Abstract
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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Notes
To be exact, this result is dependent on the tie-breaking rule of the Borda rule. However, for any tie-breaking rule, the difference between the thresholds of a Borda rule and plurality rule is always zero or minimal. This fact is discussed in the last section.
Hylland (1980) considers a more general decision scheme called a cardinal decision scheme. In this paper, only an ordinal decision scheme is discussed. Moreover, Hylland shows this result in a more restricted preference domain, in which each voter has a von Neumann-Morgenstern utility function. See Dutta et al. (2007) on the detailed explanation of these points.
If n is odd, \(\chi \left( f\right) \ge \left( n-2\right) /n\) is satisfied.
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I would like to thank an anonymous referee and an associate editor for many valuable comments and suggestions. This work was supported by JSPS KAKENHI Grant Number JP16K03612 and JP18K01513.
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Okumura, Y. What proportion of sincere voters guarantees efficiency?. Soc Choice Welf 53, 299–311 (2019). https://doi.org/10.1007/s00355-019-01184-8
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DOI: https://doi.org/10.1007/s00355-019-01184-8