# [EM] math 103 website - Arrow & Saari

Bart Ingles bartman at netgate.net
Sat Jan 5 02:30:52 PST 2002

```I had been meaning to reply to this posting, but never quite got around
to it.

Steve Barney wrote on 11/26/01:
>
> Election Methods list:
>
> Many introductory math textbooks, and the webpage <DEMOREP1 at aol.com> referred
> us to in a recent message, draw too strong a conclusion from Arrow's Theorem.
> The assertion that:
>
> "Mathematical economist Kenneth Arrow proved (in 1952) that there is NO
> consistent method of making a fair choice among three or more candidates. This
> remarkable result assures us that there is no single election procedure that
> can always fairly decide the outcome of an election that involves more than two
> candidates or alternatives"
> --http://www.ctl.ua.edu/math103/Voting/overvw1.htm#Introduction
>
> is not quite true. His theorem only proves that there is no method which can
> satisfy all of his fairness criteria.

I agree.  This quote sounds like it came from an IRV apologist.  The
general argument seems to be that "since no method can satisfy everyone,
why bother with objective criteria?"

> [deleted]
>
>         I recommend reading Donald Saari's new book,
>
>                 _Decisions and Elections_
>                 Cambridge University Press
>                 October 2001
>                 http://www.cup.org/
>
> in which he interprets and scrutinizes Arrow's Theorem in exactly this way, and
> comes up with more satisfying results. Among other things, he finds that, if
> Arrow's "binary independence" condition is slightly modified so as to require a
> procedure to pay attention to the strength of a voters preferences (he calls
> his version the "intensity of binary independence" condition), then the Borda
> Count procedure solves the problem and satisfies the theorem.

Arrow does address the subject of intensities, or "utilities", but
discounts them for various reasons.  I am curious as to how Saari uses
intensities, since Borda doesn't always do well in this regard.  For
example,

-------------------------------
49        A(100),  B(1),  C(0)
2        B(100),  A(1),  C(0)
49        C(100),  B(1),  A(0)

The Borda winner, B, is a rather poor choice when considering strength
of voter preferences.  98% of the voters appear to despise B almost as
much as their last choice.

But never fear.  Since the A and C voters apparently don't consider B to
be a worthy compromise, and since the race between A and C is a toss-up,
these voters have incentive to strategize in order to prevent B from
winning, taking their chances on the A/C lottery.

If the A and C voters swap just under half of their 2nd and 3rd choice
preferences, the final Borda scores might be something like:

A: 98 + 2 + 24 = 124
B: 25 + 4 + 25 = 53
C: 24 + 0 + 98 = 122

Of course, how many voters would be willing to strategize in such a way
depends on the sophisication of the voters, and on the actual
intensities involved, so the actual outcome would be difficult to
predict.  So much for Borda being a deterministic voting system (one of
Saari's justifications for preferring Borda over approval voting).

In this case, behavior under approval voting should be much more
predictable, since most or all of the voters could be expected to bullet
vote.

Bart

```