# [EM] My Matrix for Kemeny's Rule, n=3

barnes99 barnes99 at vaxa.cis.uwosh.edu
Fri Dec 20 13:56:03 PST 2002

```Here is my matrix for Kemeny's Rule with 3 candidates. Please let me know if I
got it right or wrong.

Here is an example of a Kemeny Rule tally for profile p, p=(1,1,0,0,0,0),
where the first thru sixth columns represent, respectively, the number of ABC,
ACB, CAB, CBA, BCA, and BAC voters.

Voting Vector:

p=[ 1 1 0 0 0 0 ]

Matrix (M) for Kemeny's Rule:

[[ 0 1 2 3 2 1 ]
[ 1 0 1 2 3 2 ]
[ 2 1 0 1 2 3 ]
[ 3 2 1 0 1 2 ]
[ 2 3 2 1 0 1 ]
[ 1 2 3 2 1 0 ]]

The KR tally is:

p(M)=[ 1 1 3 5 5 3 ], where

ABC=[0+1+0+0+0+0]=1
ACB=[1+0+0+0+0+0]=1
CAB=[2+1+0+0+0+0]=3
CBA=[3+2+0+0+0+0]=5
BCA=[2+3+0+0+0+0]=5
BAC=[1+2+0+0+0+0]=3

This is the measure of the "distance" from unanimity, so the lower the score
the better. In this example, we have a tie between the ABC and ACB outcomes,
in which case I guess the final outcome must be A>B~C. This may not be a very
interesting example, but the point is that I believe this is how to do a KR
tally with 3 candidates. Please correct me, if I'm wrong.

Thank you,
SB

Steve Barney

Richard M. Hare, 1919 - 2002, In Memoriam: <http://www.petersingerlinks.com/hare.htm>.

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