[EM] Re: Majority Criterion
Chris Benham
chrisbenham@bigpond.com
Tue Mar 16 09:16:03 2004
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James Green-Armytage wrote:
>
> majority criterion: If a majority of the voters prefers all of the members
> of a given set of candidates over all candidates outside that set, and
> they vote sincerely, then the winning candidate should come from that set.
D.R. Woodall's wording (in reference to "preferential election rules"
, on this list known as "ranked-ballot
election methods"):
"Majority: If more than half the voters put the same set of candidates
(not necessarily in the same order) at the top
of their preference listings, then at least one of those candidates
should be elected."
I regard this as equivalent to the "Mutual Majority" Criterion, as
here defined by Blake Cretney:
"If there is a majority of voters for which it is true that they all
rank a set of candidates above all others, then one of these candidates
must win."
As distinct from this, Blake Cretney also defines a "Majority"
Criterion that Bart Ingles is more familiar with:
"If an alternative is ranked first on a majority of ballots, that
alternative must win."
That version I like to call "Majority Favourite". I am happy with
Woodall's terms, but to be clear I write "(mutual) Majority"
or "Majority Favourite" (depending on which I mean).
When making technical comparisons between methods that allow voters to
fully rank the candidates and those that don't, I am
strongly of the view that the best/only way to get away from confusion
and sophistry is to consider that all the methods have as
their input ranked ballots with an approval cutoff. So we sometimes
assume that the voters' rankings exist, even if the method
doesn't allow them to be actually recorded on the ballot papers; and
also we can sometimes assume that the voters have
approval cutoffs, even if they are not indicated on the ballot papers.
In this light, Approval is a method that ignores everything except
the approval-cutoffs, and so fails even Majority Favourite.
.
Bart Ingles (Mon.Mar.15) wrote:
"Eric Gore wrote:
>
> Now, if you can present an example where the Condorcet winner, with a
> reasonable interpretation of the ballots, did not win, you may have a
> good discussion on your hands.
Nurmi gives this example (credited to Fishburn):
1 a>b>c>d>e
1 b>c>e>d>a
1 e>a>b>c>d
1 a>b>d>e>c
1 b>d>c>a>e
Condorcet winner: a
1st/2nd/3rd/4th/5th ranks
---------------------------
a 2/1/0/1/1
b 2/2/1/0/0
This is a case where the Borda (and possibly Approval) winner really
does sound more plausible than the CW based on ranks alone."
Quota Limited Weighted Approval (QLWA) in this example agrees with Condorcet.
Candidate weights- a:2 b:2 c:0 d:0 e:1
5 ballots, so Quota = 2.5.
Approvals- 1:1a>.25b, 1:1b>1c>.5e, 1:1e>.75a, 1:1a>.25b, 1:1b>1d>1c>.25a
Final scores- a:3 b:2.5 c:2 d:1 e:1.5
I give this illustration just because I like QLWA, not to dispute that plain Approval could
plausibly (or would likely) elect b.
Chris Benham
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<pre>James Green-Armytage wrote:
><i>
</i>><i> majority criterion: If a majority of the voters prefers all of the members
</i>><i> of a given set of candidates over all candidates outside that set, and
</i>><i> they vote sincerely, then the winning candidate should come from that set.</i></pre>
D.R. Woodall's wording (in reference to "preferential election rules" ,
on this list known as "ranked-ballot <br>
election methods"):<br>
<br>
"Majority: If more than half the voters put the same set of candidates
(not necessarily in the same order) at the top<br>
of their preference listings, then at least one of those candidates should
be elected."<br>
<br>
I regard this as equivalent to the "Mutual Majority" Criterion, as here
defined by Blake Cretney:<br>
"If there is a majority of voters for which it is true that they all rank
a set of candidates above all others, then one of these candidates must
win." <br>
<br>
As distinct from this, Blake Cretney also defines a "Majority" Criterion
that Bart Ingles is more familiar with:<br>
"If an alternative is ranked first on a majority of ballots, that alternative
must win."<br>
<br>
That version I like to call "Majority Favourite". I am happy with Woodall's
terms, but to be clear I write "(mutual) Majority"<br>
or "Majority Favourite" (depending on which I mean).<br>
<br>
When making technical comparisons between methods that allow voters to fully
rank the candidates and those that don't, I am<br>
strongly of the view that the best/only way to get away from confusion and
sophistry is to consider that all the methods have as <br>
their input ranked ballots with an approval cutoff. So we sometimes assume
that the voters' rankings exist, even if the method<br>
doesn't allow them to be actually recorded on the ballot papers; and also
we can sometimes assume that the voters have <br>
approval cutoffs, even if they are not indicated on the ballot papers.<br>
In this light, Approval is a method that ignores everything except the
approval-cutoffs, and so fails even Majority Favourite.<br>
.<br>
Bart Ingles (Mon.Mar.15) wrote:<br>
<pre>"Eric Gore wrote:
><i>
</i>><i> Now, if you can present an example where the Condorcet winner, with a
</i>><i> reasonable interpretation of the ballots, did not win, you may have a
</i>><i> good discussion on your hands.
</i>
Nurmi gives this example (credited to Fishburn):
1 a>b>c>d>e
1 b>c>e>d>a
1 e>a>b>c>d
1 a>b>d>e>c
1 b>d>c>a>e
Condorcet winner: a
1st/2nd/3rd/4th/5th ranks
---------------------------
a 2/1/0/1/1
b 2/2/1/0/0
This is a case where the Borda (and possibly Approval) winner really
does sound more plausible than the CW based on ranks alone."
Quota Limited Weighted Approval (QLWA) in this example agrees with Condorcet.
Candidate weights- a:2 b:2 c:0 d:0 e:1
5 ballots, so Quota = 2.5.
Approvals- 1:1a>.25b, 1:1b>1c>.5e, 1:1e>.75a, 1:1a>.25b, 1:1b>1d>1c>.25a
Final scores- a:3 b:2.5 c:2 d:1 e:1.5
I give this illustration just because I like QLWA, not to dispute that plain Approval could
plausibly (or would likely) elect b.
Chris Benham
</pre>
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