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62
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 161 (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
Junta distributions and the averagecase complexity of manipulating elections
 In AAMAS
, 2006
"... Encouraging voters to truthfully reveal their preferences in an election has long been an important issue. Recently, computational complexity has been suggested as a means of precluding strategic behavior. Previous studies have shown that some voting protocols are hard to manipulate, but used N Pha ..."
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Cited by 108 (24 self)
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Encouraging voters to truthfully reveal their preferences in an election has long been an important issue. Recently, computational complexity has been suggested as a means of precluding strategic behavior. Previous studies have shown that some voting protocols are hard to manipulate, but used N Phardness as the complexity measure. Such a worstcase analysis may be an insufficient guarantee of resistance to manipulation. Indeed, we demonstrate that N Phard manipulations may be tractable in the averagecase. For this purpose, we augment the existing theory of averagecase complexity with some new concepts. In particular, we consider elections distributed with respect to junta distributions, which concentrate on hard instances. We use our techniques to prove that scoring protocols are susceptible to manipulation by coalitions, when the number of candidates is constant. 1.
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 70 (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.
Llull and Copeland voting computationally resist bribery and control
, 2009
"... Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive con ..."
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Cited by 65 (29 self)
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Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate’s victory. An election system in which an agent can sometimes affect the result and it can be determined in polynomial time on which inputs the agent can succeed is said to be vulnerable to the given type of control. An election system in which an agent can sometimes affect the result, yet in which it is NPhard to recognize the inputs on which the agent can succeed, is said to be resistant to the given type of control. Aside from election systems with an NPhard winner problem, the only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. This paper studies a parameterized version of Copeland voting, denoted by Copeland α, where the parameter α is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive
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 63 (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.
AI’s war on manipulation: Are we winning?
 AI MAGAZINE
"... We provide an overview of more than two decades of work, mostly in AI, that studies computational complexity as a barrier against manipulation in elections. ..."
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Cited by 54 (8 self)
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We provide an overview of more than two decades of work, mostly in AI, that studies computational complexity as a barrier against manipulation in elections.
Copeland voting: Ties matter
 In To appear in Proceedings of AAMAS’08
, 2008
"... We study the complexity of manipulation for a family of election systems derived from Copeland voting via introducing a parameter α that describes how ties in headtohead contests are valued. We show that the thus obtained problem of manipulation for unweighted Copeland α elections is NPcomplete e ..."
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Cited by 51 (10 self)
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We study the complexity of manipulation for a family of election systems derived from Copeland voting via introducing a parameter α that describes how ties in headtohead contests are valued. We show that the thus obtained problem of manipulation for unweighted Copeland α elections is NPcomplete even if the size of the manipulating coalition is limited to two. Our result holds for all rational values of α such that 0 < α < 1 except for α = 1. Since it is 2 well known that manipulation via a single voter is easy for Copeland, our result is the first one where an election system originally known to be vulnerable to manipulation via a single voter is shown to be resistant to manipulation via a coalition of a constant number of voters. We also study the complexity of manipulation for Copeland α for the case of a constant number of candidates. We show that here the exact complexity of manipulation often depends closely on the α: Depending on whether we try to make our favorite candidate a winner or a unique winner and whether α is 0, 1 or between these values, the problem of weighted manipulation for Copeland α with three candidates is either in P or is NPcomplete. Our results show that ways in which ties are treated in an election system, here Copeland voting, can be crucial to establishing complexity results for this system.
How Hard Is Bribery in Elections?
"... We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by paying certain voters to change their preferences a specified candidate can be made the election’s winner? We study this problem for election systems as v ..."
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Cited by 49 (22 self)
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We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by paying certain voters to change their preferences a specified candidate can be made the election’s winner? We study this problem for election systems as varied as scoring protocols and Dodgson voting, and in a variety of settings regarding homogeneousvs.nonhomogeneous electorate bribability, boundedsizevs.arbitrarysized candidate sets, weightedvs.unweighted voters, and succinctvs.nonsuccinct input specification. We obtain both polynomialtime bribery algorithms and proofs of the intractability of bribery, and indeed our results show that the complexity of bribery is extremely sensitive to the setting. For example, we find settings in which bribery is NPcomplete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NPcomplete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, kapproval, and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simpletoevaluate condition that classifies every case as either NPcomplete or in P. 1.
A sufficient condition for voting rules to be frequently manipulable
 In Proceedings of the Ninth ACM Conference on Electronic Commerce (EC
, 2008
"... The GibbardSatterthwaite Theorem states that (in unrestricted settings) any reasonable voting rule is manipulable. Recently, a quantitative version of this theorem was proved by Ehud Friedgut, Gil Kalai, and Noam Nisan: when the number of alternatives is three, for any neutral voting rule that is f ..."
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Cited by 46 (11 self)
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The GibbardSatterthwaite Theorem states that (in unrestricted settings) any reasonable voting rule is manipulable. Recently, a quantitative version of this theorem was proved by Ehud Friedgut, Gil Kalai, and Noam Nisan: when the number of alternatives is three, for any neutral voting rule that is far from any dictatorship, there exists a voter such that a random manipulation—that is, the true preferences and the strategic vote are all drawn i.i.d., uniformly at random—will succeed with a probability of Ω ( 1), where n is the n number of voters. However, it seems that the techniques used to prove this theorem can not be fully extended to more than three alternatives. In this paper, we give a more limited result that does apply to four or more alternatives. We give a sufficient condition for a voting rule to be randomly manipulable with a probability of Ω ( 1) for at least one voter, when the number of alternatives is held n fixed. Specifically, our theorem states that if a voting rule r satisfies 1. homogeneity, 2. anonymity, 3. nonimposition, 4. a cancelingout condition, and 5. there exists a stable profile that is still stable after one given alternative is uniformly moved to different positions; then there exists a voter such that a random manipulation for that voter will succeed with a probability of Ω ( 1). We show that n many common voting rules satisfy these conditions, for example any positional scoring rule, Copeland, STV, maximin, and ranked pairs.