[EM] Re: Reverse - Symetrical Weighted Median Approval

[Chris Benham]

  Forest,
Thanks for your interest and kind words. I  may have mis-identified 
 this as  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 number of ballots.
Then the candidate with the highest approval score wins.

This is bit  different from the version that you and David Gamble have 
in mind:

"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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