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.
@cite_0 suggested lying strategies for an individual woman, and proposed an algorithm to find the best partner with the male optimal procedure. We instead focus on the complexity of determining if the procedure can be manipulated to obtain a better result. Moreover, we also provide a universal manipulation scheme that, under certain conditions on the profile, assures that the female optimal partner is returned.
@cite_0
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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. 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 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) SummaryGale and Shapley have an algorithm for assigning students to universities which gives each student the best university available in a stable system of assignments. The object here is to prove that students cannot improve their fate by lying about their preferences. Indeed, no coalition of students can simultaneously improve the lot of all its members if those outside the coalition state their true preferences. The original stable marriage problem requires all men and women to submit a complete and strictly ordered preference list. This is obviously often unrealistic in practice, and several relaxations have been proposed, including the following two common ones: one is to allow an incomplete list, i.e., a man is permitted to accept only a subset of the women and vice versa. The other is to allow a preference list including ties. Fortunately, it is known that both relaxed problems can still be solved in polynomial time. In this paper, we show that the situation changes substantially if we allow both relaxations (incomplete lists and ties) at the same time: the problem not only becomes NP-hard, but also the optimal cost version has no approximation algorithm achieving the approximation ratio of N1-Ɛ, where N is the instance size, unless P=NP. 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. This manuscript presents an alternative implementation of the truthful-in-expectation mechanism of Dughmi, Roughgarden and Yan for combinatorial auctions with weighted-matroid-rank-sum valuations. The new implementation uses only value queries and is approximately truthful-in-expectation, in the sense that by reporting truthfully each agent maximizes his utility within a multiplicative 1-o(1) factor. It still provides an optimal (1-1 e-o(1))-approximation in social welfare. We achieve this by first presenting an approximately maximal-in-distributional-range allocation rule and then showing a black-box transformation to an approximately truthful-in-expectation mechanism. Complexity of voting manipulation is a prominent research topic in computational social choice. The voting manipulation literature usually assumes that the manipulator is only concerned with improving the outcome of the election from her perspective. However, in practice, the manipulator may also be reluctant to lie, i.e., she may have a preference for submitting a vote that does not deviate too much from her true ranking of the candidates. In this paper, we study the complexity of finding a manipulative vote that achieves the manipulator's goal yet is as close as possible to her true preference order. We analyze this problem for three natural notions of closeness, namely, swap distance, footrule distance, and maximum displacement distance, and a variety of voting rules, such as scoring rules, Bucklin, Copeland, and Maximin. For all three distances, we obtain polynomial-time algorithms for all scoring rules and Bucklin and hardness results for Copeland and Maximin. We consider Stable Marriage with Covering Constraints (SMC): in this variant of Stable Marriage, we distinguish a subset of women as well as a subset of men, and we seek a matching with fewest number of blocking pairs that matches all of the distinguished people. We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. We propose a procedure for dividing indivisible items between two players in which each player ranks the items from best to worst. It ensures that each player receives a subset of items that it values more than the other player's complementary subset, given that such an envy-free division is possible. We show that the possibility of one player's undercutting the other's proposal, and implementing the reduced subset for himself or herself, makes the proposer "reasonable" and generally leads to an envy-free division, even when the players rank items exactly the same. Although the undercut procedure is manipulable, each player's maximin strategy is to be truthful. Applications of the undercut procedure are briefly discussed. We study the behavior of random walks along the edges of the stable marriage lattice for various restricted families of allowable preference sets. In the "k-attribute model," each man is valued in each of k attributes, and each woman's ranking of the men is determined by a linear function, representing her relative ranking of those attributes; men's rankings of the women are determined similarly. We show that sampling with a random walk on the marriage lattice can take exponential time, even when k = 2. Moreover, we show that the marriage lattices arising in the k-attribute model are more restrictive than in the general setting; previously such a restriction had only been shown for the sets of preference lists. The second model we consider is the "k-range model," where each person lies in a position in [i, i + k - 1], for some i, on every preference list of the opposite sex. When k = 1 there is a unique stable marriage. When k = 2 there already can be an exponential number of stable marriages, but we show that a random walk on the stable marriage lattice always converges quickly to equilibrium. However, when k ≥ 5, there are preference sets such that the random walk on the lattice will require exponential time to converge. Lastly, we show that in the extreme case where each gender's rankings of the other are restricted to one of just a constant k possible preference lists, there are still instances for which the Markov chain mixes exponentially slowly, even when k = 4. This oversimplification of the general model helps elucidate why Markov chains based on spouse-swapping are not good approaches to sampling, even in specialized scenarios. We study truthful mechanisms for matching and related problems in a partial information setting, where the agents' true utilities are hidden, and the algorithm only has access to ordinal preference information. Our model is motivated by the fact that in many settings, agents cannot express the numerical values of their utility for different outcomes, but are still able to rank the outcomes in their order of preference. Specifically, we study problems where the ground truth exists in the form of a weighted graph of agent utilities, but the algorithm can only elicit the agents' private information in the form of a preference ordering for each agent induced by the underlying weights. Against this backdrop, we design truthful algorithms to approximate the true optimum solution with respect to the hidden weights. Our techniques yield universally truthful algorithms for a number of graph problems: a 1.76-approximation algorithm for Max-Weight Matching, 2-approximation algorithm for Max k-matching, a 6-approximation algorithm for Densest k-subgraph, and a 2-approximation algorithm for Max Traveling Salesman as long as the hidden weights constitute a metric. We also provide improved approximation algorithms for such problems when the agents are not able to lie about their preferences. Our results are the first non-trivial truthful approximation algorithms for these problems, and indicate that in many situations, we can design robust algorithms even when the agents may lie and only provide ordinal information instead of precise utilities. This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multi-parameter agents. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques.Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multi-unit auctions. The best previous result for such auctions was a 2-approximation. In addition, we present a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent.The most efficient way to solve packing integer programs (PIPs) is LP-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing and multicast routing. Our approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle. We study a mechanism design version of matching computation in graphs that models the game played by hospitals participating in pairwise kidney exchange programs. We present a new randomized matching mechanism for two agents which is truthful in expectation and has an approximation ratio of 3 2 to the maximum cardinality matching. This is an improvement over a recent upper bound of 2 (, 2010 2]) and, furthermore, our mechanism beats for the first time the lower bound on the approximation ratio of deterministic truthful mechanisms. We complement our positive result with new lower bounds. Among other statements, we prove that the weaker incentive compatibility property of truthfulness in expectation in our mechanism is necessary; universally truthful mechanisms that have an inclusion-maximality property have an approximation ratio of at least 2. In this paper, we study the concepts of male optimality and uniqueness of stable marriages for partially ordered preferences. We give an algorithm to find a stable marriage that is male optimal, and a sufficient condition on the preferences, which guarantees the uniqueness of stable marriages. 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. Encouraging voters to truthfully reveal their preferences in an election has long been an important issue. Recently, computational complexity has been suggested as a means of precluding strategic behavior. Previous studies have shown that some voting protocols are hard to manipulate, but used NP-hardness as the complexity measure. Such a worst-case analysis may be an insufficient guarantee of resistance to manipulation. Indeed, we demonstrate that NP-hard manipulations may be tractable in the average-case. For this purpose, we augment the existing theory of average-case complexity with some new concepts. In particular, we consider elections distributed with respect to junta distributions, which concentrate on hard instances. We use our techniques to prove that scoring protocols are susceptible to manipulation by coalitions, when the number of candidates is constant. We present the first complete algorithm for the SMTI problem, the stable marriage problem with ties and incomplete lists. We do this in the form of a constraint programming encoding of the problem. With this we are able to carry out the first empirical study of the complete solution of SMTI instances. In the stable marriage problem (SM) [5] we have n men and n women. Each man ranks the women, giving himself a preference list. Similarly each woman ranks the men, giving herself a preference list. The problem is then to marry men and women such that they are stable i.e. such that there is no incentive for individuals to divorce and elope. This problem is polynomial time solvable. However, when preference lists contain ties and are incomplete (SMTI) the problem of determining if there is a stable matching of size n is then NP-complete, as is the optimisation problem of finding the largest or smallest stable matching [6, 10]. In this paper we present constraint programming solutions for the SMTI decision and optimisation problems, a problem generator for random instances of SMTI, and an empirical study of this problem. We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a random permutation. Our main result is a tight analysis of Greedy in this model showing that it has a competitive ratio of 1 - 1 e for maximizing the value of the assignment. We also consider the more standard i.i.d. model of input, and show that our analysis holds there as well. This is to be contrasted with the worst case analysis of [MSVV05] which shows that Greedy has a ratio of 1 2, and that the optimal algorithm presented there has a ratio of 1 - 1 e. The analysis of Greedy is important in the Adwords setting because it is the natural allocation algorithm for an auction-style process. From a theoretical perspective, our result simplifies and generalizes the classic algorithm of Karp, Vazirani and Vazirani for online bipartite matching. Our results include a new proof to show that the Ranking alforithm of [KVV90] has a ratio of 1 - 1 e in the worst case. It has been recently discovered [KV07] (independent of our results) that one of the crucial lemmas in [KVV90], related to a certain reduction, is incorrect. Our proof is direct, in that it does not go via such a reduction, which also enables us to generalize the analysis to our online assignment problem. This paper deals with the design of efficiently computable incentive-compatible mechanisms for combinatorial optimization problems with single-minded agents each possibly having multiple private parameters. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques. Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone fully polynomial time approximation scheme (FPTAS). This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multiunit auctions. The best previous result for such auctions was a 2-appro-xi-ma-tion. In addition, we present a monotone PTAS for the generalized assignment problem with any constant number of private parameters per agent. The most efficient way to solve packing integer programs (PIPs) is linear programming-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing, and multicast routing. Our primal-dual approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle.