[EM] Saari's Basic Argument

[Alex Small]

Steve Barney said:
> 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.

Decomposing a vector into its projections onto different directions
normally doesn't depend on the order in which you extract each component. 
However, when decomposing electorates we can't use negative coefficients,
we can't have a leftober profile like

-2 A>B>C
 3 B>C>A

So then the order of operations matters (not that any of Saaris examples
give us such a ridiculous profile, just that preventing such a profile is
why the order of operations matters).

> If you don't like Condorcet's example, how about this one, which I have
> looked  at before:
>
> 5 ABC
> 3 BCA
>
>
> Can you give me the decomposition profile, T(p), for this example?

We can't decompose it any futher.  The only two voter types remaining are
not reversals of one another, and with only two voter types you can't have
a Condorcet cycle.

In fact, with only two voter types any reasonable voting method will give
the same answer.  Borda is NOT a reasonable voting method.

I think Saari's work can best be assessed this way:

He has a lot of interesting insights.  Everything that he asserts as a
mathematical proposition is rigorously handled and proven.  He's a very
smart guy, and there's a lot to learn from him.

HOWEVER, when he speaks normatively on questions like "What is the best
method for society to use in elections?" he's just another John Q. Citizen
speaking his mind.  That's what we all are.  We can point out technical
properties of election methods, we can prove theorems, and we can crunch
numbers, and all of these things can be handled in an objective manner
(although we often quibble over definitions when there is not yet a widely
recognized convention).  But, when somebody says "Method X is the best" or
"Property Y is essential" then we're all just John Q. Citizens speaking.

Earlier, I said that Borda is NOT a reasonable election method.  That
subjective assertion is grounded in the assumption that when there are
only two candidates with first-place support the majority candidate should
win.  I think most people agree with me on this, but there are exceptions.
 Some people have concluded (for reasons beyond my fathoming) that Borda
provides a good alternative.  Others have concluded that a person in
Wyoming or Vermont is three times as important as a person from California
or Texas, so there's no need for the President to win the popular vote. 
Who's right?  Well, that's what the political process is there to sort
out.  The genius of democracy is that the political process can even be
used to pick a political process.

Anyway, I think I've said just about all I can think of on Saari and the
Borda Count.



Alex


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