pairwise matrices and ballots

Thu Feb 24 23:00:53 PST 2000


> Blake wrote:


> >Often people want to create examples involving pairwise methods,

> >usually to show that the method behaves badly in some situation.

> >Since not all pairwise matrices are possible, it is customary to

> >provide a set of ballots instead of just providing a pairwise matrix.


> For any pairwise preference matrix, it's possible to devise

> a set of rankings that will give that pairwise preference matrix.

> So, for pairwise methods, it's unnecessary to furnish rankings

> for an example--the pairwise preference table is sufficient.

I don't know where you got that idea.  Try the following example:

   A    B    C

A  X    1    3

B  2    X    1

C  1    3    X

The matrix produces

2   BA(C)  and/or B > (A=C)
1   AB(C)  and/or A > (B=C)
3   AC(B)  and/or A > (C=B)
1   CA(B)  and/or C > (A=B)
3   CB(A)  and/or C > (B=A)
1   BC(A)  and/or B > (C=A)

The possibility of truncated votes makes producing a matrix before having 
actual or example votes very difficult or impossible.  Roughly like taking a 
photo of a chess board after X moves and trying to go backward to get the 
exact order of the moves.

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