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88
When are elections with few candidates hard to manipulate?
 JOURNAL OF THE ACM
, 2007
"... In multiagent settings where the agents have different 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 158 (18 self)
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In multiagent settings where the agents have different 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 beneficial 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 106 (23 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.
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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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 66 (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.
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 (20 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 63 (30 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
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 57 (9 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.
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 56 (23 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.
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 53 (8 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
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.