I have been inspired by this post by Forest Simmons (Th.Feb.5, 2004)
http://groups.yahoo.com/group/election-methods-list/message/13475
which contains some discussion of Joe Weinstein's "weighted median approval method":
Voters rank the candidates, equal preferences ok.
Each candidate is given a weight of 1 for each ballot on which that candidate is ranked alone in first place, 1/2 for each ballot on which that candidate is equal ranked first with one other candidate, 1/3 for each ballot on which that candidate is ranked equal first with two other candidates, and so on so that the total of all the weights equals the number of ballots.
Then approval scores for each candidate is derived thus: each ballot approves all candidates that are ranked in first or equal first place
(and does not approve all candidates that are ranked last or equal last). Subject to that, if the total weight of the approved candidates is less than half the total of number of ballots, then the candidate/s on the second preference-level are also approved, and the third, and so on; stopping as soon as the total weight of the approved candidates equals or exceeds half the total mumber of ballots.
Then the candidate with the highest approval score wins.
This method always picks a CW if there are three candidates, and I believe that (in common with plain Approval) it meets Participation.
But unlike plain Approval, it fails Reverse Symetry.
46:A>C>B
10:B>A>C
10:B>C>A
34:C=B>A
100 ballots
The above WMA method picks C as "the most approved" candidate, but if the ballots are reversed and the same process is applied
then it also picks C as the "most disapproved" candidate.
This leads me to suggest this method, "reverse-symetrical WMA":
(1) Use the "weighted median approval method" described above to derive approval scores for each candidate
(2) Based on the reversed rankings, use the same method to derive "disapproval" scores for each candidate.
(3) Subtract the step-2 scores from the step-1 scores. The candidate with the highest resulting score wins.
This looks like a method that meets (mutual) Majority, Independence of Clones, Participation, Reverse Symetry and Woodall's
Purality criterion.
It fails Symetric Completion, IPDA (independence of Pareto-dominated alternatives), and Steve Eppley's "resistance to truncation"
criterion.
The method is highly likely to pick the CW, and I think is generally much better than the "set-intersection" methods (described by
Woodall).
Chris Benham
