Request for Forest Simmons on 1+ candidate geometry (was Re: [EM] Correction. Big CS fault?

[Craig Carey]

At 02\12\27 13:42 -0800 Friday, Forest Simmons wrote:
...
 >Put candidates A,B,C, and D at the vertices of a tetrahedron
 >whose respective edges AB, AC, AD, BC, BD, and DC have lengths of
 >5,7,8,9,4, and 6, respectively.
 >
 >If A, B, C, and D have 5, 4, 3, and 1 avid supporters, respectively,
 >located at the exact same positions as their favorite candidates, then
 >there are four factions as follows:
 >
 >5 ABCD
 >4 BDAC
 >3 CDAB
 >1 DBCA
 >

What suppose that there are 5 paper and 6 candidates.

The words "avid supporters" matches up with the mathematical entity
called "ballot papers". That is something owned by "A, B, C, and D".
The latter are 4 candidates and there are 4 vertices.

The word "position" is used in a way that has it say that "ballot papers"
are at "the exact same positions". It does not seem that the word
"positions" is referring to preferences in lists of preferences.

Again, there is the seemingly difficult to do, confusion between what
a ballot paper and what a candidate is. This is unlike Ossipoff's
idea of calling a ballot paper, a voter.

Once Forest Simmons has done that then he has a geometric theory that
is unable to plot a 2 candidate election (so the number of papers is
4 or less and the number is 0, or 1, or 2).

----

Suppose that there are 6 ballot papers and 4 candidates?.

Obviously to get that down to the 4 vertices, and we know that there
are 4 since Forest Simmons has called them candidates, there has to be
some way of reducing the 6 numbers down to 4.

The new view won't be important whilst it takes the FPTP view of
ignoring the 2nd and all subsequent preferences.

It seems to me that Forest is stuck behind a 1 candidate barrier.

As I noted at the STV list, the two winner problem was solved in
about 1952. If the problem can be solved without geometry then
geometry is optional.

----

 >Then A beats B beats C beats D beats A.
 >
 >This shows that "beat cycles" cannot be eliminated by assigning
 >preferences to voters based on the location of their favorites in
 >candidate space.
 >

There is no problem with cycles. A beat cycle is where the arrows
can't all point in the right direction.

When there are 4 winner and 4 candidates, then every candidate will
be tied for a win.

The garbage theories on Condorcet cycles get that case wrong entirely
because some of the arrows actually show a direction. That is so
dumb that a theorist would have to check twice to see if someone was
actually advancing an idea as dumb as pairwise comparing.

A quick check of the text above shows that Forest Simmons failed to
constrain the number of winners to be 1.

It is not the case that I have a minor argument.

It was Forest Simmons who wanted to get the dimensions wrong so that
the plotting of every imperfect method can't be done until there is
an arbitrary of some given method that can reduce the dimensionality.

FPTP was actually used which would be controversial for sure.

-------------------------------------------------------------------------
REQUESTS to MR Forest Simmons (not a member of one of my mailing lists):

[1]
    From where were readers to pull the restriction requiring that the
    number of winners not exceed 1 ?.

Also while answer that, tell us this:
[2]
    Was there a restriction saying that number of kinds of ballot papers
    had to be not more than the number of candidates ?.

[3]
    Since the outcome of an election can be affected by the addition of
    candidates that not a single preference names, please explain any
    otherwise-missing restrictions about negative numbers.

[4] I note that you used the word "supporters". Say if the readers
    were supposed to regard the counts as being integers. (Though only
    a given example appeared, the topic of the geometry is still around).

Also, if you have reasons for not responding to these requests then
that information is requested too (i.e. your reasoning for withholding
the information requested). If you have axioms on the withholding of
the reasoning for withholding the reasoning for withholding, then
please post that into the Election Methods List too.


 >So I withdraw my suggestion of using candidate space in that particular
 >manner.
 >

I see that the word "So" is wrong since you found a problem with some
principle that is wrong anyway. Maybe you can post in another message
that gives a correct way of finding your thinking to be wrong. Or
to post up a correction that has the word "So" be deleted may get the
deletion of the other ideas, running better.


 >My new suggestion based on moving candidate vectors into voter space via
 >the "most natural" isomorphism, looks much more promising.
 >

I object to the term "voter space". It suggests a primary importance
being attached to apparently completely irrelevant information.

We can obviously interpret voters to be a vector to be a real values.
I am not sure if Forest Simmons would want dispute the lenght of the
vector or not. Such real valued vectors would still be variables that
the solution is insensitive to (such voters are independent variables).

There could be a method that adds 10% to a candidate's score if the
candidate is a woman. However I presume it best to let all thinking of
that be delayed until the maths can be solved. Also it not exact and
naturally outside of the topic.


 >See my posting under the title "Candidate-Space Method" dated December 26,
 >2002.
 >

Can Forest Simmons post up a comment saying whether he is or is not
unable to provide a not-arbitrary (i.e. not presuming FPTP) interpretation
of a geometric nature, to 4 paper 2 candidate 1 winner elections ?.

Mr May solved the 2 candidate 1 winner election problem in 1952.
Solving the problem using axioms to create a method is lot harder than
devising a geometry into which methods can be plotted. Yet Forest Simmons
was not making progress at extending the framework of ideas out to a point
where the mere geometric interpretation could cope with 2 candidate methods.
However geometry itself could be irrelevant if replaced with algebra of
logic.

At 2002\12\25 11:35 +1300 Wednesday, Craig Carey wrote to Single-Transferable-Vote:
...
 >| [...] Kenneth May, "A Set of Independent Necessary and
 >| Sufficient Conditions for Simple Majority Decision," Econometrica
 >| 10 (1952): 680-84


I am a little confused on how Mr Simmons reasoned and I asked questions that
I have an interest in.


Craig Carey







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