[EM] Saari's Basic Argument

[Alex Small]

I've seen a lot of criticism of Saari on this list.  Last night I read some
of his latest "popular" book (if books with so much math notation can be
called popular) _Chaotic Elections_.  He hasn't converted me to the Borda
Count, but I think I finally understand the fundamental reason for why he
endorses Borda and rejects Condorcet (I haven't read his critique of
Approval yet, so I'll leave that aside).  I deeply respect his mathematics,
even though I disagree on policy.

It all comes down to whether you have the mind of a statistician or a game
theorist, and whether you think of voting as a decision made by voters or
an attempt to collect data on voters.  On this list we discuss questions
like "Will this method give me an incentive to truncate my ballot?" or "If
I switch my support to somebody else will my favorite do better? [IRV]"
or "What's my best strategy?"  Seen from that light, Condorcet does quite
well (leave aside for now arguments over resolving cyclic ambiguities).

Saari, however, makes a beautiful mathematical observation:

Suppose that we have the following profile:

33 A>B>C
33 B>C>A
33 C>A>B

This is perfectly symmetric, and a cyclic ambiguity exists.  Now perturb it
to:

34 A>B>C
33 B>C>A
32 C>A>B

The symmetry has been broken in some sense, but the basic cyclic structure
remains despite the perturbation.  Saari used to work in dynamical
systems.  There's a theorem that (crudely stated) says that for dynamical
systems the basic structure of the system in phase space can remain
unchanged despite small perturbations that break symmetry.  I'm 100% sure
that he knows of this theorem, and while I was reading his book last night
this theorem immediately jumped into my mind.  Not that this justifies
Borda as a policy, but I see where he got his inspiration.  Anyway, I
digress.

We could say that the 32 people with C>A>B, combined with 32 of those in
each of the other categories, cancel out.  It would be like saying in a 2-
way race that 40 ballots cast for A cancel out 40 of the ballots cast for
B, leaving an excess of 20 ballots for B.

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 will be
two sets of ballots to cancel out:  A>B>C and its cyclic permutations, and
C>B>A and all associated cyclic permutations.  To be concrete:

34 A>B>C
33 B>C>A
32 C>A>B

becomes

32 A>B>C          +          2 A>B>C
32 B>C>A          +          1 B>C>A
32 C>A>B          +          0 C>A>B

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 reason why
Arrow's criteria are incompatible is that most profiles of the electorate
will include those symmetric parts.

You could justify looking only at the perturbation by saying that some of
the ballots form a cycle and cancel out, so you should only look at the
excess, as in my example of a 2-way race.  Of course, the problem is that
the people with C>A>B now have NO say, whereas with Condorcet the existence
of a large C>A>B faction indicates that we have to use some other method to
resolve the ambiguity (and endless time has been spent on this list
debating the appropriate resolution).

Interestingly, Mike has pointed out on electionmethods.org that with a
slight modification of IIA, to say that candidates in the Smith Set are
not "irrelevant alternatives" the conditions of Arrow's Theorem would now
be compatible.  The basic mathematical soul of Arrow's theorem is that the
existence of cyclic ambiguities makes his conditions incompatible, and both
Mike and Saari have found ways to identify that fact.

If Borda didn't pose so many perverse problems of strategy, clone-
sensitivity, etc., then I might support Borda.  But, as Borda himself
said "My method is only for honest men!" (paraphrase).  However, Saari's
observation does suggest an interesting way to resolve cyclic ambiguities.

I confess that I don't follow (or understand) every thread, so I'm not sure
what the "Borda-seeded bubble sort" is (I hope I got the name right).  Does
it use Saari's observation to resolve cycles?

Anyway, I'd been bothered for some time that a guy as smart as Saari would
support a method that poses so many problems of strategy and practice.  Now
I think I understand, since he makes a very interesting observation about
the source of Arrow's Theorem.  Also, as a physicist I think I understand
why he would find the idea of symmetry breaking so compelling and worth
pursuing.

Alex Small