[EM] "Beatpath GMC" compliance a mistaken standard?

Kristofer Munsterhjelm km-elmet at broadpark.no
Sun Jan 11 07:05:09 PST 2009

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.

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