What proportion of sincere voters guarantees efficiency?
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This study considers a hybrid voting model where some of the voters sincerely vote but the others may not. By using the model, we discuss several voting rules: the plurality rule, Borda rule, and others. In each rule, we derive the threshold number such that a Pareto efficient alternative is always chosen if and only if the ratio of the sincere voters is more than the number. Further, we show that in any rule that satisfies strategy-proofness, a Pareto inefficient alternative may be chosen if even one voter insincerely votes.
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