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42
Marriage, honesty, and stability
 In Proceedings of the Sixteenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2005
"... Many centralized twosided markets form a matching between participants by running a stable marriage algorithm. It is a wellknown fact that no matching mechanism based on a stable marriage algorithm can guarantee truthfulness as a dominant strategy for participants. However, as we will show in this ..."
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Cited by 27 (6 self)
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Many centralized twosided markets form a matching between participants by running a stable marriage algorithm. It is a wellknown fact that no matching mechanism based on a stable marriage algorithm can guarantee truthfulness as a dominant strategy for participants. However, as we will show in this paper, in a probabilistic setting where the preference lists of one side of the market are composed of only a constant (independent of the the size of the market) number of entries, each drawn from an arbitrary distribution, the number of participants that have more than one stable partner is vanishingly small. This proves (and generalizes) a conjecture of Roth and Peranson [23]. As a corollary Ó of this result, we show that, with high probability, the truthful strategy is the best response for a given player when the other players are truthful. We also analyze equilibria of the deferred acceptance stable marriage game. We show that the game with complete information has an equilibrium in which a fraction of the strategies are truthful in expectation. In the more realistic setting of a game of incomplete information, we will show that the set of truthful strategies form a Ó
A 1.875–Approximation Algorithm for the Stable Marriage Problem
, 2007
"... We consider the problem of finding a stable matching of maximum size when both ties and unacceptable partners are allowed in preference lists. This problem is known to be APXhard, and the current best known approximation algorithm achieves the approximation ratio 2 − c 1 √ N,where c is some positiv ..."
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Cited by 14 (2 self)
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We consider the problem of finding a stable matching of maximum size when both ties and unacceptable partners are allowed in preference lists. This problem is known to be APXhard, and the current best known approximation algorithm achieves the approximation ratio 2 − c 1 √ N,where c is some positive constant. In this paper, we give a 1.875– approximation algorithm, which is the first result on the approximation ratio better than two.
Improved approximation of the stable marriage problem
 Proc. ESA 2003, LNCS 2832
, 2003
"... Abstract. The stable marriage problem has recently been studied in its general setting, where both ties and incomplete lists are allowed. It is NPhard to find a stable matching of maximum size, while any stable matching is a maximal matching and thus trivially a factor two approximation. In this pa ..."
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Cited by 14 (5 self)
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Abstract. The stable marriage problem has recently been studied in its general setting, where both ties and incomplete lists are allowed. It is NPhard to find a stable matching of maximum size, while any stable matching is a maximal matching and thus trivially a factor two approximation. In this paper, we give the first nontrivial result for approximation of factor less than two. Our algorithm achieves an approximation ratio of 2/(1+L −2) for instances in which only men have ties of length at most L. When both men and women are allowed to have ties, we show a ratio of 13/7(< 1.858) for the case when ties are of length two. We also improve the lower bound on the approximation ratio to 21
Two algorithms for the StudentProject allocation problem
 JOURNAL OF DISCRETE ALGORITHMS
, 2007
"... We study the StudentProject Allocation problem (SPA), a generalisation of the classical Hospitals / Residents problem (HR). An instance of SPA involves a set of students, projects and lecturers. Each project is offered by a unique lecturer, and both projects and lecturers have capacity constraints ..."
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Cited by 11 (3 self)
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We study the StudentProject Allocation problem (SPA), a generalisation of the classical Hospitals / Residents problem (HR). An instance of SPA involves a set of students, projects and lecturers. Each project is offered by a unique lecturer, and both projects and lecturers have capacity constraints. Students have preferences over projects, whilst lecturers have preferences over students. We present two optimal lineartime algorithms for allocating students to projects, subject to the preference and capacity constraints. In particular, each algorithm finds a stable matching of students to projects. Here, the concept of stability generalises the stability definition in the HR context. The stable matching produced by the first algorithm is simultaneously bestpossible for all students, whilst the one produced by the second algorithm is simultaneously bestpossible for all lecturers. We also prove some structural results concerning the set of stable matchings in a given instance of SPA. The SPA problem model that we consider is very general and has applications to a range of different contexts besides studentproject allocation.
A 25/17Approximation Algorithm for the Stable Marriage Problem with OneSided Ties
, 2010
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STABILITY IN MATCHING PROBLEMS WITH WEIGHTED PREFERENCES
"... stable marriages, weighted preferences The stable marriage problem is a wellknown 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 ..."
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Cited by 7 (3 self)
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stable marriages, weighted preferences The stable marriage problem is a wellknown 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 or more generally to any twosided market. In the classical stable marriage problem, both men and women express a strict preference order over the members of the other sex, in a qualitative way. Here we consider stable marriage problems with weighted preferences: each man (resp., woman) provides a score for each woman (resp., man). Such problems are more expressive than the classical stable marriage problems. Moreover, in some reallife situations it is more natural to express scores (to model, for example, profits or costs) rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, and we provide algorithms to find marriages which are stable and/or optimal according to these notions. While expressivity greatly increases by adopting weighted preferences, we show that in most cases the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem. 1
Computing With Strategic Agents
, 2005
"... This dissertation studies mechanism design for various combinatorial problems in the presence of strategic agents. A mechanism is an algorithm for allocating a resource among a group of participants, each of which has a privatelyknown value for any particular allocation. A mechanism is truthful if ..."
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Cited by 6 (1 self)
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This dissertation studies mechanism design for various combinatorial problems in the presence of strategic agents. A mechanism is an algorithm for allocating a resource among a group of participants, each of which has a privatelyknown value for any particular allocation. A mechanism is truthful if it is in each participant’s best interest to reveal his private information truthfully regardless of the strategies of the other participants. First, we explore a competitive auction framework for truthful mechanism design in the setting of multiunit auctions, or auctions which sell multiple identical copies of a good. In this framework, the goal is to design a truthful auction whose revenue approximates that of an omniscient auction for any set of bids. We focus on two natural settings — the limited demand setting where bidders desire at most a fixed number of copies and the limited budget setting where bidders can spend at most a fixed amount of money. In the limit demand setting, all prior auctions employed the use of randomization in the computation of the allocation and prices. Randomization
Cheating by men in the GaleShapley stable matching algorithm
 In ESA’06
, 2006
"... Abstract. This paper addresses strategies for the stable marriage problem. For the GaleShapley 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 ..."
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Abstract. This paper addresses strategies for the stable marriage problem. For the GaleShapley 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 GaleShapley 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. 1
A (2  c(log N/N)) Approximation Algorithm for the Stable Marriage Problem
, 2006
"... An instance of the classical stable marriage problem requires all participants to submit a strictly ordered preference list containing all members of the opposite sex. However, considering applications in realworld, we can think of two natural relaxations, namely, incomplete preference lists and ti ..."
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Cited by 5 (1 self)
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An instance of the classical stable marriage problem requires all participants to submit a strictly ordered preference list containing all members of the opposite sex. However, considering applications in realworld, we can think of two natural relaxations, namely, incomplete preference lists and ties in the lists. Either variation leaves the problem polynomially solvable, but it is known that finding a maximum cardinality stable matching is NPhard when both variations are allowed. It is easy to see that the size of any two stable matchings differ by at most a factor of two, and so, an approximation algorithm with a factor two is trivial. A few approximation algorithms have been proposed with approximation ratio better than two, but they are only for restricted instances, such as restricting occurrence of ties and/or lengths of ties. Up to the present, there is no known approximation algorithm with ratio better than two for general inputs. In this paper, we give the first nontrivial result for approximation of factor less than two for general instances. Our algorithm achieves the log N ratio 2 − c N for an arbitrarily positive constant c, where N denotes the number of men in an input.