# [EM] Re: Weighted Mean Approval

Chris Benham chrisbenham at bigpond.com
Fri Apr 9 22:34:01 PDT 2004

```Mike,
Your first impression may be a bit off.  The line I gave:

>"A candidate whose weight exceeds half the total weight wins outright."
>
is like the majority stopping rule in IRV. It has no effect on the
result. Here is another, perhaps more precise,
wording :

Weighted Mean Approval .
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.
The candidate with the highest approval score wins.

Take this recently discussed Bucklin example:

25:Brown>Jones>Davis>Smith
26:Davis>Smith>Brown>Jones
49:Jones>Smith>Brown>Davis

Weights:    Brown: 25     Davis: 26    Jones: 49    Smith: 0

WMA
25: Brown Jones
26: Davis Smith Brown
49: Jones Smith Brown

WMA scores:   Brown: 100    Davis: 26    Jones: 74    Smith: 75

Brown wins with  100% approval.  This method has in common with Bucklin
a severe failure of  Later-no-harm, combined
with meeting  Later-no-help, to create big incentives to truncate. Here
if  the  49 Jones>Smith>Brown voters had truncated
after Smith, then Smith would have won  and  if  they had  truncated
after  Jones  (bullet-voted) then  Jones would have won.

An interesting method  that  I  prefer  is  WMA-STV. The WMA scores are
used  as the fixed elimination schedule for
fractional  STV  with a majority stopping rule.  Taking the above example:

WMA-STV:  Eliminate Davis, which raises Smith's top preference score to
26 (short of  a majority), so eliminate
(next on the fixed elimination schedule) Jones, which raises Smith's top
preference score to 75 (a majority) so
Smith wins.
This time if  the  49 Jones voters bullet-vote, Smith and  Davis are
eliminated but then Brown wins (so the truncation backfires).

From what I understand  of  Forest's post "Bucklin and determining the
highest generalized median rank", Smith in the above
example is the candidate with the highest  "generalized" median rank.

http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-April/012642.html

Chris Benham
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