[EM] Proportional preferential voting

[Craig Carey]

Below is a 3 candidate one winner formula for a theory
 of preferential voting. It might be new.

When a candidate alters a vot in a voting booth, the effect
 of that change is limited. STV has an infinity of its
 own paradoxes like Condorcet, and they can be removed but
 with the result of shifting the method slightly towards
 FPP (although STV is quite strange as votes get wasted at
 most transfers of parcels).

A example of an alteration rule is:

 'permuting all preferences before a preference for a
  particular candidate, never makes a difference to
  that candidate's win/lose status'.

That is both nice and desirable.

 Unfortunately it cannot be obtained because it would cause
 about all solutions to degenerate into First Past The Post
 starting with the 3 candidate solution. I haven't a proof
 available so that could be conjecture.

Mr Saari has a duality rule, which I sent to him and which
 he may be on the verge of saying is not of value. 'll let
 that respected mathematician decide whether it is worth
 mentioning or not. One of his publications has the word
 geometry in it. That why I decided to write: his booklet
 on 'geometry' in voting.

What of Condorcet, a topic of this mailing list?:
 An example: In an election with 2,152,370 candidates and
 430,927 winners, how can it be certain that pairwise
 comparing of two candidates is an idea that ever had some
 mathematical importance on the fist day?, in France is it?.

I'd hope to see an example proving that the Condorcet method
 violates the "one man one vote" idea.

By "proprotionality", I'd mean: "adding" of votes, subject to
 contraints [its complex mathematically], and then sorting the
 candidates and selecting the x with the most votes. Sorting
 is actually not eveident at all in the formulae below.

PS. Thinking numbers is a dead end if the method is based on
 alteration rules, since election examples cannot be solved until
 an infinity of other other election examples are found.
 Quite implicit. That is known: a paper has been published.
 Lat time I checked there was only one (3-4 years ago).
 Switching to hypergeometric algreba simply removes the
 implicitness.

Mathematicians guess at methods. I'll state for the record: they
 are can't do that well. Reasoning is more succesful. Did any
 mathematician guess the following method (e.g. just before
 rejecting it)?. Not as far as is known(?)?.

The 3 candidate methods: 
  Proportional rule based preferential voting


----------------
|A.. |     |   |
|AB. | Zab | a |
|AC. | Zac |   |
---------------
|B.. |     |   |
|BC. | Zbc | b |
|BA. | Zba |   |
----------------
|C.. |     |   |
|CA. | Zca | c |
|CB. | Zcb |   |
----------------

(A wins) = (b+c<2a)[(b+Zcb<a+Zca) or (2b<a+c)][(c+Zbc<a+Zba) or (2c<a+b)]
(B wins) = (a+a<2b)[(c+Zac<b+Zab) or (2c<b+a)][(a+Zca<b+Zcb) or (2a<b+c)]
(C wins) = (b+b<2c)[(a+Zba<c+Zbc) or (2a<c+b)][(b+Zab<c+Zac) or (2b<c+a)]

({B,C} win) = (b+c>2a)[(b+Zcb>a+Zca) or (2b>a+c)][(c+Zbc>a+Zba) or (2c>a+b)]
({C,A} win) = (c+a>2b)[(c+Zac>b+Zab) or (2c>b+a)][(a+Zca>b+Zcb) or (2a>b+c)]
({C,A} win) = (a+b>2c)[(a+Zba>c+Zbc) or (2a>c+b)][(b+Zab>c+Zac) or (2b>c+a)]

The total number of papers for A is a, etc.
The number of voting papers marked 'ABC' is Zab, etc.
The duality relationship that seems to apply to this class
 of methods, is evident.

The alteration rules roughly merely define slopes of flats bounding
 the convex polytope where a single candidate wins.

The is no arbitrariness introduced into the formulae above.
There is no proof also, so please regard this message as being
 perhaps without merit since there is absence of even intent
 to provide proof.

I want another mailing list, a place where numerical examples are
 used just for destroying other methods and algebra of polytopes
 is used to build rigourously, preferential voting theory.
Which then gets implemented onto computers.

[Sent to mailing of
    http://www.eskimo.com/~robla/cpr/election-methods.html ]

16 September 1999
Mr G. A. Craig Carey
Avondale
Auckland






Mr Craig Carey

E-mail: research at ijs.co.nz

Auckland, Nth Island, New Zealand
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