Ties Matter: Complexity of Voting Manipulation Revisited

Svetlana Obraztsova, Edith Elkind, Noam Hazon

In their groundbreaking paper, Bartholdi, Tovey and Trick [1989] argued that many well-known voting rules, such as Plurality, Borda, Copeland and Maximin are easy to manipulate. An important assumption made in that paper is that the manipulator’s goal is to ensure that his preferred candidate is among the candidates with the maximum score, or, equivalently, that ties are broken in favor of the manipulator’s preferred candidate. In this paper, we examine the role of this assumption in the easiness results of [Bartholdi et al., 1989]. We observe that the algorithm presented in [Bartholdi et al., 1989] extends to all rules that break ties according to a fixed ordering over the candidates. We then show that all scoring rules are easy to manipulate if the winner is selected from all tied candidates uniformly at random. This result extends to Maximin under an additional assumption on the manipulator’s utility function that is inspired by the original model of [Bartholdi et al., 1989]. In contrast, we show that manipulation becomes hard when arbitrary polynomial-time tie-breaking rules are allowed, both for the rules considered in [Bartholdi et al., 1989], and for a large class of scoring rules.