[EM] Reverse - Symetrical Weighted Median Approval

[Chris Benham]

  Participants,

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















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