# [EM] Saari's Basic Argument

Steve Barney barnes99 at vaxa.cis.uwosh.edu
Sun Jan 19 18:16:51 PST 2003

```Alex:

SB

--- In election-methods-list at yahoogroups.com, "Alex Small" <asmall at p...>
wrote:
> Steve Barney said:
[...]
> > looked  at before:
> >
> > 5 ABC
> > 3 BCA
> >
> >
> > Can you give me the decomposition profile, T(p), for this example?
>
> We can't decompose it any futher.  The only two voter types remaining are
> not reversals of one another, and with only two voter types you can't have
> a Condorcet cycle.
[...]

Well, in fact, according to Saari's decomposition matrix, the answer goes like
this:

T=Decomposition matrix=
(1/6)*
[[2,1,-1,-2,-1,1]
[1,-1,-2,-1,1,2]
[0,1,-1,0,1,-1]
[-1,1,0,-1,1,0]
[1,-1,1,-1,1,-1]
[1,1,1,1,1,1]]

p=profile=
[[5]
[0]
[0]
[0]
[3]
[0]]

T(p)=(1/6)(7,8,3,-2,8,8)

Now that I have my worksheets along, going back to the example you were using
earlier:

66 A>B>C
34 B>C>A,

we get:

p=(66,0,0,0,34,0)
T(p)=(1/6)(98,100,34,-32,100,100).

> In fact, with only two voter types any reasonable voting method will give
> the same answer.  Borda is NOT a reasonable voting method.

Really? 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 article I referenced in my
previous post (Message # 10757).

> I think Saari's work can best be assessed this way:
>
> He has a lot of interesting insights.  Everything that he asserts as a
> mathematical proposition is rigorously handled and proven.  He's a very
> smart guy, and there's a lot to learn from him.
>
> HOWEVER, when he speaks normatively on questions like "What is the best
> method for society to use in elections?" he's just another John Q. Citizen
> speaking his mind.  That's what we all are.  We can point out technical
> properties of election methods, we can prove theorems, and we can crunch
> numbers, and all of these things can be handled in an objective manner
> (although we often quibble over definitions when there is not yet a widely
> recognized convention).  But, when somebody says "Method X is the best" or
> "Property Y is essential" then we're all just John Q. Citizens speaking.
>
> Earlier, I said that Borda is NOT a reasonable election method.  That
> subjective assertion is grounded in the assumption that when there are
> only two candidates with first-place support the majority candidate should
> win.  I think most people agree with me on this, but there are exceptions.
>  Some people have concluded (for reasons beyond my fathoming) that Borda
> provides a good alternative.
[...]

Well, I have certainly given you reasons for preferring the BC's B>A>C outcome

66 A>B>C
34 B>C>A,

and Saari has certainly given reasons in similar cases. It seems to me that
Saari is basically arguing that the problems which arise with the BC are those
caused by adding or dropping candidates. We all know that provides a means of
manipulating a BC outcome. However, let me now remind you of my original
message under this subject header - keep hitting "Up Thread", above or below,
and you will get back to message # 9198.

> Anyway, I think I've said just about all I can think of on Saari and the
> Borda Count.

Given what I and Forest Simmons said in message # 9198, perhaps it would be a
good idea to find which less manipulable methods have that greatest degree of
agreement with the BC outcome. For example, what is the likelihood that the
Approval Vote will elect the BC-winner, and how does that compare to the
Plurality Vote or the various Condorcet methods? Furthermore, are there other
ways reducing this means manipulation. Is there a reasonable way of limiting
the number of candidates in a contest, or on a ballot? How about that for new
research questions?

Steve Barney

Richard M. Hare, 1919 - 2002, In Memoriam: <http://www.petersingerlinks.com/hare.htm>.

Did you know there is an web site where, if you click on a button, the advertisers there will donate 2 1/2 cups of food to feed hungry people in places where there is a lot of starvation? See:
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