The Coalitional Manipulation (CM) problem has been studied extensively in the literature for many voting rules. The CM problem, however, has been studied only in the complete information setting, that is, when the manipulators know the votes of the non-manipulators. A more realistic scenario is an incomplete information setting where the manipulators do not know the exact votes of the non- manipulators but may have some partial knowledge of the votes. In this paper, we study a setting where the manipulators know a partial order for each voter that is consistent with the vote of that voter. In this setting, we introduce and study two natural computational problems - (1) Weak Manipulation (WM) problem where the manipulators wish to vote in a way that makes their preferred candidate win in at least one extension of the partial votes of the non-manipulators; (2) Strong Manipulation (SM) problem where the manipulators wish to vote in a way that makes their preferred candidate win in all possible extensions of the partial votes of the non-manipulators. We study the computational complexity of the WM and the SM problems for commonly used voting rules such as plurality, veto, k-approval, k-veto, maximin, Copeland, and Bucklin. Our key finding is that, barring a few exceptions, manipulation becomes a significantly harder problem in the setting of incomplete votes.
Two closely related problems that have been extensively studied in the context of incomplete votes are and @cite_19 . In the PW problem, we are given a set of partial votes @math and a candidate @math , and the question is whether there exists an extension of @math where @math wins, while in the NW problem, the question is whether @math is a winner in all extensions of @math . Following the work of , a number of special cases and variants of the PW problem have been studied in the literature @cite_3 @cite_5 @cite_11 @cite_16 @cite_4 @cite_23 @cite_31 . Clearly, the WM problem is a specialization of the PW problem where there exist at least one empty vote (see proposition pw_hard ). On the other hand, the SM problem is significantly different from the NW problem --- the NW problem is known to belong to the complexity class , whereas the SM problem does not belong to unless @math (since, the SM problem is for some voting rules, for example the Borda voting rule). However, the SM problem clearly belongs to the complexity class @math .
@cite_4 @cite_3 @cite_19 @cite_23 @cite_5 @cite_31 @cite_16 @cite_11
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In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations. We prove the following theorem: if we fix the number of alternatives, generate @math votes i.i.d. according to a distribution @math , and let @math go to infinity, then for any @math , with probability at least @math , the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) @math , (3) @math , and (4) @math . This theorem holds for any set of vote operations, any individual vote distribution @math , and any integer generalized scoring rule, which includes (but is not limited to) almost all commonly studied voting rules, e.g., approval voting, all positional scoring rules (including Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs. We also show that many well-studied types of strategic behavior fall under our framework, including (but not limited to) constructive destructive manipulation, bribery, and control by adding deleting votes, margin of victory, and minimum manipulation coalition size. Therefore, our main theorem naturally applies to these problems. A possible winner of an election is a candidate that has, in some kind of incomplete-information election, the possibility to win in a complete extension of the election. The first type of problem we study is the Possible co-Winner with respect to the Addition of New Candidates (PcWNA) problem, which asks, given an election with strict preferences over the candidates, is it possible to make a designated candidate win the election by adding a limited number of new candidates to the election? In the case of unweighted voters we show NP-completeness of PcWNA for a broad class of pure scoring rules. We will also briefly study the case of weighted voters. The second type of possible winner problem we study is Possible Winner co-Winner under Uncertain Voting System (PWUVS and PcWUVS). Here, uncertainty is present not in the votes but in the election rule itself. For example, PcWUVS is the problem of whether, given a set C of candidates, a list of votes over C, a distinguished candidate c ∈ C, and a class of election rules, there is at least one election rule from this class under which c wins the election. We study these two problems for a class of systems based on approval voting, the family of Copelandα elections, and a certain class of scoring rules. Our main result is that it is NP-complete to determine whether there is a scoring vector that makes c win the election, if we restrict the set of possible scoring vectors for an m-candidate election to those of the form (α1,..., αm−4, x1, x2, x3, 0), with xi = 1 for at least one i ∈ 1, 2, 3 . In computational social choice, one important problem is to take the votes of a subelectorate (subset of the voters), and summarize them using a small number of bits. This needs to be done in such a way that, if all that we know is the summary, as well as the votes of voters outside the subelectorate, we can conclude which of the m alternatives wins. This corresponds to the notion of compilation complexity, the minimum number of bits required to summarize the votes for a particular rule, which was introduced by [IJCAI-09]. We study three different types of compilation complexity. The first, studied by , depends on the size of the subelectorate but not on the size of the complement (the voters outside the subelectorate). The second depends on the size of the complement but not on the size of the subelectorate. The third depends on both. We first investigate the relations among the three types of compilation complexity. Then, we give upper and lower bounds on all three types of compilation complexity for the most prominent voting rules. We show that for l-approval (when l ≤ m 2), Borda, and Bucklin, the bounds for all three types are asymptotically tight, up to a multiplicative constant; for l-approval (when l > m 2), plurality with runoff, all Condorcet consistent rules that are based on unweighted majority graphs (including Copeland and voting trees), and all Condorcet consistent rules that are based on the order of pairwise elections (including ranked pairs and maximin), the bounds for all three types are asymptotically tight up to a multiplicative constant when the sizes of the subelectorate and its complement are both larger than m1+∊ for some ∊ > 0. The voting rules proposed by Dodgson and Young are both designed to find an 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: a combinatorial, greedy algorithm and an LP-based algorithm, both of which yield an approximation ratio of H"m"-"1, where m is the number of alternatives and H"m"-"1 is the (m-1)st harmonic number. We also prove that our algorithms are optimal within a factor of 2, unless problems in NP have quasi-polynomial-time algorithms. Despite the intuitive appeal of the greedy algorithm, we argue that the LP-based algorithm has an advantage from a social choice point of view. Further, we demonstrate that computing any reasonable approximation of the ranking produced by Dodgson@?s rule is NP-hard. This result provides a complexity-theoretic explanation of sharp discrepancies that have been observed in the social choice theory literature when comparing Dodgson elections with simpler voting rules. Finally, we show that the problem of calculating the Young score is NP-hard to approximate by any factor. This leads to an inapproximability result for the Young ranking.