[EM] Most elimination methods based on weighted positional systems are nonmonotone

[Kristofer Munsterhjelm]

Consider an elimination method based on a weighted positional system. 
WLOG, when dealing with three candidates, the weighted positional system 
can be defined so that the candidate ranked first on a ballot gets one 
point, the candidate ranked second gets w points, and the candidate 
ranked last gets zero, where 0 <= w <= 1.

Furthermore, define that in the two candidate case, the weighted 
positional method reduces to Plurality - it elects the candidate which 
is ranked first most often, the candidate with a majority.

Then the following example should work to show nonmonotonicity for all 
values of w where 0 <= w <= 0.9999:

45: A>B>C
19: A>C>B
14: B>A>C
30: B>C>A
40: C>A>B
  9: C>B>A.

The outcome is that, first, B is eliminated and C wins, but when 14 
voters change their opinion from A>B>C to B>A>C (lowering A), C is 
eliminated and A wins.

Let's calculate this for the edge values (0 and 0.999). First, 0:

  Base situation:
	A: 64
	B: 44
	C: 49

  So B is eliminated. Then:
	A: 78
	C: 79

  So C wins.

  Then, 14 voters lower A as mentioned above and:
	A: 50
	B: 58
	C: 49

  So C is eliminated. Then:
	A: 90
	B: 67

  and A wins.


Second, 0.9999:

  Base situation:
	A: 117.9950
	B:  97.9946
	C:  97.9951

  So B is eliminated. Then:
	A: 78
	C: 79

  So C wins.

  Then, 14 voters lower A as mentioned above and:
	A: 117.9930
	B:  97.9960
	C:  97.9951

  So C is eliminated. Then:
	A: 90
	B: 67

  and A wins.

It seems that this example can be used on w = 1-eps as well, for 
arbitrarily small eps > 0, but I have not proved that. If that is true, 
then all we need to know is to show an example of Coombs failing 
monotonicity and all WPS-based loser elimination methods that reduce to 
"majority winner" in the two-candidate case will have been shown to fail 
monotonicity.