[EM] Catchy goes Taxi Driver (watch my slow descent into frustration and brutality, prompted by the madness of another!)

[David Catchpole]

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"Being in politics is like being a football coach. You have to be smart
enough to understand the game, and dumb enough to think it's important"
							-Eugene McCarthy

---------- Forwarded message ----------
Date: Tue, 5 Sep 2000 08:54:53 +1000 (GMT+1000)
From: David Catchpole <s349436 at student.uq.edu.au>
To: Craig Carey <research at ijs.co.nz>
Subject: Re: Sets of vertices leads nowhere; Mike Ossipoff

On Tue, 5 Sep 2000, Craig Carey wrote:

> At 14:36 00.09.04 +1000 Monday, David Catchpole wrote:
>  >On Mon, 4 Sep 2000, Craig Carey wrote:
>  >
>  >> At 02:30 00.09.04 +0000 Monday, s349436 at student.uq.edu.au wrote:
>  >>  >
>  >>  >I now have a nifty little DOS program that translates the edge-points
>  >>  >of an n-l votes surface to the end-points of an n-m-l sub-set votes
>  >>  >surface (Yeah I know, makes no sense). Anyway, Craig, are you
>  >>  >interested in having a look at it?
>  >>  >
>  >> I guess not.
>  >> That "translate" sentence maybe does not make sense. Do you mean
>  >> 'tilt so that the lower dimensional flats lie in it?.
>  >
>  >"Transform" is probably the more mathematically orthodox term I'm looking
>  >for. It's the transform from, for instance, the voting plane as defined by
>  >x[A>B>C]+x[A>C>B]+x[B>A>C]+x[B>C>A]+x[C>A>B]+x[C>B>A]=1 into the space
>  >(y[A>B],y[B>A],y[A>C],y[C>A],y[B>C],y[C>B]). In that case, it's the "Saari
>  >Cube." Please forgive me- I was in a rush, and "translate" makes
>  >algorithmic sense even if it doesn't mean what I meant it to mean in a
>  >context of geometry.
>  >
> 
> SO you have a space containing a simplex. It has 6 x points as vertices
> and 6 y points as vertices. It has an unstated number of other vertices.
> 
> In matrices language, a point would be shifted with this multiplication:
> 
> ( x1 )       ( 0  0  0  0        )
> ( y1 )       ( 1  1  0  0        )
> ( x2 )   =   ( 0  0  0  0        )  .   Old_Z
> ( y2 )       ( 0  0  1  1 ...    )
> ( ...)       (         ...       )
> 
> In Redlog, a concave/convex region would be projected onto the Y flat
> this way:
> 
>   New_Z := sub({x1=y1, x2=y2,   ...}, Old_Z);  % so a.x1+b.y1 -> (a+b).y1
> 
> Alternatively, one could use (There Exists newx[j], newy[j], such that
> newx[j]+newy[j] = oldx[j]+oldy[j] and newx[j] = 0). That would be like
> this:
> 
>   New_Z := sub( {x1b=x1, y1b=y1, ...},
>            rlqe ex( {x1b, y1b, x2b, y2b, ...},
>            (x1b+y1b = x1+y1) and (0 = x1b) and
>            (x2b+y2b = x2+y2) and (0 = x2b) and ...
>            and Old_Z ) );
> 
> That seems to contain matrices
> 
> ( xkb )   =   ( 0  0 ) . ( xk )
> ( ykb )       ( 1  1 )   ( yk )
> 
> 
> 'sub' is substitute, so: sub ({p,q}, R(p,q,t)) = R(q,q,t)
> rlqe is solve. It removes any "For All"s from rlall and all, and any
> "There Exists" from "ex" and "rlex". rlex is like ex, except it
> removes every single free variable and thus returns a boolean saying
> whether the region is empty.
> 
> 
>  >>  >The end-points map a convex surface. One day I might get around to
>  >>  >the
>  >>  >second part of the slog- deriving the (in)equality conditions that
>  >>  >define the translated surface (The conditions that define the n-l
>  >>  >votes surface are sum(i:xi)=1 , 0<=xi<=1 all i). This would entail a
>  >>  >program that could go through all combinations of 2 to (nCm)*(mCl)!
>  >>  >points in a set of (nCl)! points, work out the vector set of planes
>  >>  >that pass through every one of those points, and find the relation of
>  >>  >all points in the set to those planes ("above," "below," "equal").
>  >>
>  >> It is easier to use REDLOG. More complex manipulations can be done,
>  >> and that may be essential if the problem is to be solved. Programs
>  >> can be written in REDUCE. Non-convex problems can be solved. You do
>  >> not mean "planes" I presume
>  >
>  >I do indeed mean planes. A whole-number-sided convex surface or solid
>  >is a surface or solid which is an "and" junction of planes.
>  >
> 
> You can't simply find "above", "below", and "equal", unless the space
> has been cut into 2. In n dimensional space, only a n-1 dimensional flat
> will do that. In 3-d space, to say a point is above a 1-d line, requires
> another 1-d object to be named to get the dimensionality up to 3-1, i.e.
> a plane.

Very true. Hence the "apostrophes." One can choose, for instance, a
privileged axis against which "above," "below," "equal" had meaning. If
one were using a linear algebraic approach, as I intend to, one would have
worked out from each ensemble of points a vector space of planes a that
solve-

point 1 - [1,x1(1),x2(1),x3(1),...]	a = 0
point 2 - [1,x1(2),x2(2),x3(2),...]	~
		....

In the case of 3-2-1, let's choose 3 points in my dinky little program's
output that I already know produce useful results (the indexation goes
CBA,CAB...)

(101001)
(010101)
(011010)

So the matrix on the far left of the equation above is

[1101001]
[1010101]
[1011010]

And using the algebraic technique that I favour most,
a =	[0 0-1 0-1 0-1]	a
~	[0 0 1 0 0 1-1]	~
	[0 0 1 0 0 0 0]
	[0 0 0 0 1-1 1]
	[0 0 0 0 1 0 0]
	[0 0 0 0 0 1 0]
	[0 0 0 0 0 0 1]

that is, a is an element of the vector set-

<	[-1],	[-1],	[0],	[-1] >
	[1]	[0]	[1]	[-1]
	[1]	[0]	[0]	[0]
	[0]	[1]	[-1]	[1]
	[0]	[1]	[0]	[0]
	[0]	[0]	[1]	[0]
	[0]	[0]	[0]	[1]

This means the three points are intersected by the planes-
-1+x1+x2=0
-1+x3+x4=0
x1-x3+x5=0
-1-x1+x3+x6=0

The thing is now to work out whether, for each of these equations, whether

(1)	All points satisfy the equation.
(2)	The value of the left side of the equation for all points is not
	less than, or not more than, zero.
(3)	The value of the left side of the equation is positive for some
	point and negative for another.

If (1), then LS=0 is a universal constraint on the surface.
If (2), then LS>=0 or LS<=0 is a universal constraint on the surface.
If (3), then because the surface is convex, the plane presents no
constraint.

Testing this is the same as multiplying the row vectors describing each
point, with a "1" tagged to the left side, by the particular basis vector
for a that corresponds to the given plane. e.g.

[1,x1(1),x2(1),...] 	[-1] =	[?] =	[LS(1)]
[1,x1(2),x2(2),...] 	[1]	[?]	[LS(2)]
	...		[1]	[?]	[LS(3)]
			...	...	...

Turns out that in my example,

-1+x1+x2=0
-1+x3+x4=0
x1-x3+x5>=0
-1-x1+x3+x6<=0

In this way, we've worked out "above," "below," "equal" without having to
work out above and below. If you get my drift.

> 
> I can't be speak against what you are saying too strongly. In real
> research, an equation can be 40 lines line and concave in places. The

You mean there are 40 (in)equations, some of which in your case may be
nonlinear. In the case I'm considering, it can be demonstrated that the
surfaces involved are convex and the (in)equations that construct them are
all linear planes.

> truth can be that the number one need is to simplify the equation. It
> might simplify hugely. You might have a plan to write polytopes as
> a union of a set of small polytopes that are defined by their vertices.
> Suppose there are 30 polytopes and 10 are interior to all the others.

No, that would make it harder. There's no need to introduce new conditions
to convex polytopes (in my case, surfaces).(more further down)

> Simplification is the number one problem. I solved that by finding
> REDLOG was the best starting point, and then 3 subroutines to simplify
> polytopes equations.
> 
> 
> I once read a little book on algebra on higher dimensions. There is a
> lot I can't recall. But calculating volumes is perhaps not needed.
> 
> There are the researches of Hermann Gunther Grassman.
> 
> Manipulating Quaternion equations is hard, probably because there is
> wrap around. Matrices are much easier to work with (problems don't
> wrap around but instead tensor terms would appear). Quaternion
> equations can be written as matrix equations, solved, and then
> converted back into a Quaternion form.
> That v.hard vs easier feature between quaternions and matrices, occurs
> here, where it is between matrices and the algebra of polytopes.
> If you want, you can solve you matrix problems using algebra and then
> rewrite the solutions in a matrix O.R. form.

I'm glad Hermann agrees with me on the use of matrices. Now, what was your
point, Craig?(more further down)

> 
> This is a new way of solving these things, and it is better and it
> being promoted by the 2 men at the University of Passau and by Fujitsu.
> The promoters are small enough in number that maths and industry may
> stick with the old inferior computer-ish methods for a long time.

REDLOG is useful where one wants to preserve logical relations between
constraints, for instance in the case of _concave_ polytopes. Not so
useful when the only logical relation is going to be "AND" and the
computational and conceptual difficulties involved exceed those of a
simple linear algebra approach. And they do.

> It is not all that simple, but maybe finding vertices of concave
> polytopes is not something that is wanted.

No, it is not.

> Saari's theory is all linear, so REGLOG ought be able to handle it.
> None of his theories respect truncation resistance I guess. All of
> his ideas have to be rejected, don't they. He said in his message to
> the list he was surveying the field of Borda-like methods.

I seem to recall that he was interested in some form of truncation
resistance. You might want to look up his papers or his book. They might
have some nuggets of interest.

Saari can't be taken credulously, just like all the opposing camps of
voting theory- the Bordites, the Approvalists, the Condorcans, the
silly-little-boys-who-think-single-winner-debates-are-fruitless-and-are
-interested-in-multimember-elections. You have to decide yourself what
he's about.(more further down)

> A reason for getting REDLOG, is so problems can actually be solved,
> especially when the major hindrance to finding a solution is the
> human's ability to understand how to get a solution out. I got a
> REDLOG subroutine to written shadow any polytope using any polytope
> light source. It 9?-ish lines long. Suppose you have two collections
> of a list of points representing 2 concave polytopes:
> 
>   (Light_Region1, Light_Region2, ...), (Obj_Region1, Obj_Region2, ...)
> 
> Light_Region1 = a list of the vertices defining convex polytope #1
>                  that is inside the light source.
> (problem 1)
> 
> You might say that shadowing is not needed by you. But a rule that
> puts voters first and that excludes bad behaviour appearing in a
> voting booth where the outcome is known, is infinitesimal, so if
> constrains slopes. A linear tangent-polytope constraint on the normal
> to all flats bounding where B wins, casts a shadow using a polytope
> light source. So it is not really all that IFPP-ish, but instead
> shadowing will be an aspect of any good preferential voting method.
> 
> Maybe I should say, don't get REDLOG, and tell me how you'd solve
> '(problem 1)'.

The way I described "casting a shadow" in the dim dark distant past is
probably conceptually close to how REDLOG chugs out an
answer. Wire-frames, baby, wire-frames! 

> 
> I can give you a task: write a REDLOG program that creates voting
> method formulae, and tests them. It generates all possible voting
> methods. We assume that coefficients in the inequalities are

Ye canna dae tha', cap'n!(more further down)

> integers (so STV is rejected). Then the deriving part, which is a
> slow step, can be omitted. Another problem is to put quotas into
> STV or IRV. STV is nonlinear so it has to be handled numerically
> and the software would have to be complex and efficient. But IRV
> has integer coefficients and is linear, and it should not reject
> only 1 candidate at each stage. It should use quotas instead. But
> I never found out what.

I'm thinking one day I'll get around to writing a program that uses a game
theory approach to work out the strategy equilibrium for any
situation. Such a program would need adjunct subroutines that would go
through most voting ensembles and work out the outcome given a particular
election method. Not really the same, but there are some similarities with
working out the response of a method over a continuous range of variables.

> 
> 
> The EM list is a huge zero. It is puddle hole for water loving
> slaters.

"Are not!"
"Are too!"

Jaysus H. Feck, Craig, Fecking Get The Feck Over It.

> 
> Quoting from a Demorep message:
> 
>  >Mr. Ossipoff wrote in part-
>  >
>  > The problem is when all that a certain majority agrees on is
>  > that there's someone whom they don't want to elect
>
> How does Mike know, when he wasn't e-mailed all the details?. It

What? I see a statement by Mike Ossipoff, apparently in original context
and well like a statement you would expect Mike to make without external
reference. Mike believes in something called the "lesser evil." Then I see
a statement by Craig Carey that is out of context. What am I supposed to
think? Huh?

> is the Ossipoff spirit investigators again. I recently sent in a
> message to Usenet supporting the Pope (and DejaNews didn't index
> it, perhaps because I used a US machine to post). Are the spirits
> of God, above or below those of Mike Ossipoff?. It is not a lie:
> there has to be Ossipoff or else the man is lying. A laboured
> explanation would be needed to correct me over that. But it is safe
> get the shit and smudge it into Schulze and the rest. I think they
> fail the test.

OK, you're just getting weird. Stop being an arsehole and people will like
you. It works for me. Try it on for size.

> I wrote that society is not a part of voting theory.
> Why is Mike Ossipoff so very distant from rejecting the notion that
> humans can be referred to when taking about voting methods. It is
> a cover bullshitting. I want to suggest to YOU that a right view is

Of course it has to relate to society- it's a fucking social theory! So it
is perfectly valid for Mike to say "we expect this of the society that is
voting." If you're trying to demonstrate him wrong, you will have to
demonstrate that you cannot expect "this" of the society that is voting.

> to delete Mike Ossipoff from consideration and regard the tolerance
> of being bulshitted at, as being contrary to the interests of
> politicians. Mathematicians are interested in politicians?. Cast

You are a member of a political party, amis? So am I.

I will ignore the stuff about Mike, 'cause it's just web warrior bullshit
(why don't you set up a "Mike Ossipoff sucks dick 100%" or "I know
about the Protocols of the Elders of Mike Ossipoff!" site? You could have
animated GIFs _proving_ that Mike is the _antichrist_. Feck.). Suffice
to say- your list is small and seems to have only one active member, it
is, as someone I don't like once said, an "ego petting zoo." And one
Australian public servant subscribing doesn't make you a great academic
hero. I'll leave you with a lovely little quote on applied mathematics-

The use of applied mathematics in its relation to a physical problem
involves three steps:
i)	a dive from the world of reality into the world of mathematics;
ii)	a swim in the world of mathematics;
iii)	a climb from the world of mathematics back into the world of reality,
	carrying the prediction in our teeth.

That was a quote from C.A. Coulson's acceptance speech to the Chair of
Applied Mathematics at Oxford. It's quoted by Robin Farquharson in the
introduction to his book "Theory of Voting."