# [EM] Proportional preferential voting

Craig Carey research at ijs.co.nz
Wed Sep 15 22:37:27 PDT 1999

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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.

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
Pages: Snooz Metasearch: http://www.ijs.co.nz/info/snooz.htm,