New Approximations for Coalitional Manipulation in General Scoring Rules
نویسندگان
چکیده
We study the problem of coalitional manipulation—where k manipulators try to manipulate an election on m candidates—under general scoring rules, with a focus on the Borda protocol. We do so both in the weighted and unweighted settings. For these problems, recent approaches to approximation tried to minimize k, the number of manipulators needed to make the preferred candidate win (thus assuming that the number of manipulators is not limited in advance), we focus instead on minimizing the maximum score obtainable by a non-preferred candidate. In the strongest, most general setting, we provide an algorithm for any scoring rule as described by a vector ~ α = (α1, . . . , αm): for some β = O( √ m logm), it obtains an additive approximation equal to W · maxi|αi+β − αi|, where W is the sum of voter weights. In words, this factor is the maximum difference between two scores in ~ α that are β entries away, multiplied by W . The unweighted equivalent is provided as well. For Borda, both the weighted and unweighted variants are known to be NP-hard. For the unweighted case, our simpler algorithm provides a randomized, additive O(k √ m logm) approximation; in other words, if there exists a strategy enabling the preferred candidate to win by an Ω(k √ m logm) margin, our method, with high probability, will find a strategy enabling her to win (albeit with a possibly smaller margin). It thus provides a somewhat stronger guarantee compared to the previous methods, which implicitly implied (with respect to the original k) a strategy that provides an Ω(m)-additive approximation to the maximum score of 1 ar X iv :1 70 8. 04 86 2v 1 [ cs .D S] 1 6 A ug 2 01 7 a non-preferred candidate: when k is o( √ m/ logm), our strategy thus provides a stronger approximation. For the weighted case, our generalized algorithm provides anO(W √ m logm)additive approximation, where W is the sum of voter weights. This is a clear advantage over previous methods: some of them do not generalize to the weighted case, while others—which approximate the number of manipulators—pose restrictions on the weights of extra manipulators added. We note that our algorithms for Borda can also be viewed as a (1 + o(1))-multiplicative approximation since the values we approximate have natural Ω(km) (unweighted) and Ω(Wm) (weighted) lower bounds. Our methods are novel and adapt techniques from multiprocessor scheduling by carefully rounding an exponentially-large configuration linear program that is solved by using the ellipsoid method with an efficient separation oracle. We believe that such methods could be beneficial in social choice settings as well.
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ورودعنوان ژورنال:
- CoRR
دوره abs/1708.04862 شماره
صفحات -
تاریخ انتشار 2017