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When are elections with few candidates hard to manipulate?
 JOURNAL OF THE ACM
, 2007
"... In multiagent settings where the agents have di®erent preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using protoco ..."
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Cited by 141 (18 self)
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In multiagent settings where the agents have di®erent preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using protocols where determining a bene¯cial manipulation is hard. Especially among computational agents, it is reasonable to measure this hardness by computational complexity. Some earlier work has been done in this area, but it was assumed that the number of voters and candidates is unbounded. Such hardness results lose relevance when the number of candidates is small, because manipulation algorithms that are exponential only in the number of candidates (and only slightly so) might be available. We give such an algorithm for an individual agent to manipulate the Single Transferable Vote (STV) protocol, which has been shown hard to manipulate in the above sense. This motivates the core of this paper, which derives hardness results for realistic elections where the number of candidates is a small constant (but the number of voters can be large). The main manipulation question we study is that of coalitional manipulation by weighted voters. (We show that for simpler manipulation problems, manipulation cannot be hard with few candidates.) We study both constructive manipulation (making a given candidate win) and de
Nonexistence of voting rules that are usually hard to manipulate
 In AAAI
, 2006
"... Aggregating the preferences of selfinterested agents is a key problem for multiagent systems, and one general method for doing so is to vote over the alternatives (candidates). Unfortunately, the GibbardSatterthwaite theorem shows that when there are three or more candidates, all reasonable votin ..."
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Cited by 82 (9 self)
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Aggregating the preferences of selfinterested agents is a key problem for multiagent systems, and one general method for doing so is to vote over the alternatives (candidates). Unfortunately, the GibbardSatterthwaite theorem shows that when there are three or more candidates, all reasonable voting rules are manipulable (in the sense that there exist situations in which a voter would benefit from reporting its preferences insincerely). To circumvent this impossibility result, recent research has investigated whether it is possible to make finding a beneficial manipulation computationally hard. This approach has had some limited success, exhibiting rules under which the problem of finding a beneficial manipulation is NPhard, #Phard, or even PSPACEhard. Thus, under these rules, it is unlikely that a computationally efficient algorithm can be constructed that always finds a beneficial manipulation (when it exists). However, this still does not preclude the existence of an efficient algorithm that often finds a successful manipulation (when it exists). There have been attempts to design a rule under which finding a beneficial manipulation is usually hard, but they have failed. To explain this failure, in this paper, we show that it is in fact impossible to design such a rule, if the rule is also required to satisfy another property: a large fraction of the manipulable instances are both weakly monotone, and allow the manipulators to make either of exactly two candidates win. We argue why one should expect voting rules to have this property, and show experimentally that common voting rules clearly satisfy it. We also discuss approaches for potentially circumventing this impossibility result.
Generalized scoring rules and the frequency of coalitional manipulability
 In Proceedings of the Ninth ACM Conference on Electronic Commerce (EC
, 2008
"... We introduce a class of voting rules called generalized scoring rules. Under such a rule, each vote generates a vector of k scores, and the outcome of the voting rule is based only on the sum of these vectors—more specifically, only on the order (in terms of score) of the sum’s components. This clas ..."
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Cited by 66 (22 self)
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We introduce a class of voting rules called generalized scoring rules. Under such a rule, each vote generates a vector of k scores, and the outcome of the voting rule is based only on the sum of these vectors—more specifically, only on the order (in terms of score) of the sum’s components. This class is extremely general: we do not know of any commonly studied rule that is not a generalized scoring rule. We then study the coalitional manipulation problem for generalized scoring rules. We prove that under certain natural assump), then tions, if the number of manipulators is O(n p) (for any p < 1 2 the probability that a random profile is manipulable is O(n p − 1 2), where n is the number of voters. We also prove that under another set of natural assumptions, if the number of manipulators is Ω(n p) (for any p> 1) and o(n), then the probability that a random pro2 file is manipulable (to any possible winner under the voting rule) is 1 − O(e −Ω(n2p−1)). We also show that common voting rules satisfy these conditions (for the uniform distribution). These results generalize earlier results by Procaccia and Rosenschein as well as even earlier results on the probability of an election being tied.
Determining Possible and Necessary Winners under Common Voting Rules Given Partial Orders
"... Usually a voting rule or correspondence requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a profile of part ..."
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Cited by 58 (13 self)
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Usually a voting rule or correspondence requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a profile of partial orders and a candidate c, two important questions arise: first, is c guaranteed to win, and second, is it still possible for c to win? These are the necessary winner and possible winner problems, respectively. We consider the setting where the number of alternatives is unbounded and the votes are unweighted. We prove that for Copeland, maximin, Bucklin, and ranked pairs, the possible winner problem is NPcomplete; also, we give a sufficient condition on scoring rules for the possible winner problem to be NPcomplete (Borda satisfies this condition). We also prove that for Copeland and ranked pairs, the necessary winner problem is coNPcomplete. All the hardness results hold even when the number of undetermined pairs in each vote is no more than a constant. We also present polynomialtime algorithms for the necessary winner problem for scoring rules, maximin, and Bucklin.
Elections Can be Manipulated Often
"... The GibbardSatterthwaite theorem states that every nontrivial voting method between at least 3 alternatives can be strategically manipulated. We prove a quantitative version of the GibbardSatterthwaite theorem: a random manipulation by a single random voter will succeed with nonnegligible probab ..."
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Cited by 57 (1 self)
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The GibbardSatterthwaite theorem states that every nontrivial voting method between at least 3 alternatives can be strategically manipulated. We prove a quantitative version of the GibbardSatterthwaite theorem: a random manipulation by a single random voter will succeed with nonnegligible probability for every neutral voting method between 3 alternatives that is far from being a dictatorship.
A short introduction to computational social choice
 Proc. 33rd Conference on Current Trends in Theory and Practice of Computer Science
, 2007
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Improved bounds for computing kemeny rankings
 In AAAI’06, 620–626
, 2006
"... Voting (or rank aggregation) is a general method for aggregating the preferences of multiple agents. One voting rule of particular interest is the Kemeny rule, which minimizes the number of cases where the final ranking disagrees with a vote on the order of two alternatives. Unfortunately, Kemeny r ..."
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Cited by 51 (8 self)
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Voting (or rank aggregation) is a general method for aggregating the preferences of multiple agents. One voting rule of particular interest is the Kemeny rule, which minimizes the number of cases where the final ranking disagrees with a vote on the order of two alternatives. Unfortunately, Kemeny rankings are NPhard to compute. Recent work on computing Kemeny rankings has focused on producing good bounds to use in searchbased methods. In this paper, we extend on this work by providing various improved bounding techniques. Some of these are based on cycles in the pairwise majority graph, others are based on linear programs. We completely characterize the relative strength of all of these bounds and provide some experimental results.
Algorithms for the coalitional manipulation problem
 In The ACMSIAM Symposium on Discrete Algorithms (SODA
, 2008
"... 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 Scoring rules, Maximin and Plurality with Runoff, and analyze their windows of error. Specifically, given an instance on ..."
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Cited by 48 (10 self)
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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 Scoring rules, 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. 1
Computing Slater rankings using similarities among candidates
 In Proceedings of the 21st National Conference on Artificial Intelligence (AAAI
, 2006
"... Voting (or rank aggregation) is a general method for aggregating the preferences of multiple agents. One important voting rule is the Slater rule. It selects a ranking of the alternatives (or candidates) to minimize the number of pairs of candidates such that the ranking disagrees with the pairwi ..."
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Cited by 47 (7 self)
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Voting (or rank aggregation) is a general method for aggregating the preferences of multiple agents. One important voting rule is the Slater rule. It selects a ranking of the alternatives (or candidates) to minimize the number of pairs of candidates such that the ranking disagrees with the pairwise majority vote on these two candidates. The use of the Slater rule has been hindered by a lack of techniques to compute Slater rankings. In this paper, we show how we can decompose the Slater problem into smaller subproblems if there is a set of similar candidates. We show that this technique suffices to compute a Slater ranking in linear time if the pairwise majority graph is hierarchically structured. For the general case, we also give an efficient algorithm for finding a set of similar candidates. We provide experimental results that show that this technique significantly (sometimes drastically) speeds up search algorithms. Finally, we also use the technique of similar sets to show that computing an optimal Slater ranking is NPhard, even in the absence of pairwise ties.