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
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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.