# [EM] Saari's Basic Argument

Steve Barney barnes99 at vaxa.cis.uwosh.edu
Tue Feb 25 12:13:39 PST 2003

```[I am resending this, because nobody replied yet.]

Forest:

How do you decompose my example (from my last email, #10873), and what do you
get?:

3:A>B>C
5:A>C>B
0:C>A>B
5:C>B>A
0:B>C>A
5:B>A>C

SB

>From: Forest Simmons <fsimmons at pcc.edu>
>To: EM-list <election-methods-list at eskimo.com>
>Subject: RE: [EM] Saari's Basic Argument
>
>On Tue, 18 Feb 2003, Steve Barney wrote:
>
>> Here is a simpler example to illustrate the difference that the order in
which
>> cyclic and reversal terms are canceled does not matter when using the
strictly
>> correct method - as opposed to the method used by Forest Simmons and Alex
>> Small, and in some of Saari's popular expositions where he is merely trying
to
>> illustrate basic concepts to a more general audience.
>
>It's true that in my earlier examples I didn't take the decomposition all
>of the way to its logical conclusion.  But if you look at my last few
>examples, you'll see that I finally hit on a simple method of adding and
>subtracting the two kinds of symmetries that reduce each ballot set to a
>canonical ballot set consisting of either one faction or two adjacent
>factions with positive multiplicity, and all other factions zero.
>
>[Two factions are adjacent if one of them follows the other in the cyclic
>order ABC->ACB->CAB->CBA->BCA->BAC->ABC.]
>
>It is easy to prove that this decomposition is unique, since it preserves
>the Borda count of each candidate, and the Borda counts determine the
>number in each faction when there are two or fewer adjacent factions.  So
>the order of application of the symmetries doesn't matter.
>
>As my examples show, this can be done very simply without matrices.
>
>Presumably Saari uses matrices because he wants to develop tools that will
>generalize to more than three candidates.
>
>But worrying about the details of symmetry cancellations is to bark up the
>wrong tree.
>
>The fact is that in the ballot set 65*ABC+35*BCA the candidates A, B, and
>C have the same respective average ranks as they do in the simpler ballot
>set 30*ABC+35*BAC , so according to Borda they are equivalent ballot sets,
>and B should be the winner of the first election as decisively as in the
>second (according to Borda and Saari).
>
>This result may make sense in the context of dispassionate decision making
>such as in robotics when a robot is trying to decide what movement to make
>or whether a visual image represents the letter U or V.
>
>But in the context of public elections, this supposed equivalence is
>almost ludicrous.
>
>So the question is not, "Why is Borda such a great method for public
>elections?"
>
>The question is, "Why does the symmetry argument lead us down the wrong
>path?"
>
>At least that is the question I was trying to answer (and did answer to
>my own satisfaction).
>
>In a nutshell the answer to this question is that the symmetries in the
>distribution of ballots are at odds (more often than not) with Saari's
>symmetries.
>
>In other words, Saari's transformations do not preserve the natural axes
>of symmetry that may (and do) exist in the ballot distributions.
>
>In the above example, the first ballot set 65*ABC+35*BCA consists of two
>factions that determine an axis of symmetry.  The second ballot set
>35*BAC+30*ABC also consists of two factions along an axis of symmetry, but
>this axis is rotated thirty degrees relative to the original axis.
>
>So there is an essential change in the symmetry of the distribution that
>the Borda count doesn't detect, and Saari's symmetry transformations
>cannot preserve.
>
>[The center of gravity of the distribution is preserved, but the principal
>axes of rotation and the radii of gyration are changed.]
>
>For me this insight is sufficient to explain the fallacy of the symmetry
>arguments.
>
>Most non-mathematicians don't care one whit about what went wrong with the
>symmetry arguments; rather than watch the gory details of an autopsy, they
>prefer to move onward and upward.
>
>Forest

----