method. And, in contrast to Donald Knuth's earlier work which primarily focused on the application of mathematics to the analysis of algorithms, this book illustrates the productive and almost inseparable relationship between mathematical insight and the design of efficient algorithms.Dan Gusfield is Associate Professor of Computer Science at the University of California, Davis. Robert W. Irving is Senior Lecturer in Computing Science at the University of Glasgow. The Stable Marriage Problem is included in the Foundations of Computing Series, edited by Michael Garey and Albert Meyer.