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.
Coalition manipulation is considered in @cite_6 . Huang shows how a coalition of men can get a better result in the men-proposing Gale-Shapley algorithm. By contrast, we do not consider a coalition but just a single manipulator, and do not consider just the Gale-Shapley algorithm.
@cite_6
1605637365 2033898593 2963788595 2060522636
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. 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. 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 investigate the problem of coalitional manipulation in elections, which is known to be hard in a variety of voting rules. We put forward efficient algorithms for the problem in Borda, Maximin and Plurality with Runoff, and analyze their windows of error. Specifically, given an instance on which an algorithm fails, we bound the additional power the manipulators need in order to succeed. We finally discuss the implications of our results with respect to the popular approach of employing computational hardness to preclude manipulation.