[EM] Greatest Majority Consent (GMC)

[Forest W Simmons]

This method has the same relationship to Beatpath that MMPO has to 
MinMax.

Let's call the opposite of opposition, consent.  Then MMPO which is an 
abbreviation for Min Max (pairwise opposition) could be characterized 
as Max Min (pairwise consent).

To be definite, the pairwise consent for A relative to B is the number 
of ballots on which A is ranked at least as high as B.

Note that under this definition either A has majority pairwise conset 
relative to B, or B has majority pairwise consent relative to A, OR 
they are tied 50/50.

In large scale public elections, even if there is a pairwise tie, we 
don't expect it to be 50/50 since even one ballot that ranks the tied 
candidates equal or truncates them both would change the tie to x/x 
where x is greater than 50%.

So to make things easier, I am going to assume that every candidate 
pair has a majority pairwise consent in at least one of the two 
directions.

Here's the GMC method:

1.  Construct the pairwise majority consent graph whose vertices 
represent the candidates and whose weighted, directed edges from vertex 
i to vertex j with weight w represent majority consent w of candidate i 
relative to candidate j.

2.  The strength of a directed path in this graph (i.e. a majority 
consent path) is the weight of the weakest directed edge in the path.

3.  Elect the candidate X for which, for each candidate Y not equal to 
X, there is a majority consent path from X to Y that is stronger than 
any such path from Y to X.

That's it.

The existence of the winner depends on the following two facts:

1. For any two candidates X and Y, if there is no majority consent path 
from Y to X, then there is one from X to Y.  In fact there is a path of 
one step, under the above assumption that we have adopted for large 
scale public elections.

2. Majority consent paths induce a transitive relation on the 
candidates in the same way that beatpaths do.

Note that if every ballot fully ranks the candidates, the method is 
identical to beatpath, just as MMPO is identical to ordinary MinMax in 
that context.

I wonder if this method preserves the FBC property of MMPO.

Any thoughts?

Forest