Let's agree that when we say an Election Method selects the "correct" winner, what we mean by that is that is the winner is consistent with the majority's opinion.
When there are only two candidates, this is unambiguous.
With three or more candidates it is possible that there are different "majorities", so there are different "election methods" that select one of the majorities to determine the winner.
We are all familiar with candidateXcandidate matrix that sums voter's "pairwise" prrefences, but I would suggest that such a matrix is not very useful for determining which of the "majorities" in the pairwise comparisons is the ONE that that is most reflective of the majority of the voters' preferences.
Let's forget a moment the pairwise comparisons and instead of looking at things as
>31:� B>A>E>C>D
>23:� C>B>A>E>D
>25:� D>A>C>E>B
>11:� D>C>B>A>E
>10:� E>A>C>B>D
we examine the ordinal rankings of each candidate by each of the sets of ballots:
V A B C D E
31: 2 1 4 5 3
23: 3 2 1 5 4
25: 2 5 3 1 4
11: 4 3 2 1 5
10: 2 4 3 5 1
Now define the Score for a candidate as SUM ( function_of_ordinal_by_voter( candidate ) ).
We can describe Plurality by defining function_of_ordinal_by_voter as F(candidate) = 1 for candidates ranked 1 by the voter and F(candidate)=0 for all others. B wins with 31 to 25 to 23 to 11 to 10.
Borda is sum over voters with F(candidate) = (number of candidates)-(ordinal_by_voter). A gets: 255, B gets 225, C gets 226, D gets 144, and E 150. By Borda, the plurality winner comes in third and the winner is A even though A got no first place votes.
My point is that in discussing the merits of Election Methods, translating the examples into an anlytical form makes it easier to compare results from different methods.
It is slightly harder to deal with iterative methods, but it appears to me on the face of it that iterative methods need to be preference-preserving: i.e. if I ranked X and Y on my ballot, when the higher of them is eliminated by an iterative method, the other should get the same weight in the method's vote-counting process as the one that was eliminated. (I.e. when I said X>Y>Z I want my Y>Z to count as much as my X>Y did.)
Regards,
Paul Kislanko
