[EM] Multiwinner participation rule. Geometric descriptions

[David Catchpole]

> Mr Schulze, are satisfied with my comments on geometry?.
> I haven't heard much about Condorcet in quite a while (in this
>  mailing list).

Condorcet. Condorcet, Condorcet, Condorcet, Condorcet. But seriously-
there hasn't been much mention of _any_ single-winner election method for
a while. Doesn't necessarily mean people are retracting...(more further
down)

> Mr Catchpole wrote "Regularity is cool". I believe that that indicates
>  Mr Catchpole ignored or did not get (or read) my proof that his
>  "Regularity" idea is a useless and valueless and impossible to satisfy.
>  I shall repeat the comments I made:

I wrote the Regularity was jiggy with examples given by yourself
and Demorep! Don't take my gear out of context, hipster!

Now- you ignored the basic properties of the election methods to which
Regularity has pertinence. These are probabilistic election methods which
may have a degree of uncertainty in who wins despite certainty about
votes. Why extend a consideration of election methods to incorporate
these? Three good answers-

(i) Because they're interesting
(ii) Because deterministic election methods have drawbacks and paradoxes
which can be ameliorated by the use of probabilistic methods
(iii) Because, rather than making assumptions about how likely certain
voter configurations are, we can use degree of similarity to a
probabilistic analogue which satisfies a rule, to judge "closeness of
fit."

Note- you _must_ acknowledge probabilistic voting systems to have an
impact on this debate.(more further down)

> >No, here we're talking about non-deterministic voting systems where some
> >form of "lottery" is involved. For instance, an example of a
> 
> By message 4, probability theory seemed to be redundant.

Wha the? (more further down)

> Message 3: ---------------------------------------------------------------
> Subject: RE: The family of "regular" probabilistic (stochastic) electoral systems
> At 13:48 12.12.99 , David Catchpole wrote:
> >On Wed, 8 Dec 1999 DEMOREP1 at aol.com wrote: ...
> >> D-
> >> A simple example-
> >> 
> >> 2 A>B>C
> >> 1 C>B>A
> >> 
> >
> >well, without C: p(A)=2/3, p(B)=1/3
> >without B: p(A)=2/3, p(C)=1/3
> >without A: p(B)=2/3, p(C)=1/3
> ...
> 
> Six unexplained assertions. They need to be explained.

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)

> 
> Message 4: ---------------------------------------------------------------
> Subject: RE: The family of "regular" probabilistic (stochastic)  electoralsystems
> At 14:08 12.12.99 , David Catchpole wrote:
> >Ja, vat ist das problem? Obviously, A's removal will _increase_  B's
> >probability of winning to unity. Regularity is cool with this- it would
> ...
> >> 
> >> 1 A
> >> 1 B
> >>...
> 
> 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"?. I presume every
>  subscriber has this figured out.

_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)

> 
> Message 5: ---------------------------------------------------------------
> Subject: RE: The family of "regular" probabilistic (stochastic)   electoralsystems
> At 14:12 12.12.99 , David Catchpole wrote:
> ...
> >
> >Well, it's not a cube per se, but rather an octahedron formed by the
> >intersection of 
> >
> >1<=x+y+z<=2
> >
> >and
> >
> >0<=x,y,z<=1
> 
> What are x and y and z ? (they were not defined).

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) )

A visit to Saari would help clear up what the Saari octahedron is.(more
further down)

> Message 6: ---------------------------------------------------------------
> Subject: Re: FPTP family theory, REDLOG shadowing
> At 14:17 12.12.99 , David Catchpole wrote:
> >Saari also uses a triangle, but the triangle doesn't go far enough in an
> >analysis of what's a possible configuration of votes and what's not.
> 
> The words "possible configuration of votes" suggests a simplex (and nothing
>  else). But Mr Catchpole makes a blunder, uses the word octahedron.
>  Surely Mr Saari would prefer student Catchpole tried harder.
> 
> >That's where the Saari octahedron kicks in in his analysis.

"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?"

Grrr.

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Nothing is foolproof given a talented fool.