[EM] Re: Majority Criterion

[Chris Benham]

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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