[EM] SD example--incomplete. Has error?
Markus Schulze
schulze at sol.physik.tu-berlin.de
Sun Jul 9 10:44:33 PDT 2000
Dear Norman,
you wrote (8 July 2000):
> The only remaining uncertainty is whether or not this is a
> *possible* pairwise matrix. It's somewhat difficult to work
> backwards from a 9-candidate example to a set of ballots, so
> I haven't attempted it.
It can be demonstrated that Blake's example is possible without
having to give an explicite set of ballots:
Step1: Blake demonstrated that whenever a given pairwise matrix
isn't possible then there is a number M such that if you add M
to each element of the pairwise matrix then this pairwise matrix
is possible (23 Feb 2000). Therefore it is possible to create
a set of ballots for what Mike calls "Act I" (8 July 2000)
of Blake's example (7 July 2000). You can add 7 voters who
vote A>G>B>C>D>E>F>H>I and 7 voters who vote I>H>F>E>D>C>B>G>A
to Act I of Blake's example without changing the pairwise matrix.
Step2: After having created a set of ballots for Act I of
Blake's example, you can create a set of ballots for Act II
of Blake's example simply by substituting the 7 A>G>B>C>D>E>F>H>I
voters with 7 G>A>B>C>D>E>F>H>I voters.
Therefore it is clear that Blake's example showing SD violates
monotonicity is possible.
Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de
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