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Forest,<br>
Thanks for your interest and kind words. I may have mis-identified this
as Joe Weinstein's "weighted median approval" method:<br>
<br>
Voters rank the candidates, equal preferences ok. <br>
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.<br>
Then approval scores for each candidate is derived thus: each
ballot approves all candidates that are ranked in first or equal first
place<br>
(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 number of ballots.<br>
Then the candidate with the highest approval score wins.<br>
<br>
This is bit different from the version that you and David Gamble have
in mind:<br>
<pre wrap="">"Once the weights have been calculated, then on each ballot determine
which candidates will be approved: Suppose that candidate K is marked at
level L on ballot B. If the total weight of all the candidates marked
above level L (on ballot B) is less than the total weight of all
candidates marked below level L (on ballot B), then ballot B contributes
full approval of candidate K (as well as all of the other candidates
marked at level L). If the total weights above and below are equal, then
the candidates at level L receive half approval from ballot B. Otherwise,
the candidates at level L receive no approval from ballot B."
In my version the sum of the weights of the candidates approved by each ballot is always at least half the total weight of all the candidates.
My version will always pick a CW if there are three candidates.
40:A>B>C
25:B>A>C
35:C>B>A
100 ballots. B is the CW.
Weights: A:40 B:25 C:35
In my version, all the ballots approve B and then B wins with 100% approval.
In yours, the 40 A>B>C voters don't approve B because A has a greater weight than C.
Your final approval scores are A:65, B:60, C:35, so A wins.
In this case the symetrified versions give the same results.
Reverse weights: A35 B0 C65. Both versions give these reverse approval scores A:35, B:35, C:65.
So in your symetrified version the final scores are A:30, B:25, C:-30 , whereas in mine they are
A:30, B:65, C:-30.
You mentioned this example (from David Gamble):
215:A>B>C
169:A>C>B
109:B>A>C
177:B>C>A
115:C>A>B
215:C>B>A
1000 ballots. B is the CW.
Weights: A:384 B:286 C:330
All ballots (in my version) approve their top two (because each of these weights is less than half the total weight, and any two add up to more than half) to give these approval scores
A:608 B:716 C:676
Reverse weights: A:392 B:284 C:324
Again in my version all ballots reverse approve their bottom two, to give these reverse approval scores
A:616 B:714 C:670
Net final scores
A:-8 B:2 C:6
Interestingly, in this case the symetrifying process causes the CW to lose to the Borda winner C.
(Your version picks C both ways.)
I find this surprising with only three candidates, but not neccessarily a disaster.
Of greater concern to me is that the method very clearly fails Later-no-harm while meeting Later-no-help, so zero-information voters might be disadvantaged if they vote purely according to their
rankings (taking no account of their ratings). To put it more plainly, the method is susceptible to
strategic truncation, and might generally encourage truncation.(Maybe in this respect your version is
better.)
Chris Benham
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