[EM] I think Bishop's deconstruction algorithm fails

[Paul Kislanko]

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: election-methods-bounces at electorama.com 
> [mailto:election-methods-bounces at electorama.com] On Behalf Of 
> Warren Smith
> Sent: Sunday, November 27, 2005 5:24 PM
> To: election-methods at 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
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