[EM] I think Bishop's deconstruction algorithm fails

Dan Bishop daniel-j-bishop at neo.tamu.edu
Sun Nov 27 18:01:09 PST 2005

Warren Smith wrote:
> Well, one reply to that is "duh."  Another reply is, there are matrices 
> which do not arise from ballots.  It is an interesting question which
> matrices are achievable and which are not.
> Bishop's algoorithm if it works (which I doubt) would answer that question.
> I have a method involving solving an integer program which does answer the
> question, but only at heavy computational cost.  Bishop's method
> if it works would have mild computational cost.
> My method works and I doubt Bishop's works, but it would be nice
> to produce an explicit counterexample to Bishop's algorithm.

I think I've found one:

   A B C D
A - 0 0 1
B 3 - 1 2
C 3 2 - 3
D 2 1 0 -

This is a valid Condorcet matrix, created by the ballots C>B>A>D, 
C>D>B>A, and B>C>D>A.

My CMD proposal creates two C>B>D>A ballots, which would force the 
remaining ballot to contain a cycle.

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