On the manipulability of approval voting and related scoring rules
研究了在三种偏好扩展下批准投票规则可被操纵的偏好分布，并比较了不同k值下的k-批准规则，发现当k约为候选人半数时操纵性最低。
We characterize all preference profiles at which the approval (voting) rule is manipulable, under three extensions of preferences to sets of candidates: by comparison of worst candidates, best candidates, or by comparison based on stochastic dominance. We perform a similar exercise for k -approval rules, where voters approve of a fixed number k of candidates. These results can be used to compare ( k -)approval rules with respect to their manipulability. Analytical results are obtained for the case of two voters, specifically, the values of k for which the k -approval rule is minimally manipulable—has the smallest number of manipulable preference profiles—under the various preference extensions are determined. For the number of voters going to infinity, an asymptotic result is that the k -approval rule with k around half the number of candidates is minimally manipulable among all scoring rules. Further results are obtained by simulation and indicate that k -approval rules may improve on the approval rule as far as manipulability is concerned.