# [EM] Saari's Basic Argument

Alex Small asmall at physics.ucsb.edu
Sun Jan 19 22:53:42 PST 2003

```First, I tried to get to the paper that you referenced but the link was
bad.  Rather than the full link, maybe it's best to send me instructions
on how to search for the paper.

Steve Barney said:
> p=profile=
> [[5]
>  [0]
>  [0]
>  [0]
>  [3]
>  [0]]
>
> T(p)=(1/6)(7,8,3,-2,8,8)

I'll have to look again at Saari's books (I have two of them here) before
commenting on that.  I need to recall what each coordinate represents.

> Try decomposing a UNANIMITY profile, such as 1 A>B>C, into it's
> reversals and cycles and basic and kernal terms. Here's what you get:
>
> p=(1,0,0,0,0,0)
> T(p)=(1/6)(2,1,0,-1,1,1).
>
>
> And you must notice that there IS disagreement over the correct outcome
> for  this profile. The Borda Count and Condocet-based outcomes for this
> unanimity  profile are A>B>C, of course, but the plurality outcome is
> A>B~C! For Saari's  comments on this, see section 8.3 in the online

Who cares?  It's unanimous:  A wins!  (Unless, of course, this is the
Electoral College, in which case we have to figure out who loses so we can
elect him ;)  Don't get me wrong, Saari provides useful insights to keep
in mind when evaluating election methods, but the more I study the BC the
more I'm convinced that it's a bad election method.  So, he does a great
job until right before the finish line.

> Well, I have certainly given you reasons for preferring the BC's B>A>C
> outcome  in your example,
>
> 66 A>B>C
> 34 B>C>A,
>
>
> and Saari has certainly given reasons in similar cases.

Saari has explained why the Borda count _does_ give such results, but I'm
not sure how well he's explained why we shouldn't care that the BC gives
such results.

On page 162 of my copy of _Basic Geometry of Voting_ Saari tries to
justify his reasons for disregarding the first choice of the majority
(when such a candidate exists).  At the risk of sounding like a whiner,
the explanation is hard to follow and makes heavy reference to technical
points made earlier in the chapter.  As somebody who has TAed for
undergrads I'm normally unimpressed by such complaints.

However, he's ultimately making a normative judgement rather than a
technical judgement.  The short and very technical discussion fails to
address the basic social and political reasons for favoring majority rule.
So, my own lack of effort is canceled out by his lack of any effort to
address the heavy social and political arguments against his assertion.

In all fairness, I will at some point try to study his argument and make
sense of it.  However, when a person's published explanations of
controversial political claims require serious study for even an
afficionado to understand them, one can't really say that he's adequately
addressed the matter.

In the end, if nearly 2/3 of the voters say "This is my favorite
candidate", well, I'm sold.  Unless Saari can prove to me that most of the
electorate would actually prefer the second choice, I'm sticking with A in
the example

66 ABC
34 BCA

> hitting "Up Thread", above or below,  and you will get back to message #
> 9198.

In message 9198, you said:

> This seems to beg a question of balance. At what point is the BC's
> accuracy at determining the "correct" winner overcome by it's
> manipulability? Your qualification about sincere votes seems to imply
> that there must be some point at which another method is more likely
> then the BC to determine the winner(s) which would have been elected by
> the BC under sincere voting conditions. It seems to me that this may be '
> a quantifiable problem.

Your concern seems to be that since Borda encourages insincerity, one
should try to find a voting scheme that will usually pick the _sincere_
Borda winner, even if people try to vote strategically.

I keep going back to 66 ABC, 34 BCA.  With sufficient information the ABC
faction will divide into two groups, one voting ACB and the other voting
ABC, so that their support for B is cut in half but their support for C is
still kept reasonably low (otherwise the BCA faction could insincerely
list CBA).  (ABC might also truncate, if the Borda implementation doesn't
declare truncated ballots to be spoiled).

Now, suppose we could design a method that elected B anyway, despite the
ABC faction's strategic efforts.  Why would we?  Nearly 2/3 of the
electorate is trying hard to keep B out of office, why are we trying to
stop them?

OK, maybe it would be hard to design a method that picks B in that case.
But one might envision a method that's less prone to strategy and picks B
in the case 51 ABC, 49 BCA.  Once again, why?  Besides, we already have a
perfectly horrid method for thwarting the will of the voters:  The
Electoral College.  It was designed around the time of Borda, and with the
best of intentions, and it's turned out to be almost as flawed.

Anyway, that's all I have to say about that.

Alex

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