Paul Kislanko wrote:
Arrrgggg. Explain, someone, anyone, how MM can change an (A B) to an (A B C)
possible winner set by adding voters for A.
Juho has already tried, but I'll try another way.
The first election is:
26 A>B
25 B>A
49 C
where mutual majority elects {A B}, and the second is
26 A>B
25 B>A
49 C
5 A
where mutual majority elects {A B C}, that is, has no opinion.
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
The majority is 51.
Now list the possible sets of candidates:
A B C: 100 prefer {A B C} to the empty set
A B : 26 + 25 = 51 prefer {A B} to C
A C : 0 prefer {A C} to B
B C : 0 prefer {B C} to A
A : 26 prefer A to {B C}
B : 25 prefer B to {A C}
C : 49 prefer C to {A B}
Whether a voter votes A > B or B > A, that counts as preferring the set
{A B} to C, because both are (one of A or B) > (the other of A or B) >
C. On the other hand, the C > A = B votes don't count towards {A C}
because they don't rank both A and C above B.
To satisfy mutual majority, we then ask if there's a set that's
preferred by a majority (more than 50) to all the members outside the
set. There is: the set {A B}. Thus, any method that satisfies mutual
majority must pick from either A or B.
--
Now let's consider the second election, similarly extended for the sake
of visibility:
26: A > B > C
25: B > A > C
49: C > A = B
5: A > B = C
The majority is 53, so anything above 52 is a majority.
Again, list the possible sets of candidates:
A B C: preferred by 100 to {}
A B : preferred by 26+25 = 51 to {C}
A C : preferred by 0 to {B}
B C : preferred by 0 to {A}
A : preferred by 26+5 = 31 to {B C}
B : preferred by 25 to {A C}
C : preferred by 49 to {A B}
To repeat, when we say that 51 prefer {A B} to C, it means that 51
voters vote both A and B strictly above C - in this case, those are the
26 A > B voters and the 25 B > A voters.
-
Since the majority threshold has increased from 51 to 53, the {A B} set
no longer has a majority. Neither does any of the others, so mutual
majority has no opinion (unless you count the unanimity for all candidates).
To answer your question directly: MM changes from {A B} to {A B C} (or
rather, from {A B} to no opinion or constraint) because the majority
threshold increases while the support for {A B} does not increase to the
same degree.
----
Election-Methods mailing list - see http://electorama.com/em for list info
