[EM] Probability theory has nothing to do with preferential voting

[David Catchpole]

On Mon, 13 Dec 1999, Craig Carey wrote:
> 
> I gave an old list of axioms and I stated that the absence of
>  randomness was an axiom for me. Anyway, statisticians would say
>  that randomness in findings needs to be "minimised". That is surely
>  opposed to the basis of Mr Catchpole's probailistic voting theory.

I don't approve of non-deterministic systems being used in real life
(don't approve of Condorcet being used either, so you can't use that
against me either), but that doesn't mean they should be dismissed as an
analytical tool- hopefully, especially, as a vindication of
multiple-winner and PR election systems against single-winner and
small-winner systems (in the long run I hope to prove that the level of
randomness required to fully satisfy rules reduces with the number of
candidates elected). (more further down)

PS "Probabilistic election methods" don't mean randomness in how people
vote- rather, they impute randomness in the result thereof. So, a coin
toss between the two front runners is a probabilistic election method, for
instance.

> 
> ...
> >p(X) represents the probability of X being the winner. From the restraints
> >from regularity, in an X-Y contest, if X>Y:Y>X = 2:1 then p(X)=2/3,
> >p(Y)=1/3. (more further down)
> 
> I note the word "regularity" is used again. Regularity theory was
>  destroyed for all eternity by me, and I used mathematical reasoning
>  and noted that a B-wins region was not closed. Mr Catchpole, can you
>  please either stop implying that your regularity idea has a reality,
>  or find a real fault in my argument. Have yuou redefined the word
>  "regularity" and if so what are all the definitions?.
> 

You misunderstood the definition of regularity and applied it only to some
deterministic formalism. So- you're using circular reasoning- you
misinterpret regularity, you go on in turn to ignore the substance of the
initial e-mail, in turn you dismiss any non-deterministic formalism, and
in turn you feel vindicated when you misinterpret regularity. The basic
substance of the first message was this- in order for regularity to be
satisfied, the probability of A winning versus B as a function of
x=n(A>B)/( n(A>B)+n(B>A) ) has to obey the following restraints...

p(1/2)=1/2

p(2/3)=2/3

p(3/4)=3/4

(more further down)

> ...
> >> If the probability of a a single candidate winning a 'one winner one
> >>  candidate' election is exactly one ("unity"), then isn't the probability
> >>  of a winner winning any election exactly "unity"?. ...
> >
> >_WE ARE TALKING ABOUT PROBABILISTIC ELECTION METHODS_ Grrr! Talk to
> >Markus!
> >
> >Example: when both A and B are running, it may well be we have an equal
> >coin toss between A and B- that is, p(A)=1/2, p(B)=1/2. when B withdraws,
> >and A is the only candidate, p(A)=1.(more further down)
> 
> What is this "when both A and B are running, it may well be we have an
>  equal coin toss". Consider the implications:
> 
>         In 7 years, 5 UK jail inmates, voted on whether they would
>          vote for the Tory party or the Liberal party.
>         This is a 2 candidate election.

I took it that you would understand that I was referring to your example
(immediately above the paragraph) of 1 A>B , 1 B>A. We've already taken as
given the distributions of votes- equal. Obviously in such a tied
situation a coin-toss would be justified! (more further down)

> 
> ------------
> ...
> >Well, say n(A>B) is the number of voters who prefer A over B.
> >
> >x=n(A>B)/( n(A>B)+n(B>A) )
> >y=n(C>A)/( n(A>C)+n(C>A) )
> >z=n(B>C)/( n(B>C)+n(C>B) )
> >
> 
> Pairwise comparing is a presumption that destroys the credibility of
>  the results. Proponents of STV can find little of value in parwise
>  comparing. It is very difficult to see just by how much pairwise
>  comparing expells optimality. 

Pairwise comparison in this case does not constitute a presumption
(though, yes, it does in others)- rather, it states the states of votes
upon the retraction of a candidate. Whoah- this is part of the approach
needed- to consider the removal of candidates! Think of it as mama
three-way and 3 babies pairwise. No conditions are placed on the three-way
contest results by the octahedron. Borda is as acceptable as Condorcet as
FPTP without the imposition of a rule. The rule places restraints on the
probability functions related to the two-way contests in order to ensure
that the addition of one candidate will never contradict regularity by
requiring that the probability of some candidate (other than the new one)
winning would increase with the addition. (more further down)

> 
> At 10:54 13.12.99 , David Catchpole wrote:
> >On Mon, 13 Dec 1999, David Catchpole wrote:
> >> "possible configuration of votes" means a likely schema of votes. Where
> >> the **** does "simplex" always come from? How can a simplex on the
> >> triangle be interpreted as a "possible configuration of votes?"
> >
> >Should have written "possible schema of votes." Slipped my mind for a
> >second that I was referring to possible existence, not likely existence.
> 
> If there are 230 counts of papers (i.e. numbers), then the point will
>  be placed in space having 229 or 230 dimensions.
> 
> That x,y,z you wrote about does not have that many dimensions.

Correct- because at this stage this consideration is only one of
three-candidate systems and their relation to any two-candidate systems
which result upon the withdrawal of a candidate. If you like, the Saari
octahedron is a useful transformation into three dimensions of a
6-dimension (12-dimension if you allow for indifference) space such that

[ x,0,0 ]AX=BX
[ 0,y,0 ]
[ 0,0,z ]

where X is the vector in the 6(12)-dimension space and A and B are two
constant 3x6(3x12) matrices whose values I can't be bothered to look up.
It doesn't purport to state every single fact about a voting schema-
simply the state of the vote upon the withdrawal of one candidate or
another. For the sake of the first message, it suited its purpose. It's
slightly easier to construct a three-dimensional object from graph paper
and trace a Nikko marker between "neighbourhoods."

You may notice that the triangle you keep on referring to has only two
real dimensions and a pseudo-dimension.

Getting your hands on Saari would be a really awesome idea right now.

The reaon why I objected to your relation of "simplex" to "possible
configuration of votes" is that, obviously, in any useful geometrical
consideration, _each_ "possible configuration of votes" will be
represented by a single point, not by a line, a shape, or a solid. What I
think you're trying to say is that "all possible configurations of votes
may be found within a simplex." Got no problem with that...

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Nothing is foolproof given a talented fool.