When are elections with few candidates hard to manipulate?

Abstract

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 destructive manipulation (making a given candidate not win). We characterize the exact number of candidates for which manipulation becomes hard for the plurality, Borda, STV, Copeland, maximin, veto, plurality with runoff, regular cup, and randomized cup protocols. We also show that hardness of manipulation in this setting implies hardness of manipulation by an individual in unweighted settings when there is uncertainty about the others' votes (but not vice versa). To our knowledge, these are the first results on the hardness of manipulation when there is uncertainty about the others' votes.

Elections with Few Candidates: Prices, Weights, and Covering Problems

“…We test the running times of our algorithms empirically. Algorithmic problems that model the manipulation of elections include, among others, strategic voting problems [1,6] (where we are given an election with honest voters Supported by DFG project PAWS (NI 369/10). Supported by a DFG Mercator fellowship within project PAWS (NI 369/10) while staying at TU Berlin, and by AGH University grant 11.11.230.124 afterward.…”

“…We focus on the case where we have a few candidates but (possibly) many voters. As pointed out by Conitzer et al [6], this is a very natural setting and it models many real-life scenarios such as political elections or elections among company stockholders.…”

“…On the one hand, if the elections are weighted (as is the case for the elections held by company stockholders), then our problems are typically NP-hard even if the number of candidates is a small fixed constant [6,12,14]; these results typically follow by reductions from the well-known NP-hard problem PARTITION. One particular example where we do not have NP-hardness is control by adding/deleting voters under the Approval and k-Approval voting rules.…”

An NTU Cooperative Game Theoretic View of Manipulating Elections

“…Formally, in the coalitional manipulation problem, introduced by Conitzer, Sandholm, and Lang [5], the voters are divided into two groups, the manipulators and the nonmanipulators. The votes of the nonmanipulators are assumed to be known, and the problem is to determine whether the manipulators can use their votes to achieve a given goal.…”

“…As in most of the literature on voting, we assume that the agents do not have the ability to make or receive payments, so that we are in the nontransferable utility (NTU) case of cooperative game theory. (Recently, Bachrach, Elkind, and Faliszewski [2] studied a similar problem in the transferable utility setting, and obtained results linking coalitional manipulation [5] and bribery [6] problems with their cooperative gametheoretic model.) Moreover, in this setting, what one (sub)coalition of manipulators can achieve depends on the actions (votes) of the manipulators outside the coalition.…”