On the manipulability of approval voting and related scoring rules

We characterize all preference pro¯les at which the approval (voting) rule is manipulable, under three extensions of preferences to sets of alterna-tives: by comparison of worst alternatives, best alternatives, or by com-parison based on stochastic dominance. We perform a similar exercise for k-approval rules, where voters approve of a ¯xed number k of alternatives. 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, speci¯cally, the values of k for which the k-approval rule is mini-mally manipulable { has the smallest number of manipulable preference pro¯les { under the various preference extensions are determined. For the number of voters going to in¯nity, an asymptotic result is that the k-approval rule with k around half the number of alternatives is minimally manipulable among all scoring rules. Further results are obtained by sim-ulation and indicate that k-approval rules may improve on the approval rule as far as manipulability is concerned.