[EM] Intuitive argument that FPTP manipulability approaches certainty in impartial culture

[Kristofer Munsterhjelm]

A continuation of this:

It seems that (again, if my calculations are correct) that, for three 
candidates, every weighted positional method (1, w, 0) with 0 <= w < 1, 
basically everything but antiplurality, is almost surely manipulable 
under the impartial culture as the number of voters approaches infinity.

Antiplurality converges to 1/2.

Explanation for the latter: If WLOG, we let candidate A be the one with 
highest score, then candidate B, then candidate C in sorted descending 
order of score, so that A is the honest winner; then if B's score is 
closer to A's than to C's, strategy by B>A voters can make B win, but 
not otherwise. This because the manipulators are forced to increase C's 
score (by voting BCA) when they increase B's relative to A. So burying A 
makes C's score approach B's; B can only win if B overcomes A before C 
overcomes B, and that happens if B's score is closer to A's than to C's.

The math for other weighted positional methods is pretty hairy, but I 
can provide it if anyone's curious.

-km