We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2+e)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.
Several works have addressed the last question with experiments. Quinn @cite_34 observes experimentally that a matching with only a fraction of unstable edges emerges long before the Gale--Shapley algorithm converges. Lu and Zheng @cite_13 propose a parallel algorithm that outperforms the Gale--Shapley algorithm in practice. Theorem gives theoretical support to the findings in Quinn @cite_34 . Theorem also addresses the concern expressed in the conclusions of Lu and Zheng @cite_13 where it is claimed that Most of existing parallel stable matching algorithms cannot guarantee a matching with a small number of unstable pairs within a given time interval.'' Theorem suggests that if the number of acceptable partners for each participant is bounded, the Gale--Shapley algorithm guarantees a small relative number of unstable edges.
@cite_34 @cite_13
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We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2+e)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching. This paper addresses strategies for the stable marriage problem. For the Gale-Shapley algorithm with men proposing, a classical theorem states that it is impossible for every cheating man to get a better partner than the one he gets if everyone is truthful. We study how to circumvent this theorem and incite men to cheat. First we devise coalitions in which a nonempty subset of the liars get better partners and no man is worse off than before. This strategy is limited in that not everyone in the coalition has the incentive to falsify his list. In an attempt to rectify this situation we introduce the element of randomness, but the theorem shows surprising robustness: it is impossible that every liar has a chance to improve the rank of his partner while no one gets hurt. To overcome the problem that some men lack the motivation to lie, we exhibit another randomized lying strategy in which every liar can expect to get a better partner on average, though with a chance of getting a worse one. Finally, we consider a variant scenario: instead of using the Gale-Shapley algorithm, suppose the stable matching is chosen at random. We present a modified form of the coalition strategy ensuring that every man in the coalition has a new probability distribution over partners which majorizes the original one. The stable marriage problem is a well-known problem of matching men to women so that no man and woman who are not married to each other both prefer each other. Such a problem has a wide variety of practical applications ranging from matching resident doctors to hospitals to matching students to schools. A well-known algorithm to solve this problem is the Gale-Shapley algorithm, which runs in polynomial time. It has been proven that stable marriage procedures can always be manipulated. Whilst the Gale-Shapley algorithm is computationally easy to manipulate, we prove that there exist stable marriage procedures which are NP-hard to manipulate. We also consider the relationship between voting theory and stable marriage procedures, showing that voting rules which are NP-hard to manipulate can be used to define stable marriage procedures which are themselves NP-hard to manipulate. Finally, we consider the issue that stable marriage procedures like Gale-Shapley favour one gender over the other, and we show how to use voting rules to make any stable marriage procedure gender neutral. Lying in order to manipulate the Gale-Shapley matching algorithm has been studied by Dubins and Freedman (1981) and by Gale and Sotomayor (1985), and was shown to be generally more appealing to the proposed-to side (denoted as the women in Gale and Shapley's seminal paper (1962)) than to the proposing side (denoted as men there). It can also be shown that in the case of lying women, for every woman who is better off due to lying, there exists a man who is worse off. In this paper, we show that an even stronger dichotomy between the goals of the sexes holds, namely, if no woman is worse off then no man is better off, while a form of sisterhood between the lying and the "innocent" women also holds, namely, if none of the former is worse off, then neither is any of the latter. These results are robust: they generalize to the one-to-many variants of the algorithm and do not require the resulting matching to be stable (i.e. they hold even in out-of-equilibria situations). The machinery we develop in our proofs sheds new light on the structure of lying by women in the Gale-Shapley matching algorithm. This paper is based upon an undergraduate thesis (2007) by the first author. 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, the Mendelsohn-Dulmage theorem, the Kundu-Lawler theorem, Tarski's fixed-point theorem, the Cantor-Bernstein theorem, Pym's linking theorem, or the monochromatic path theorem of 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 and Rothblum on the bipartite stable matching polytope. We study strategic issues in the Gale-Shapley stable marriage model. In the first part of the paper, we derive the optimal cheating strategy and show that it is not always possible for a woman to recover her women-optimal stable partner from the men-optimal stable matching mechanism when she can only cheat by permuting her preferences. In fact, we show, using simulation, that the chances that a woman can benefit from cheating are slim. In the second part of the paper, we consider a two-sided matching market found in Singapore. We study the matching mechanism used by the Ministry of Education (MOE) in the placement of primary six students in secondary schools, and discuss why the current method has limited success in accommodating the preferences of the students, and the specific needs of the schools (in terms of the “mix” of admitted students). Using insights from the first part of the paper, we show that stable matching mechanisms are more appropriate in this matching market and explain why the strategic behavior of the students need not be a major concern. (Stable Marriage; Strategic Issues; Gale-Shapley Algorithm; Student Posting Exercise) We study a labor market with finitely many heterogeneous workers and firms to illustrate the decentralized (myopic) blocking dynamics in two-sided one-to-one matching markets with continuous side payments (assignment problems, Shapley and Shubik, 1971). A labor market is unstable if there is at least one blocking pair, that is, a worker and a firm who would prefer to be matched to each other in order to obtain higher payoffs than the payoffs they obtain by being matched to their current partners. A blocking path is a sequence of outcomes (specifying matchings and payoffs) such that each outcome is obtained from the previous one by satisfying a blocking pair (i.e., by matching the two blocking agents and assigning new payoffs to them that are higher than the ones they received before). We are interested in the question if starting from any (unstable) outcome, there always exists a blocking path that will lead to a stable outcome. In contrast to discrete versions of the model (i.e., for marriage markets, one-to-one matching, or discretized assignment problems), the existence of blocking paths to stability cannot always be guaranteed. We identify a necessary and sufficient condition for an assignment problem (the existence of a stable outcome such that all matched agents receive positive payoffs) to guarantee the existence of paths to stability and show how to construct such a path whenever this is possible. In sponsored search, a number of advertising slots is available on a search results page, and have to be allocated among a set of advertisers competing to display an ad on the page. This gives rise to a bipartite matching market that is typically cleared by the way of an automated auction. Several auction mechanisms have been proposed, with variants of the Generalized Second Price (GSP) being widely used in practice. There is a rich body of work on bipartite matching markets that builds upon the stable marriage model of Gale and Shapley and the assignment model of Shapley and Shubik. This line of research offers deep insights into the structure of stable outcomes in such markets and their incentive properties. In this paper, we model advertising auctions in terms of an assignment model with linear utilities, extended with bidder and item specific maximum and minimum prices. Auction mechanisms like the commonly used GSP or the well-known Vickrey-Clarke-Groves (VCG) can be interpreted as simply computing a bidder-optimal stable matching in this model, for a suitably defined set of bidder preferences, but our model includes much richer bidders and preferences. We prove that in our model the existence of a stable matching is guaranteed, and under a non-degeneracy assumption a bidder-optimal stable matching exists as well. We give an algorithm to find such matching in polynomial time, and use it to design truthful mechanism that generalizes GSP, is truthful for profit-maximizing bidders, correctly implements features like bidder-specific minimum prices and position-specific bids, and works for rich mixtures of bidders and preferences. Our main technical contributions are the existence of bidder-optimal matchings and strategyproofness of the resulting mechanism, and are proved by induction on the progress of the matching algorithm. In any two-sided matching market, a stable matching can be found by a central agency using the deferred acceptance procedure of Gale and Shapley. But if the market is decentralized and information is incomplete then stability of the ensuing matching is not to be expected. Despite the prevalence of such matching situations, and the importance of stability, little theory exists concerning instability. We discuss various measures of instability and analyze how they interact with the structure of the underlying preferences. Our main result is that even the outcome of decentralized matching with incomplete information can be expected to be “almost stable” under reasonable assumptions. The original work of Gale and Shapley on an assignment method using the stable marriage criterion has been extended to find all the stable marriage assignments. The algorithm derived for finding all the stable marriage assignments is proved to satisfy all the conditions of the problem. Algorithm 411 applies to this paper. 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. We consider a two-sided market under incomplete preference lists with ties, where the goal is to find a maximum size stable matching. The problem is APX-hard, and a 3 2-approximation was given by McDermid [1]. This algorithm has a non-linear running time, and, more importantly needs global knowledge of all preference lists. We present a very natural, economically reasonable, local, linear time algorithm with the same ratio, using some ideas of Paluch [2]. In this algorithm every person make decisions using only their own list, and some information asked from members of these lists (as in the case of the famous algorithm of Gale and Shapley). Some consequences to the Hospitals Residents problem are also discussed. We analyze the Gale-Shapley matching problem within the context of Rawlsian justice. Defining a fair matching algorithm by a set of 4 axioms (Gender Indifference, Peer Indifference, Maximin Optimality, and Stability), we show that not all preference profiles admit a fair matching algorithm, the reason being that even this set of minimal axioms is too strong in a sense. Because of conflict between Stability and Maximin Optimality, even the algorithm which generates the mutual agreement match, paradoxically, has no chance to be fair. In this paper, we propose a new approach, parallel iterative improvement (PII), to solving the stable matching problem. This approach treats the stable matching problem as an optimization problem with all possible matchings forming its solution space. Since a stable matching always exists for any stable matching problem instance, finding a stable matching is equivalent to finding a matching with the minimum number (which is always zero) of unstable pairs. A particular PII algorithm is presented to show the effectiveness of this approach by constructing a new matching from an existing matching and using techniques such as randomization and greedy selection to speedup the convergence process. Simulation results show that the PII algorithm has better average performance compared with the classical stable matching algorithms and converges in n iterations with high probability. The proposed algorithm is also useful for some real-time applications with stringent time constraint. For f a weighted voting scheme used by n voters to choose between two candidates, the nShapley-Shubik Indices (or Shapley values) of f provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 [SS54] and are widely studied in social choice theory as a measure of the "influence" of voters. The Inverse Shapley Value Problem is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant e>0 our algorithm runs in fixed poly(n) time (the degree of the polynomial is independent of e) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error e. If there is a "reasonable" voting scheme in which all voting weights are integers at most poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error e=n−1 8. Abstract : A two sided matching model is a game in which there are two types of agents, and the essential coalitions are singletons and doubletons containing one agent of each type. Over the years, they have become an important part of economic theory. One reason for this is that they have been deemed worthy models of economic markets with indivisible goods. Gale and Shapley modeled college admissions as a matching market. Shapley and Shubik adopted a similar model for their housing market. Crawford and Knoer defined labor markets in these terms. In all of these instances, the relevant solution concept is that of the core which is the set of economic allocations where no coalition of agents can improve their lot on their own. Herein lies another reason for the study of these games; the fact that their cores have many nice properties. For instance, their cores are always nonempty. Much of the literature relate relatively simple algorithms which calculate core points. Another important idea is that in some of these games, the core and the set of economic equilibria are equivalent and both are generally sets of positive measure. This contrasts other economic models (with completely divisible goods) in which the set of equilibria is a proper subset of the core and is of measure zero. Thus, given a sets of agents' preferences, there is usually some latitude in choosing equilibrium prices. Intuitively, the amount of flexibility would be measured by the core's volume. This paper quantifies this last notion in the setting of Shapley and Shubik's housing market. Given a set of n men represented by n points lying on a line, and n women represented by n points lying on another parallel line, with each person having a list that ranks some people of opposite gender as his her acceptable partners in strict order of preference. In this problem, we want to match people of opposite genders to satisfy people’s preferences as well as making the edges not crossing one another geometrically. A noncrossing blocking pair w.r.t. a matching M is a pair (m, w) of a man and a woman such that they are not matched with each other but prefer each other to their own partners in M, and the segment (m, w) does not cross any edge in M. A weakly stable noncrossing matching (WSNM) is a noncrossing matching that does not admit any noncrossing blocking pair. In this paper, we prove the existence of a WSNM in any instance by developing an (O(n^2) ) algorithm to find one in a given instance. Scholars are increasingly citing electronic “web references” which are not preserved in libraries or full text archives. WebCite is a new standard for citing web references. To “webcite” a document involves archiving the cited Web page through www.webcitation.org and citing the WebCite permalink instead of (or in addition to) the unstable live Web page. This journal has amended its “instructions for authors” accordingly, asking authors to archive cited Web pages before submitting a manuscript. Almost 200 other journals are already using the system. We discuss the rationale for WebCite, its technology, and how scholars, editors, and publishers can benefit from the service. Citing scholars initiate an archiving process of all cited Web references, ideally before they submit a manuscript. Authors of online documents and websites which are expected to be cited by others can ensure that their work is permanently available by creating an archived copy using WebCite and providing the citation information including the WebCite link on their Web document(s). Editors should ask their authors to cache all cited Web addresses (Uniform Resource Locators, or URLs) “prospectively” before submitting their manuscripts to their journal. Editors and publishers should also instruct their copyeditors to cache cited Web material if the author has not done so already. Finally, WebCite can process publisher submitted “citing articles” (submitted for example as eXtensible Markup Language [XML] documents) to automatically archive all cited Web pages shortly before or on publication. Finally, WebCite can act as a focussed crawler, caching retrospectively references of already published articles. Copyright issues are addressed by honouring respective Internet standards (robot exclusion files, no-cache and no-archive tags). Long-term preservation is ensured by agreements with libraries and digital preservation organizations. The resulting WebCite Index may also have applications for research assessment exercises, being able to measure the impact of Web services and published Web documents through access and Web citation metrics. @PARASPLIT [J Med Internet Res 2005;7(5):e60] Purpose – The purpose of this paper is to suggest an alternative to the widely used Times Cited criterion for analysing citation networks. The approach involves taking account of the natures of the papers that cite a given paper, so as to differentiate between papers that attract the same number of citations.Design methodology approach – ArticleRank is an algorithm that has been derived from Google's PageRank algorithm to measure the influence of journal articles. ArticleRank is applied to two datasets – a citation network based on an early paper on webometrics, and a self‐citation network based on the 19 most cited papers in the Journal of Documentation – using citation data taken from the Web of Knowledge database.Findings – ArticleRank values provide a different ranking of a set of papers from that provided by the corresponding Times Cited values, and overcomes the inability of the latter to differentiate between papers with the same numbers of citations. The difference in rankings between Times Cited ... Abstract We give an on-line deterministic algorithm for the weighted bipartite matching problem that achieves a competitive ratio of (2 n −1) in any metric space (where n is the number of vertices). This algorithm is optimal - there is no on-line deterministic algorithm that achieves a competitive ratio better than (2 n −1) in all metric spaces. We also study the stable marriage problem, where we are interested in the number of unstable pairs produced. We show that the simple “first come, first served” deterministic algorithm yields on the average O( n log n ) unstable pairs, but in the worst case no deterministic or randomized on-line algorithm can do better than ω( n 2 ) unstable pairs. This appears to be the first on-line problem for which provably one cannot do better with randomization; for most on-line problems studied in the past, randomization has helped in improving the performance.