[EM] Saari's Basic Argument

[Steve Barney]

Here is a simpler example to illustrate the difference that the order in which 
cyclic and reversal terms are canceled does not matter when using the strictly 
correct method - as opposed to the method used by Forest Simmons and Alex 
Small, and in some of Saari's popular expositions where he is merely trying to 
illustrate basic concepts to a more general audience.

Let's say we have 18 voters who start out completely indifferent or tied, as 
in (sometimes you have to consider fractional votes, as in a case with 15 
voters):

3:A>B>C
3:A>C>B
3:C>A>B
3:C>B>A
3:B>C>A
3:B>A>C


And then, after becoming informed, change they minds and acquire the following 
preferences (it is useful to think of a voter profile as going from a 
completely tied one to whatever the ballots indicate in the end):

3:A>B>C
5:A>C>B
0:C>A>B
5:C>B>A
0:B>C>A
5:B>A>C

or

p=[[3],[5],[0],[5],[0],[5]] (This is supposed to be put in the form of a 
vertical column matrix.)


"T" is The decomposition matrix:

T=(1/6)*
[[2	1	-1	-2	-1	1] Basic-A Term
[1	-1	-2	-1	1	2] Basic-B Term
[0	1	-1	0	1	-1] Reversal-A Term
[-1	1	0	-1	1	0] Reversal-B Term
[1	-1	1	-1	1	-1] Condorcet Term
[1	1	1	1	1	1]] Kernal Term

The columns represent the voter types A>B>C, A>C>B, C>A>B, C>B>A, B>C>A, 
B>A>C, respectively, and the rows define the differentials into which the 
profile is decomposed. The Basic-A term, for example, is the differential 
where one C>A>B, 2 C>B>C and one B>C>A voters become A>B>C, A>C>B and B>A>C 
voters (in any order) - notice these numbers add up to zero. (Strictly 
speaking, the Kernal term is not a profile "differential," since it does not 
add up to zero.)


The decomposition profile, T(p), is the product of matrixes "T" and "p" (read 
"T compose p," or "T of p"):

T(p)=(1/6)[[6],[3],[0],[-3],[-12],[18]] (Again, this is supposed to be put in 
the form of a vertical column matrix.)


Before I try to explain the decomposition profile T(p), above, let's look at 
the unanimity A>B>C profile, p', and it's T(p'):

1:A>B>C
0:A>C>B
0:C>A>B
0:C>B>A
0:B>C>A
0:B>A>C

or

p'=[[1],[0],[0],[0],[0],[0]]

which decomposes into:

T(p')=[[2],[1],[0],[-1],[1],[1]].


Now, again, here's the decomposition of my first profile, p:

p=[[3],[5],[0],[5],[0],[5]]
T(p)=[[6],[3],[0],[-3],[-12],[18]]


Ok, now let me try to explain. This decomp, T(p), may make more sense if you 
think of it as multiplying the unanimity decomp, T(p'), by 3, and then 
subtracting 15 cyclic Condorcet terms. You subtract 15 from 3 to get -12, 
because the 5:A>C>B, 5:C>B>A, 5:B>A>C voters make up a cycle in the negative 
direction. IF YOU DO THE DECOMPOSITION WITH THESE MATRIXES, IT DOES NOT MATTER 
IF YOU FIND THE REVERSAL TERMS OR THE CONDORCET (CYCLIC) TERMS FIRST.

On the other hand, if you decompose p in the way Forest and Alex did earlier 
in this thread - see:

http://groups.yahoo.com/group/election-methods-list/message/10747

and my reply:

http://groups.yahoo.com/group/election-methods-list/message/10757


the order does make a difference, but that is NOT THE CORRECT WAY to decompose 
a profile, according to Saari's mathematics - again, some of his expositions 
are another story, which merely tries to roughly describe the general 
concepts.

Anyway, if you use the method of decomposition used earlier by Alex Small and 
Forest Simmons, it would give different answers, depending on the order of 
cancellations. First let try removing Condorcet triplets first:

p=[[3],[5],[0],[5],[0],[5]]

A>B>C: 3
A>C>B: 5-5=0
C>A>B: 0
C>B>A: 5-5=0
B>C>A: 0
B>A>C: 5-5=0

This leaves us with the unanimity profile with 3 voters A>B>C.


If you remove the reversals first, and then the Condorcet triplets, you get 
the following:

A>B>C: 3-3=0
A>C>B: 5-2=3
C>A>B: 0
C>B>A: 5-3=2,2-2=0
B>C>A: 0
B>A>C: 5-2=3


Obviously, these are not the same decompositions. That is granted, but that is 
not the mathematically correct decomposition method, according to Saari, in 
spite of the fact that he does use something like that in some of his more 
popular expositions. For example, compare his decomposition of Condorcet's 
example, p=(30,1,10,1,10,29), in his following publications:

"EXPLAINING ALL THREE-ALTERNATIVE VOTING OUTCOMES"
DONALD G. SAARI
see "8.1. Condorcet's example," electronic page 25
<http://citeseer.nj.nec.com/cache/papers/cs/11095/http:zSzzSzwww.math.nwu.eduz
Sz~d_saarizSzvotezSztriple.pdf/saari99explaining.pdf>

_Chaotic Elections_ (book)
Donald G. Saari
page 117
http://www.amazon.com/exec/obidos/tg/detail/-/0821828479



SB

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