It depends upon what you mean by "fail". There are many different ballot configurations that result in the same pairwise-matrix, so there is not likely to be a unique solution to the Diophantine equations involved in the decomposition. Whether that is a problem or not is a matter of taste.
Bishop's decomposition algorithm finds ONE set of ballots that results in the given pairwise-matrix. That there are other possibilities is an obvious consequence of Arrow's theorem (if it isn't actually part of the proof). This is my philosophical objection to vote-counting methods that use the pairwise-matrix as input. You cannot map the pairwise matrix to the voters' ballots unambiguously, so any such method is by definition "not transparent". I believe any valid Condorcet-method should be able to be defined without reference to the pairwise matrix. Begin with the array of ballots = voters x alternatives, and work from there. If you don't begin with ballots, then it's not an election-method. > -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] On Behalf Of > Warren Smith > Sent: Sunday, November 27, 2005 5:24 PM > To: election-methods@electorama.com > Subject: [EM] I think Bishop's deconstruction algorithm fails > > And I think you can construct a counterexample of this form: > for some fairly large number C of candidates, but only 2 voters, > make a Condorcet CxC matrix out of 2 random votes. > > Can anybody conform or deny this? > wds > ---- > election-methods mailing list - see http://electorama.com/em > for list info > ---- election-methods mailing list - see http://electorama.com/em for list info
