[EM] math 103 website - Arrow & Saari

[Steve Barney]

Bart:

In this instance, I was referring to Saari's discussion (in his book,
_Decisions and Elections_) of the intensity of voter preferences strictly as it
relates to Arrow's Impossibility Theorem. Saari shows that Arrow's Theorem can
be resolved by slightly modifying the Independence from Irrelevant Alternatives
criterion, so as to require a procedure to take account of the intensity of a
voter's preferences, as can be determined by the Borda Count be noting the
number of candidates who are ranked in between each pair. Perhaps Arrow
discounts the subject of intensities, as you say, because his IIA criterion
requires a procedure to do so. This criterion eliminates all but pairwise
voting, which ignores intensity. This criterion, coupled with the condition
that cyclic outcomes are not allowable, is the source of the impossibility in
this theorem. Those two conditions, alone, are incompatible with each other,
and thus eliminate all chances of finding a voting method which will satisfy
all of his fair criteria and conditions.

There is much, much more to Saari's book, of course. I am only touching on it
here.

Steve Barney

--- In election-methods-list at y..., Bart Ingles <bartman at n...> wrote:
> 
> 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 a...> 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,
> 
> votes     candidate(intensity)
> -------------------------------
> 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

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