Voting with Rubber Bands, Weights, and Strings

It follows that Borda’s behavior in the Crossing Boundaries example is duplicated by that of every scoring rule other than plurality — one observes two weak manipulations as the votes are transferred one-at-a-time, while M^cBorda exhibits a single strong manipulation. In the case of Plurality voting, the transfers have no effect, with all profiles in the sequence yielding the same two-way tie. The situation suggests why it may be misleading to compare two voting rules according to how frequently they can be manipulated, unless one looks at decisiveness at the same time.

(We are making a standard assumption here about scoring rules — that scoring weights for first, second, and third place satisfy w₁ ≥ w₂ ≥ w₃ with w₁ > w₃.)