Juho Laatu wrote:
--- On Sun, 11/1/09, Kristofer Munsterhjelm <km-el...@broadpark.no> wrote:
Let's consider the first election first, with
truncation extended to full preference:
26: A > B > C
25: B > A > C
49: C > A = B
A B C: 100 prefer {A B C} to the empty set
This case is interesting (not that it
would have any impact on the ongoing
mutual majority discussion but just
for theoretical interest).
The number of candidates was not exactly
stated in the example. If there are e.g.
four candidates then the votes would be:
26: A > B > C = D
25: B > A > C = D
49: C > A = B = D
Set {A, B, C} has in this case no support.
Let's assume that there are also other
citizens (=potential candidates who
are however not candidates) than the
named candidates. The opinions of the
first 26 voters could be as follows.
26: X1 > A > B > X2 > C = D = X3 > X4
The point here is that the voters have
not said that they would prefer A, B, C
and D to the other citizens / potential
candidates (X1, X2,...).
It is ok to say that if there are no
"mutual majorities" the winner can be
elected from the whole set of candidates
{A, B, C} or {A, B, C, D} or whatever set.
One can not say that the voters would
prefer the all the candidates (or those
that are named on the ballots) to other
citizens. What is the meaning of saying
that they prefer these candidates to an
empty set?
There is no real meaning - it's just an artifact of taking the process
to its conclusion. The only thing it means is that all voters who voted,
voted for the candidates they voted for, which is a tautology.
Smaller unanimity sets can only exist if there's a candidate or a
candidate set that everybody ranks last.
Also note that changing a vote from A > B to X1 > A > B can dissolve
what would otherwise be a majority for {A B}. Mutual majority isn't
complete - it only says that in certain cases (majority support for a
set), certain things should happen (the method should elect from the
set). In that respect, it's kind of like independence of clones. You can
make a method that technically passes mutual majority yet wouldn't be
any good, just like you can prefix a method with "remove clones" yet it
would be a bad method if a single voter didn't vote clones in strict
clone order.
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