Efficiently Negative: Complexity and Approximations of Targeted Negative Campaigning

Given the ubiquity of negative campaigning in recent political elections, we find it important to study its properties from a theoretical computational perspective. To this end, we present a model where elections can be manipulated by convincing voters to demote specific non-favored candidates, and study its properties in the classic setting of scoring rules. When the goal is constructive (making a preferred candidate win), we prove that finding such a demotion strategy is easy for Plurality and Veto, while generally hard for t-approval and Borda. We also provide a min(t,m− t)-factor approximation for t-approval for every tε {1,…,m − 1} (where m is the number of candidates), and a 3-factor approximation algorithm for Borda. Interestingly enough—following recent trends in political science that show that the effectiveness of negative campaigning depends on the type of candidate and demographic—when assigning varying prices to different possible demotion operations, we are able to provide inapproximability results. When the goal is destructive (making the leading opponent lose), we show that the problem is easy for a broad class of scoring rules and provide an FPTAS for the general case.