[EM] Greatest Majority Consent (GMC)

[Dave Ketchum]

I think I see Condorcet hidden in these new words.

Not clear how cycles get resolved - need more clarity?

DWK

On Thu, 22 Mar 2007 13:37:21 -0700 (PDT) Forest W Simmons wrote:

> 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

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  Dave Ketchum   108 Halstead Ave, Owego, NY  13827-1708   607-687-5026
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