[EM] Re: Weighted Median Approval

[Chris Benham]

Forest is right. I  am the one who started mixing up means and medians. 
Below is my previous mis-named
post "Weighted Mean Approval", with some corrections and additions.

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 Median 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).
If  they instead truncate after Smith, Davis and then Brown are 
eliminated and then Jones wins. So we have an example of  the
method failing Later-no-help (desirable, in my view, so as to balance 
failing Later-no-harm.)

Plain  WMA, as  I  have defined it,  is descended from an earlier 
version (from Joe Weinstein, Forest tells us) in which each ballot
approves  as many of  the highest-ranked candidates as possible without 
their combined weight exceeding half the total weight,
and then only approves the next ranked candidate if the weight of 
candidates ranked below this (pivot) candidate is greater  than
the weight of candidates ranked above it. If the two weights are equal, 
then the ballot half-approves that candidate.
The problem with this is that  it fails 3-candidate Condorcet. To 
distinguish it, this earlier version could perhaps be called
"Above Median Weighted Approval" (AMWA). In the example above the 
different rule has no effect.


 >From what I understand  of  Forest's post "Bucklin and determining the 
highest generalized median rank", Jones 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

I  am pretty sure that the method  that always picks the candidate with 
the highest  "generalised median rank", is Woodall's
"Quota-Limited Trickle-Down" (QLTD)  rule. The simplest definition is 
that it is just like Bucklin, except that  when more than
one candidate has a majority, the winner is the candidate who had more 
votes at the end of  the previous round (when  the tallies
were highest before any candidate had a majority). In the above example, 
Jones is the QLTD winner.

Woodall splits the Independence of  Clones Criterion into 
 "Clone-Winner"  and "Clone-Loser".

"Clone-Winner: cloning a candidate who has a positive probability of 
 election should not help any other candidate"
"Clone-Loser: cloning a candidate who has a zero probabilty of election 
should not change the result of the election."

Woodall  lists QLTD  as failing both of these. He rejects it (mainly) 
because it fails Mono-add-top (which he demonstrates).
Going down the list, he has it meeting  Majority, Plurality, fails all 
his Condorcet-related criteria, meets  Mono-raise,
Mono-remove-bottom, Mono-raise-delete, Mono-sub-plump, Mono-add-plump, 
 Mono-append; but fails Mono-add-top,
Mono-remove-bottom, Participation, Mono-raise-random, Mono-sub-top, 
Later-no-harm and Symetric Completion.
It  meets Later-no-help.
There is some discussion of  QLTD, and those interested can brush up on 
those monotoicity criteria definitions here:
http://groups.yahoo.com/group/election-methods-list/files/wood1996.pdf

Chris Benham











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