[EM] math 103 website - Arrow & Saari

[Forest Simmons]

On Mon, 26 Nov 2001, Steve Barney wrote:

> 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. In other words, Arrow proved that his
> criteria are inconsistent with one another. We must remember that "fairness" is
> not a strictly objective thing. It necessarily involves an evaluative judgment,
> and is based on questionable intuitions. Fairness goes outside of the realm of
> factual truth and falsity, and into the realm of the good and bad. The realm of
> the good and bad is not a matter of mere mechanics. Fairness cannot be
> mathematically proven one way or the other. Arrow's Theorem may be interpreted
> as providing a good reason to subject his fairness criteria to further
> scrutiny, to try to understand why his particular criteria are inconsistent
> with each other, and to come up with more satisfactory results with other
> elementary fairness criteria or axioms.

Bien dicho (well said)!

Just because we have a second law of thermodynamics (ruling out perpetual
motion machines) it doesn't necessarily follow that it cannot be
determined whether a BMW performs better than a Citroen. 

Some methods are better than others, and some criteria are more important
than others in various applications.

Still, there are many pitfalls in judging the relative importance of
various criteria in various applications. 

Take Saari, for example.  He doesn't seem to worry about the Favorite
Betrayal Criterion or the Independence From Clones Criterion, two very
important criteria for public elections.

The first of these is important because voters strongly object to voting a
ballot that puts their second choice higher than their first. This
violence to the voter's conscience is required more often in Borda Choice
than it is in IRV, and it is one of the main reasons that IRVies consider
Borda an unacceptable method for public elections.

The second of these is important in order to keep the interest groups that
can run the most candidates from gaining advantage by running clone
candidates, especially when primaries are eliminated.  Again, Borda is
inferior to IRV.

The fact that Borda violates the Majority Rule Criterion is not as
important in itself as these other two violations, but it is related to
both of them:

Clones can convert a majority candidate from win to lose status, and when
a majority candidate is not able to win by sincere votes, a majority is
tempted to carry out order reversals to salvage the win (except zero
information cases). 

When there is neither potential for clone problems nor enough information
or disinformation to encourage favorite betrayal (or other order
reversals), then Borda is uniformly better than IRV.  But these conditions
are not often met in public elections. 

I'm not advocating IRV.  I'm comparing Borda to IRV to show how bad Borda
is as a public proposal. 

Borda's response to these difficulties was that his method was only
intended for honest voters, those who would not stoop to strategic order
reversals.

In my opinion, no defense better than this inadequate one has been put
forth to date, despite Saari's great mathematical work showing the value
of Borda Count in situations where strategic incentives for order reversal
are not a problem (as in mechanical applications, sports rankings, etc.)

Forest

> 	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. 
> 	I am no professional voting theorist, but I have studied this subject and his
> work in some depth, and I think this is a very important book. I wouldn't be
> surprised if Saari is rewarded with a Noble Prize in Economics for his work -
> at least two other ground breaking voting theorists, Sen and Arrow, have
> received them.
> 
> Steve Barney, student
> University of Wisconsin Oshkosh
>