Forest: Apparently, as I thought, your method of decomposition is to simply to remove cycles first, and then reversals. My point remains, then, that your decomposition method does NOT NECESSARILY yield the same outcome as Saari's matrix decomposition method.
There is a lot of disagreemnt over the correct outcome, in this example. Even Kemeny's Rule (which is also Condorcet's method, according to Hannu Nurmi, as I quoted previously in message # 10496) disagrees with both your decomp and Saari's. According to my calculations, Kemeny's Rule goes like this: Voting Vector: p=[ [3] [5] [0] [5] [0] [5] ] [This is a vertical column matrix.] Matrix (M) for finding the tally for Kemeny's Rule: M= [[ 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: M(p)=[ [25] [23] [31] [29] [31] [23] ], or 25:A>B>C [0(3) + 1(5) + 2(0) + 3(5) + 2(0) + 1(5) = 25] 23:A>C>B [1(3) + 0(5) + 1(0) + 2(5) + 3(0) + 2(5) = 23] 31:C>A>B [2(3) + 1(5) + 0(0) + 1(5) + 2(0) + 3(5) = 31] 29:C>B>A [3(3) + 2(5) + 1(0) + 0(5) + 1(0) + 2(5) = 29] 31:B>C>A [2(3) + 3(5) + 2(0) + 1(5) + 0(0) + 1(5) = 31] 23:B>A>C [1(3) + 2(5) + 3(0) + 2(5) + 1(0) + 0(5) = 23] With the KR, the lower tally is the better, so we have a KR tie between A>C>B and B>A>C. Interesting enough, that is similar to the end result that you get when you first remove the reversals, and then the cycles: 0:A>B>C: 3-3-0=0 3:A>C>B: 5-2-0=3 0:C>A>B: 0-0-0=0 0:C>B>A: 5-3-2=0 0:B>C>A: 0-0-0=0 3:B>A>C: 5-0-2=3 SB >----- Forwarded Message ----- >From: Forest Simmons <[EMAIL PROTECTED]> >To: <[EMAIL PROTECTED]> >Subject: RE: [EM] Saari's Basic Argument > >On Fri, 21 Feb 2003, Steve Barney wrote: > >> Forest: >> >> How do you decompose my example (from my last email, #10873), and what do you >> get?: >> >> 3:A>B>C >> 5:A>C>B >> 0:C>A>B >> 5:C>B>A >> 0:B>C>A >> 5:B>A>C > >Subtract out five copies of the cycle ACB+CBA+BAC. > >That leaves 3*ABC. > >Forest Steve Barney Richard M. Hare, 1919 - 2002, In Memoriam: <http://www.petersingerlinks.com/Hare/>. Did you know there is a 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>. ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
