In social choice settings with linear preferences, random dictatorship is known to be the only social decision scheme satisfying strategyproofness and ex post efficiency. When also allowing indifferences, random serial dictatorship (RSD) is a well-known generalization of random dictatorship that retains both properties. RSD has been particularly successful in the special domain of random assignment where indifferences are unavoidable. While executing RSD is obviously feasible, we show that computing the resulting probabilities is #P-complete and thus intractable, both in the context of voting and assignment.

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Abstract. Let I be a stable matching instance with N stable matchings. For each man m, order his N stable partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as pi(m). Let αi consist of the man-woman pairs where each man m is matched to pi(m). Teo and Sethuraman proved this surprising result: for i = 1 to N, not only is αi a matching, it is also stable. The αi’s are called the generalized median stable matchings of I. In this paper, we present a new characterization of these stable matchings that is solely based on I’s rotation poset. We then prove the following: when i = O(log n), where n is the number of men, αi can be found efficiently; but when i is a constant fraction of N, finding αi is NPhard. We also consider what it means to approximate the median stable matching of I, and present results for this problem. 1

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