... problem for multiagent systems, and one general method for doing so is to vote over the alternatives (candidates). Unfortunately, the Gibbard-Satterthwaite 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 NP- hard, #P-hard, or even PSPACE-hard. 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.

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 gener-alized 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 pro-2 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.

The Gibbard-Satterthwaite theorem states that every non-trivial voting method between at least 3 alternatives can be strategically manipulated. We prove a quantitative version of the Gibbard-Satterthwaite theorem: a random manipulation by a single random voter will succeed with non-negligible probability for every neutral voting method between 3 alternatives that is far from being a dictatorship.

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

We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by a certain amount of bribing voters 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 the nature of the voters, the size of the candidate set, and the specification of the input. We obtain both polynomial-time bribery algorithms and proofs of the intractability of bribery. Our results indicate that the complexity of bribery is extremely sensitive to the setting. For example, we find settings where bribing weighted voters is NP-complete in general but if weights are represented in unary then the bribery problem is in P. We provide a complete classification of the complexity of bribery for the broad class of elections (including plurality, Borda, k-approval, and veto) known as scoring protocols.

The Gibbard-Satterthwaite 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. non-imposition, 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.

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 head-to-head contests are valued. We show that the thus obtained problem of manipulation for unweighted Copeland α elections is NP-complete 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 NP-complete. 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.

Abstract not found

The coalitional manipulation problem is one of the central problems in computational social choice. In this paper we focus on solving the problem under the important family of positional scoring rules, in an approximate sense that was advocated by Zuckerman et al. [SODA 2008]. Our main result is a polynomial-time algorithm with (roughly speaking) the following theoretical guarantee: given a manipulable instance with m alternatives the algorithm finds a successful manipulation with at most m − 2 additional manipulators. Our technique is based on a reduction to the scheduling problem known as Q|pmtn|Cmax, along with a novel rounding procedure. We demonstrate that our analysis is tight by establishing a new type of integrality gap. We also resolve a known open question in computational social choice by showing that the coalitional manipulation problem remains (strongly) NP-complete for positional scoring rules even when votes are unweighted. Finally, we discuss the implications of our results with respect to the question: “Is there a prominent voting rule that is usually hard to manipulate?”

Recent results have established that a variety of voting rules are computationally hard to manipulate in the worst-case; this arguably provides some guarantee of resistance to manipulation when the voters have bounded computational power. Nevertheless, it has become apparent that a truly dependable obstacle to manipulation can only be provided by voting rules that are average-case hard to manipulate. In this paper, we analytically demonstrate that, with respect to a wide range of distributions over votes, the coalitional manipulation problem can be decided with overwhelming probability of success by simply considering the ratio between the number of truthful and untruthful voters. Our results can be employed to significantly focus the search for that elusive average-case-hard-to-manipulate voting rule, but at the same time these results also strengthen the case against the existence of such a rule. 1