[EM] Greatest Majority Consent (GMC)
Dave Ketchum
davek at clarityconnect.com
Thu Mar 22 21:14:40 PDT 2007
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
--
davek at clarityconnect.com people.clarityconnect.com/webpages3/davek
Dave Ketchum 108 Halstead Ave, Owego, NY 13827-1708 607-687-5026
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