[EM] My Matrix for Kemeny's Rule, n=3
Forest Simmons
fsimmons at pcc.edu
Tue Dec 31 14:02:57 PST 2002
Your example is correctly done.
Despite the intractability of the method for large numbers of candidates,
it seems like an ideal method for some situations.
One application could be in choosing between several orders that have been
found by other means.
[The main computational difficulty of finding the Kemeny order is not in
calculating the mean Kemeny distance to any particular order, but in the
sheer magnitude of the number of possible orders.]
Here's a homely example:
Start with the candidates ranked by Approval, Borda, Ranked Pairs, etc.
Do both a sink sort and a bubble sort to each of these preliminary orders.
[These sortings are ways of achieving "local Keminization." Ranked Pairs
is already locally Kemeny optimal.]
See which of the resulting locally optimum orders is best (among those
considered) globally by calculating the mean distance of each to the
preference orders on the ballots.
Forest
On Fri, 20 Dec 2002, barnes99 wrote:
> 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>.
>
> Did you know there is an web site where, if you click on a button, the advertisers there will donate 2 1/2 cups of food to feed hungry people in places where there is a lot of starvation? See:
> <http://www.thehungersite.com>.
>
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