[EM] Saari's Basic Argument

Forest Simmons fsimmons at pcc.edu
Fri Mar 15 08:59:12 PST 2002

At best Saari proves that Borda is the best choice method based on
rankings in situations where there can be no stacking of the deck (clones)
or insincere rankings.

Unfortunately these two conditions are usually important considerations in
elections. So Borda's method should be restricted to applications in
sports, robotics, etc. where other kinds of choices are made on the basis
of rankings.

In my opinion Borda should not be thought of as a serious election method
but rather as a decision process for other kinds of choices.

In the context of elections it can serve as a benchmark of limitations in
expected utility for methods based on ranked ballots, like the idealized
Carnot engine operating in an absolute zero temperature environment as an
unattainable benchmark in the context of industrial engine design.

While many applications of Borda to sports are known, here is an
interesting sports one that may have been overlooked:

In a round robin tournament (where every team plays every other team) give
the championship to the team with the greatest difference in its sum of
victory margins and sum of defeat margins, i.e. the team with the greatest
row sum in the margins matrix, i.e. the team with the greatest mean
margin.  [This can be justified by considering this sample mean as an
estimate of the expected margin in a random match.]

How is this Borda?  Where are the rankings?

Well, if anybody really cares, you can work backwards from the margins
matrix (as Blake explained a few months ago) to get rankings that produce
the margins matrix.  There are many ways to do this, but no matter how you
do it, the Borda winner according to the rankings will be the same as the
tournament champion picked from the margins matrix by the above rule.


On Sat, 9 Mar 2002, [iso-8859-1] Alex Small wrote:

> 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.


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