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<div class="moz-text-html" lang="x-western">Forest is right. I am the one
who started mixing up means and medians. Below is my previous mis-named <br>
post "Weighted Mean Approval", with some corrections and additions.<br>
<br>
Mike,<br>
Your first impression may be a bit off. The line I gave:<br>
<blockquote type="cite">
<pre>"A candidate whose weight exceeds half the total weight wins outright."</pre>
</blockquote>
is like the majority stopping rule in IRV. It has no effect on the result.
Here is another, perhaps more precise,<br>
wording :<br>
<pre>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
</pre>
<blockquote type="cite"> </blockquote>
WMA <br>
25: Brown Jones<br>
26: Davis Smith Brown<br>
49: Jones Smith Brown<br>
<br>
WMA scores: Brown: 100 Davis: 26 Jones: 74 Smith: 75 <br>
<br>
Brown wins with 100% approval. This method has in common with Bucklin
a severe failure of Later-no-harm, combined<br>
with meeting Later-no-help, to create big incentives to truncate. Here
if the 49 Jones>Smith>Brown voters had truncated<br>
after Smith, then Smith would have won and if they had truncated after
Jones (bullet-voted) then Jones would have won.<br>
<br>
An interesting method that I prefer is WMA-STV. The WMA scores are
used as the fixed elimination schedule for <br>
fractional STV with a majority stopping rule. Taking the above example:<br>
<br>
WMA-STV: Eliminate Davis, which raises Smith's top preference score to
26 (short of a majority), so eliminate <br>
(next on the fixed elimination schedule) Jones, which raises Smith's
top preference score to 75 (a majority) so<br>
Smith wins.<br>
This time if the 49 Jones voters bullet-vote, Smith and Davis are eliminated
but then Brown wins (so the truncation backfires).<br>
If they instead truncate after Smith, Davis and then Brown are eliminated
and then Jones wins. So we have an example of the <br>
method failing Later-no-help (desirable, in my view, so as to balance failing
Later-no-harm.)<br>
<br>
Plain WMA, as I have defined it, is descended from an earlier version
(from Joe Weinstein, Forest tells us) in which each ballot<br>
approves as many of the highest-ranked candidates as possible without
their combined weight exceeding half the total weight,<br>
and then only approves the next ranked candidate if the weight of candidates
ranked below this (pivot) candidate is greater than<br>
the weight of candidates ranked above it. If the two weights are equal,
then the ballot half-approves that candidate.<br>
The problem with this is that it fails 3-candidate Condorcet. To distinguish
it, this earlier version could perhaps be called <br>
"Above Median Weighted Approval" (AMWA). In the example above the different
rule has no effect.<br>
<br>
<br>
>From what I understand of Forest's post "Bucklin and determining
the highest generalized median rank", Jones in the above<br>
example is the candidate with the highest "generalized" median rank.<br>
<a
href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-April/012642.html"><br>
http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-April/012642.html</a><br>
<br>
I am pretty sure that the method that always picks the candidate with
the highest "generalised median rank", is Woodall's <br>
"Quota-Limited Trickle-Down" (QLTD) rule. The simplest definition is that
it is just like Bucklin, except that when more than<br>
one candidate has a majority, the winner is the candidate who had more votes
at the end of the previous round (when the tallies<br>
were highest before any candidate had a majority). In the above example,
Jones is the QLTD winner.<br>
<br>
Woodall splits the Independence of Clones Criterion into "Clone-Winner"
and "Clone-Loser".<br>
<br>
"Clone-Winner: cloning a candidate who has a positive probability of election
should not help any other candidate"<br>
"Clone-Loser: cloning a candidate who has a zero probabilty of election should
not change the result of the election."<br>
<br>
Woodall lists QLTD as failing both of these. He rejects it (mainly) because
it fails Mono-add-top (which he demonstrates).<br>
Going down the list, he has it meeting Majority, Plurality, fails all his
Condorcet-related criteria, meets Mono-raise, <br>
Mono-remove-bottom, Mono-raise-delete, Mono-sub-plump, Mono-add-plump, Mono-append;
but fails Mono-add-top,<br>
Mono-remove-bottom, Participation, Mono-raise-random, Mono-sub-top, Later-no-harm
and Symetric Completion.<br>
It meets Later-no-help.<br>
There is some discussion of QLTD, and those interested can brush up on those
monotoicity criteria definitions here:<br>
<a class="moz-txt-link-freetext" href="http://groups.yahoo.com/group/election-methods-list/files/wood1996.pdf">http://groups.yahoo.com/group/election-methods-list/files/wood1996.pdf</a><br>
<br>
Chris Benham<br>
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