[EM] Method S92 and the new P4 rule

Craig Carey research at ijs.co.nz
Tue Sep 25 12:25:09 PDT 2001

My latest notes to the Politicians and Polytopes mailing list
produced information showing that the idea of requiring a 1/4
quota in 4 candidate 1 winner elections, was being moved closer
to being rejected. I had found a monotonic (i.e. P1 complaint)
method that solved the 1 winner (AB),(B),(C),(D) problem.
The method did not obey the 1/4 quota rule and it was passed by
the P4 (one man, one vote) axiom. However the absence of (AC)
and (AD) papers weakened the testing against P4, making it less
clear whether that quota must be removed or not.

This new simple 4 candidate method (arbitrarily I named the
method S92):

Here is yet another picture of STV being rejected for all time,
because of its bugs:
 >p41ck2(a,aW);  % = true
 >p41ck2(b,bW);  % = true
 >p41ck2(c,cW);  % = false
 >p41ck2(d,dW);  % = false
 >% %%%%%
 >Apparently the 4 candidate Alternative Vote fails tests of test case 1.
 >Test case 1 is the most important, and is considering only bad design
 >affecting the first preference: "1:(A) --[umbral]--> s:(*)".
 >REDLOG takes a bit under 1.85 secs to reject the Alternative Vote
 >(= 1 winner STV). If politicians wonder if it is a good method, an
 >answer is that it was rejected at GMT 24 Sept 2001.

It can't fail that test like that and be a good method that should
be used in open fair public elections. It violates the right of
suffrage. The lawyers are nonmathematical thinkers and some patching
of their principles could be expected to occur.

Also out there is a method that is different from both that
1/n quota method ("Q method") and the attempted fix I posted up
here recently.

Persons wanting a practical method can try embed that into their
best ideas. That would be a possible waste of time if handling
(AC) and (AD) papers results in noncompliance with the P4 rule.

The P4 rule is this:

(P4) = (All t) (All H) (All S)
     (Exists Z)(Z = H+S)
     (Exists T)
     (T is an allowed LHS simplex vector)
     (Exists Y)(Y = H+T)
     (All k)(0<=T[k])
     [  [  (S is an allowed RHS vector)
           (All k)(0<=S[k])
           ((Sum k)S[k] <= t)
        ]  implies
        [  (t = (Sum k)T[k])
           [  [ (Satis(Y, SL) known) . (Satis(Z, SL) known)
              ] implies
              [ Satis(Winners(Y), SL) >= Satis(Winners(Z), SL)] ]]]

(The textual error of putting the Winners function inside the Satis
function is not repeated here. Satis is defined at the PaP list).

The definition of P4 is online here:


P4 is designed to permit the derivation of a perfect preferential
voting method. [That method that Mr Shulze commented on, the 2nd
version of my Q method, was flawed and it certainly was not IFPP
which is based upon P4.]

P4 has a point Z on the RHS and a point in the simplex H+T on the
LHS with requirement that a not worse satisfaction value point must
exist in the LHS simplex, subject to the constraints including that
not less votes are added to the LHS. It means what is says and a
clarifying implementation of it is online.

P4 implements the "one man, one vote" principle. It is certainly a
rejected idea at the Election Methods List. Let's approach it
indirectly and have clear statements on whether there are "many
rules" that shift and change their infinitesimal requirements thus
altering the reasonableness of the method using considerations
unknown to voters.

P4 defined above only considers alterations to papers,k because of
the way that H appears.

The Election Methods List rejects the idea of limiting the power
of voting papers. Political principles are rejected so their is no
protection against bad men and the list could not ever achieve
understanding major P1 rule which can be defined in 2 line, so
voters are unprotected against bad methods. Proportionality seemed
to be not aimed for.


At 01.09.23 13:44 -0700 Sunday, Blake Cretney wrote:
 >On Sun, 23 Sep 2001 12:50:17 -0700
 >Bart Ingles <bartman at netgate.net> wrote:
 >> Craig Carey wrote:
 >> Explorer.
 >We often seem to assume that a voter's chief problem with a system is
 >the ballot itself.  That is, the voter neatly assigns a utility to
 >each candidate, but has trouble translating this into a ranked ballot

Voter's chief problem could be said to be researchers. So it is the
use of STV papers?.

What is a utility and why would any voter want to even think of it.
Irrespective of what it is, the method can be plotted as if a
hard coloured polytope inside a simplex (e.g. in 5000 dimensions)
and the voter has the problem of shifting the point into a more
desired region. Utility is a replacement for what is reasonable.

Perhaps Mr Blake Cretney could comment further on the theory that the
voters have a problem filling out preferences. Suppose the problem was
so bad that over 89% died with heart attacks?. The old mistake of
"utility" has returned. I don't suppose people here ever got far with

Why not have this list imply that it understands its topic of
preferential voting and reject the P4 formula. It certainly seems
that the posters are evasive when a complete or sufficient set of
defining axioms is sought.

So help us, Mr Cretney, is that writing in a style of attempting to
communicate understanding?.


Here is the 4 candidate method. I never got around to rewriting it
so that it looked like it had stages. A point to note is that
that (a+c<b+d) terms could be rewritten to be more like (a+c<2b) or

At 01.09.22 08:29 +1200 Saturday, Craig Carey wrote to PaP:
 >    AB  ab     :  a = ab
 >    B   b
 >    C   c
 >    D   d
 >After quite some casting of shadows using REDLOG, this was obtained:

Method S92 (which passed all P4 tests that could be applied):
 >   aW = (b<a)(c<a)(d<a)
 >   bW = (a<b)(c<b+ab)(d<b+ab)(a+c<b+d or a+c<2b)(a+d<b+c or a+d<2b)
 >   cW = (a<c)(d<c)(b+ab<c or (b+d<a+c)(2b<a+c))
 >   dW = (a<d)(c<d)(b+ab<d or (b+c<a+d)(2b<a+d))
 >bW = (B wins), etc.
 >Note:  B can win when it has less than 1/4 of the votes:
 >    AB  2
 >    B   3
 >    C   4
 >    D   4   : B wins with under 1/4 of the vote.

The candidate with the smallest number of 1st-preference votes loses.

Tell us more about these voters. Not important enough to model,
and not so insignificant that the ideations of their existence, can't
be the gloves that will touch and constrain polytopes.

G. A. Craig Carey


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