On the manipulability of approval voting and related scoring rules

We characterize all preference profiles at which the approval (voting) rule is manipulable, under three extensions of preferences to sets of alternatives: by comparison of worstalternatives, best alternatives, 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 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, 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 alternatives 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.