How Hard Is Bribery in Elections?

Abstract

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 homogeneous-vs.-nonhomogeneous electorate bribability, bounded-size-vs.-arbitrary-sized candidate sets, weighted-vs.-unweighted voters, and succinct-vs.-nonsuccinct input specification. We obtain both polynomial-time 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 NP-complete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NP-complete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, k-approval, and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simple-to-evaluate condition that classifies every case as either NP-complete or in P.

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

“…Definition 1 (Bartholdi et al [2], Faliszewski et al [12,21]). In each of the problems Approval-$BRIBERY (priced bribery), Approval-$CCAV (priced control by adding voters), and Approval-$CCDV (priced control by deleting voters), we are given an approval election E = (C, V ) with C = {p, c 1 , .…”

“…We obtain the following corollary. Contrarily, it is either shown explicitly by Faliszewski et al [12], or follows trivially, that these problems with both prices and weights are NP-hard already for two candidates (that is, Para-NP-hard with respect to the number of candidates).…”

“…These include, for example, various bribery problems [10,12] and priced control problems [21] that were known to be in XP since nearly 10 years ago, but were neither known to be fixed-parameter tractable (in FPT), nor to be W[1]-hard. We develop a general technique for showing their membership in FPT, which also applies to weighted voter control for Approval voting, improving results of Faliszewski et al [14].…”

“…and we ask whether a group of manipulators can cast votes to ensure their preferred candidate's victory), election control problems [2,17] (where we are given an election and ask if we can ensure a given candidate's victory by adding/deleting candidates/voters), or bribery [10,12,24] and campaign management problems [3,5,8,23] (where we want to ensure a given candidate's victory by changing some of the votes, but where each vote change comes at a price and we are bound by a budget). We focus on the case where we have a few candidates but (possibly) many voters.…”

“…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.…”