# [EM] Saari's Basic Argument

Steve Barney barnes992001 at yahoo.com
Sat Mar 9 16:59:35 PST 2002

Alex:

--- In election-methods-list at y..., "Alex Small" <asmall at p...> wrote:
[...]
> So, if you "subtract out" the ballots that form a perfectly symmetric
> cycle, Saari has proven that applying the Borda count to what remains will
> satisfy the conditions of Arrow's Theorem.  In a 3-way race there [...]
I must say that your statement of Saari's observation seems a bit misleading,
if not inaccurate. Saari shows that the BC is the only positional (or pairwise)
voting method which (AUTOMATICALLY) cancels out those symmetrical ballots
(reversals and cyclic triplets, in the case of 3 candidates). It is not as if
the BC only works if you first cancel out the symmetrical ballots, and then
apply it to only the remaining ballots. To the contrary, he shows that all
positional and pairwise methods reach the same conclusion when applied only to
the ballots which remain (the "perturbation") after that cancellation. That is
a very important point, in his view.

> The voter profile is in some sense the sum of a symmetric part plus a
> perturbation, and the Borda Count satisfies the conditions Arrow's Theorem
> when applied to the perturbation.  He has also proven that the Borda Count
> is the only positional method to satisfy the conditions of Arrow's theorem
> when you consider the perturbation.  In other words, the only
No, not quite. Again, as all methods reach the same outcome when applied to the
perturbation, they all would satisfy Arrow's Theorem when applied only to the
perturbation (or what Saari calls the "Basic" profile). The important thing is
that the BC AUTOMATICALLY cancels out the symmetrical ballots and bases it
conclusion on ONLY the remainder.

[...]

BTW, did you see my review of this book on Amazon.com?

Chaotic Elections! : A Mathematician Looks at Voting
by Donald G. Saari ; Mass Market Paperback
http://www.amazon.com/exec/obidos/ASIN/0821828479/

I hope you will write and publish your own review when you finish it.

SB

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