# [EM] Saari's Basic Argument

Steve Barney barnes99 at vaxa.cis.uwosh.edu
Sat Jan 18 16:06:54 PST 2003

Forest and Alex:

I don't have my worksheets with me, but I am not sure that you are doing the
decompositions correctly. You should note that Saari's exposition is a
simplified intuitive description of his decomposition method, and what he
describes in hard math is not exactly the same. I suspect that this is why you
get two different decompositions when you do it in different orders. Try using
Saari's decomposition matrix with your examples, and see if you get the same
decomposition profile as you get with your method.

I said something like this previously in message # 9251, and in private
correspondence with Alex (if I am not mistaken).

Anyway, perhaps we could focus on Saari's analysis of Condorcet's example -
see Section 8, in:

EXPLAINING ALL THREE-ALTERNATIVE VOTING OUTCOMES
DONALD G. SAARI
http://citeseer.nj.nec.com/cache/papers/cs/11095/http:zSzzSzwww.math.nwu.eduzSz~d_saarizSzvotezSztriple.pdf/saari99explaining.pdf

at before:

5 ABC
3 BCA

Can you give me the decomposition profile, T(p), for this example?

SB

>From: Forest Simmons <fsimmons at pcc.edu>
>To:  <election-methods-list at eskimo.com>
>Subject: RE: [EM] Saari's Basic Argument
>
>I hate to beat a dead horse, but in order to see the fallacy of Saari's
>symmetry arguments let's take this example a little further:
>
>66 ABC
>34 BCD
>
>The 12 o'clock and 8 o'clock positions representing these two factions are
>non adjacent on the clock face.  Between them is the fully ranked order
>BAC (at 10 o'clock).
>
>To get them adjacent with Saari's symmetries, first add 34 copies of the
>canceling pair {BAC, CAB} to get
>
>66 ABC, 34 BAC, 34 BCD, 34 CAB.
>
>Now remove 34 copies of the cycle {ABC,BCD,CAB} to get
>
>32 ABC, 34 BAC.
>
>According to Saari this ballot set is equivalent to the original.
>
>How could these perfect symmetries bring about such a ridiculous
>equivalence?
[...]

Steve Barney

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