 Section 4.1 discussed election scenarios in which voters choose their ballots strategically and briefly introduced the Gibbard-Satterthwaite Theorem. This theorem shows that every voting method satisfying natural properties has profiles in which there is some voter, called a manipulator, that can achieve a better outcome by selecting a ballot that misrepresents her preferences. Importantly, in order to successfully manipulate an election, the manipulator must not only know which voting method is being used but also how the other members of society are voting. Although there is some debate about whether manipulation in this sense is in fact a problem (Dowding and van Hees 2008; Conitzer and Walsh, 2016, Section 6.2), there is interest in mechanisms that incentivize voters to report their “truthful” preferences. In a seminal paper, Bartholdi et al. (1989) argue that the complexity of computing which ballot will lead to a preferred outcome for the manipulator may provide a barrier to voting insincerely. See Faliszewski and Procaccia 2010, Faliszewski et al. 2010, Walsh 2011, Brandt et al. 2013, and Conitzer and Walsh 2016 for surveys of the literature on this and related questions, such as the the complexity of determining the winner given a voting method and the complexity of determining which voter or voters should be bribed to change their vote to achieve a given outcome. 