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.

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 NP-complete; also, we give a sufficient condition on scoring rules for the possible winner problem to be NP-complete (Borda satisfies this condition). We also prove that for Copeland and ranked pairs, the necessary winner problem is coNP-complete. All the hardness results hold even when the number of undetermined pairs in each vote is no more than a constant. We also present polynomial-time algorithms for the necessary winner problem for scoring rules, maximin, and Bucklin.

Voting is a general method for aggregating the preferences of multiple agents. Each agent ranks all the possible alternatives, and based on this, an aggregate ranking of the alternatives (or at least a winning alternative) is produced. However, when there are many alternatives, it is impractical to simply ask agents to report their complete preferences. Rather, the agents’ preferences, or at least the relevant parts thereof, need to be elicited. This is done by asking the agents a (hopefully small) number of simple queries about their preferences, such as comparison queries, which ask an agent to compare two of the alternatives. Prior work on preference elicitation in voting has focused on the case of unrestricted preferences. It has been shown that in this setting, it is sometimes necessary to ask each agent (almost) as many queries as would be required to determine an arbitrary ranking of the alternatives. In contrast, in this paper, we focus on single-peaked preferences. We show that such preferences can be elicited using only a linear number of comparison queries, if either the order with respect to which preferences are single-peaked is known, or at least one other agent’s complete preferences are known. We show that using a sublinear number of queries does not suffice. We also consider the case of cardinally single-peaked preferences. For this case, we show that if the alternatives ’ cardinal positions are known, then an agent’s preferences can be elicited using only a logarithmic number of queries; however, we also show that if the cardinal positions are not known, then a sublinear number of queries does not suffice. We present experimental results for all elicitation algorithms. We also consider the problem of only eliciting enough information to determine the aggregate ranking, and show that even for this more modest objective, a sublinear number of queries per agent does not suffice for known ordinal or unknown cardinal positions. Finally, we discuss whether and how these techniques can be applied when preferences are almost single-peaked. 1 1

Abstract. We show that, other than for standard complexity theory with known NP-completeness results, the computational complexity of the Dodgson and Young election systems is completely different from a parameterized complexity point of view. That is, on the one hand, we present an efficient fixed-parameter algorithm for determining a Condorcet winner in Dodgson elections by a minimum number of switches in the votes. On the other hand, we prove that the corresponding problem for Young elections, where one has to delete votes instead of performing switches, is W[2]-complete. In addition, we study Dodgson elections that allow ties between the candidates and give fixed-parameter tractability as well as W[2]-hardness results depending on the cost model for switching ties. 1

The voting rules proposed by Dodgson and Young are both designed to find the alternative closest to being a Condorcet winner, according to two different notions of proximity; the score of a given alternative is known to be hard to compute under either rule. In this paper, we put forward two algorithms for approximating the Dodgson score: an LP-based randomized rounding algorithm and a deterministic greedy algorithm, both of which yield an O(log m) approximation ratio, where m is the number of alternatives; we observe that this result is asymptotically optimal, and further prove that our greedy algorithm is optimal up to a factor of 2, unless problems in N P have quasi-polynomial time algorithms. Although the greedy algorithm is computationally superior, we argue that

Electoral control refers to attempts by an election’s organizer (“the chair”) to influence the outcome by adding/deleting/partitioning voters or candidates. The groundbreaking work of Bartholdi, Tovey, and Trick [BTT92] on (constructive) control proposes computational complexity as a means of resisting control attempts: Look for election systems where the chair’s task in seeking control is itself computationally infeasible. We introduce and study a method of combining two or more candidateanonymous election schemes in such a way that the combined scheme possesses all the resistances to control (i.e., all the NP-hardnesses of control) possessed by any of its constituents: It combines their strengths. From this and new resistance constructions, we prove for the first time that there exists an election scheme that is resistant to all twenty standard types of electoral control.

Dodgson’s election system elegantly satisfies the Condorcet criterion. However, determining the winner of a Dodgson election is known to be Θ p 2-complete ([HHR97], see also [BTT89]), which implies that unless P = NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates (although the number of voters may still be polynomial in the number of candidates), a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner. 1

We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant—optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm. The problem of determining Dodgson winners is known to be complete for parallel access to NP [HHR97]. Homan and Hemaspaandra [HH06] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is a “frequently self-knowingly correct algorithm. ” We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein’s junta distributions [PR07]. Key words: approximation, Dodgson elections, election systems, frequently selfknowingly correct algorithms, greedy algorithms, optimal lobbying, preference aggregation.

The aggregation of conflicting preferences is an important issue in human society and multiagent systems. Due to its universality, voting among a set of alternatives has a central role among preference aggregation mechanisms. We consider the most general case of voting in which the voters ’ rankings of alternatives are mapped to a collective ranking of alternatives by a so-called social welfare functional (SWF). Maintaining privacy of individuals ’ preferences is crucial in order to guarantee freedom of choice (e.g., lack of vote coercing and reputation effects), and to not facilitate strategic voting. We investigate whether unconditional full privacy can be achieved in preference aggregation, that is, privacy that relies neither on trusted third parties (or on a certain fraction of the voters being trusted), nor on computational intractability assumptions. More precisely, we study the existence of distributed protocols that allow voters to jointly determine the collective preference ranking without revealing further information. We prove that there exists no SWF that is non-dictatorial, Paretian, monotonic, and privately computable (any three of these properties can be achieved). Moreover, we show that replacing privacy with anonymity enables the joint computation of arbitrary symmetric SWFs. 1

In a combinatorial auction, a set of items is for sale, and agents can bid on subsets of these items. In a voting setting, the agents decide among a set of alternatives by having each agent rank all the alternatives. Many of the key research issues in these two domains are similar. The aim of this paper is to give a convenient side-by-side comparison that will clarify the relation between the domains, and serve as a guide to future research. 1