I've decided to bin (i.e.I now withdraw) my suggested "Add-Top Proofed
Mutual Majority" as something highly desirable or a real improvement on
plain Mutual Majority.
I defined it thus:
*If the number of ballots on which some set S of candidates is voted
strictly above all the candidates outside S is greater than the number
of ballots on which all the members of S are voted below equal-top
(i.e. strictly below some/any outside-S candidate), then the winner must
come from S.*
I've discovered that it's incompatible with the Condorcet criterion.
2: A>B=C=D=E
1: A>X
1: B>X
1: C>X
1: D>X
1: E>X
The criterion says that the winner must come from {A B C D E}, but X is
the CW (pairwise beating all the other candidates 4-3).
Following a suggestion from Kevin Venzke, I now instead propose
"Add-Plump Proofed Mutual Majority" (APPMM):
*If the number of ballots on which some set S of candidates is voted
strictly above all the candidates outside S is greater than the number
of ballots on which any outside-S candidate is voted strictly above any
member of S, then the winner must come from S.*
Kevin gives this demonstration of how it differs from regular Mutual
Majority:
28: A>B
27: B>A
45: C>D
Both MM and APPMM say that the winner must come from {A B}, but if we
add 12A ballots APPMM says the same thing while MM now says nothing.
I haven't found or thought of any method that meets both of
Mono-add-Plump and regular Mutual Majority (aka Majority for Solid
Coalitions) but fails APPMM.
Chris Benham
I wrote (13 Jan 2012):
On 21 Dec 2011 I proposed this criterion:
> *The winner must come from the smallest set S of candidates about which
> the following is true: the number of ballots on which all the members
> (or sole member) is voted strictly above all the non-member candidates
> is greater than the number of ballots on which a (any) non-member
> candidate is voted strictly above all the members of S.*
That is fairly clear, but the wording could perhaps be improved, say:
*If the number of ballots on which some set S of candidates is voted
strictly above all the candidates outside S is greater than the number
of ballots on which all the members of S are voted below equal-top
(i.e. strictly below some/any outside-S candidate), then the winner must
come from S.*
I tentatively suggested the name "Add-Top Proofed Solid Coalition
Majority". A bit less clumsy would be "Add-Top Proofed Mutual
Majority". Maybe there is a better name that either does without the
word "Majority" or includes another word that qualifies it. For the
time being I'll stick with Add-Top Proofed Mutual Majority (ATPMM)
I gave this example:
45: A>B
20: A=B
32: B
03: D
My criterion says that the winner must be A, but Mike Ossipoff's MTA
method elects B.
I did endorse MTA as an improvement on MCA, but since it (and not MCA)
fails this (what I consider to be very important) criterion (and is also
a bit more complicated than MCA) I now withdraw
that endorsement. I still acknowledge that MTA may be a bit more
"strategically comfortable" for voters, but I can't give that factor
enough weight to make MTA acceptable or win its comparison with MCA.
Chris Benham
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