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Whereas the literature on mechanism design typically takes the view that a well-designed social choice mechanism should ideally implement the outcome that is selected when individuals report their preferences truthfully, this paper considers an alternative metric to use in comparing alternative voting mechanisms, namely, the proportion of the population which benefits from manipulation of the mechanism. An important feature of the voting populations that we study is that some proportion of the voters may be highly partisan, and will always vote for their preferred candidate, whereas the remaining voters may find that it is payoff-maximizing to vote strategically. We study two notions of manipulation: Gibbard manipulations, in which any voter who does not vote sincerely must prefer the outcome of the manipulated vote to the sincere voting outcome, and a new concept of manipulation, Nash manipulations, which requires all voters to be choosing best responses to the votes of the other candidate, and any candidate who does not vote for their preferred candidate must weakly prefer the manipulated outcome to the outcome which would prevail if they voted sincerely. We observe that both notions of manipulation lead to outcomes that are Pareto non-comparable with respect to the sincere voting outcome, and we calculate exact minimum and maximum bounds on the number of voters who benefit from strategic voting as a function of the number of voters, the number of candidates, and the number of highly partisan voters. Subsequently, we disaggregate the overall effect and calculate exact bounds for both the sincere and on the strategic voting populations, which provides us with additional insight into how these gains are shared. In some sense, our analysis can be viewed as a cautionary tale against being overly focused on designing collective choice procedures which always select the outcome which prevails when electors vote sincerely, because this may not be an outcome that is particularly worth protecting. In effect, the plurality rule becomes vulnerable to manipulation when it selects an outcome that is the preferred outcome of only a minority of voters, but it is precisely in such circumstances that manipulation leads to an outcome which is Pareto non-comparable to that selected under sincere voting. A voting procedure with respect to which manipulations typically benefit a large proportion of the voting population might therefore be seen as more desirable than those for which manipulations are typically only advantage a narrow minority of citizens.
