[EM] Election Methods Lists online archives: URLs
Craig Carey
research at ijs.co.nz
Sat Aug 17 07:45:37 PDT 2002
Despite the brevity of Mr Simmons communication, there was no definition
of what a "Condorcet Method" was. So no one here has even said that
Condorcet itself can be repaired, yet they write about a class of
repaired or similar methods. It seems impossible to believe that
members of this mailing list have a way of computing whether a method
(finding 1 winner only) is like the Condorcet method.
Every reader does without algebra just to hear the deep noises on how
Condorcet can be respected or fixed: a method that could be good but
that is bad, but it is all saved by the vague cloud of similar
methods that is around it. Vagueness seems OK, but I presume that those
that wish the belief to arise that pairwise comparing is neat and not
trash, want to both include and exclude, some preferential voting
methods. Knowledge and progress has been lost to get the concept of
what is good about pairwise comparing as vague and tenous as it
currently is, but more is on offer:
give a specific method, the promoters of the ideal can't decide if
the method is one of the good ones that is tainted with goodness,
or whether it is a bad method.
I do hope the Forest Simmons can take some time and answer (with
exactness) my request for the definition of a "Condorcet Method".
It looks like Mr Simmons may both want to include and exclude some
preferential voting methods.
What we are not seeing is definition just for the fun of it.
At 02\08\17 19:35 +1200 Saturday, Craig Carey wrote:
>
>
>
>Election Methods Lists online archives are these two websites:
>
>http://www.mailarchive.com/electionmethodslist@eskimo.com/index.html
>
>http://groups.yahoo.com/group/electionmethodslist/messages
>
>
>
>_____
>
>
>At 2002\08\16 15:56 0700 Friday, Forest Simmons wrote:
>...
> >The simplest of these, PAV and Sequential PAV, are based on Approval style
> >ballots.
> >
> >More complicated are pairwise PR methods (based on the D'Hondt rule, for
> >example) that reduce to Condorcet methods in the single winner case.
> >
>
>
>
>I ask Mr Simmons to email into that mailing list the definition of a
>"Condorcet method", subject to the constraint that the definition
>take account only of the polytopes (or shapes) that are [its own]
>definition. Anything unique that you have. You have not ruled [out] that any
>method that is _not_ what you would call a Condorcet method can
>nevertheless be constructed from some ingenious use of pairwise
>comparing. What is a a "Condorcet method" ?.
...
=====================================================================
Notes leading to a Borda method that is (unless a mistake was made)
obtainable from the three 3candidate pairwise comparing numbers.
Also REDLOG computations are shown.
__________________________
load_package rl; % load_package redlog;
rlset ofsf; % reals (not discrete, not complex numbers)
clear p,q,r;
% b is the subtotal for A minus the subtotal for B in a general Borda method.
b := mat((1, pq, pr, 1, qp, rp, 0, qr, rq));
% votes and their subtotal weightings:
% (A) : (1,0,0)
% (AB) : (p,q,r)
% a is parwise comparing matrix, for an A before B pairwise comparing sum.
a := mat(
( 1, 1, 0), % A
( 1, 1, 1), % AB
( 1, 1,1), % AC
(1, 0, 1), % B
(1, 1, 1), % BA
(1,1, 1), % BC
( 0,1,1), % C
( 1,1,1), % CA
(1,1,1)); % CB
a;
tp a;
% the nullspace is the set of perpendicular vectors
nullspace a;
n := nullspace tp a;
h := for k:=1:length(n) do write (0 = b * part(n,k));
% 4 equations result:
% 0 =  p  q + 2*r + 2
% 0 = p + q  2*r  2
% 0 = p  2*q + r
% 0 =  p + 2*q  r
% Simplifies to
% 0 =  p  q + 2*r + 2
% 0 = p  2*q + r
% Equivalent to 0 = f * (1,p,q,r), so (1,p,q,r) is a linear combination
% of the nullspace of f.
f := mat ((2, 1, 1, 2), (0, 1, 2, 1));
% the nullspace of f is the 2 vectors: (3,0,2,4), (0,1,1,1)
% Hence (1,p,q,r) = (1,0,2/3,4/3) + t*(0,1,1,1)
% Hence p,q, and r are found: (p, q, r) = (t, t2/3, t4/3)
% An alternative method:
rlqea rlex ((0 =  p  q + 2*r + 2) and
(0 = p  2*q + r) and
(0 = p + q  2*r  2) and
(0 =  p + 2*q  r));

For the subtotals of candidates A, B, and C, these are the weights for
the shown papers (preference lists):
(A) : (1, 0, 0)
(AB) : (4/3, 2/3, 0)
Did Forest Simmons mean that that generalised Borda method was a
"Condorcet Method".


1: load_package rl; % load_package redlog;
*** enlarging fasl space by 175000 items

2: rlset ofsf; % reals (not discrete, not complex numbers)

{}

3:
3: clear p,q,r;

4: % b is the subtotal for A minus the subtotal for B in a general Borda method.
4: b := mat((1, pq, pr, 1, qp, rp, 0, qr, rq));

b := [1 p  q p  r 1  p + q  p + r 0 q  r  q + r]


5:
5: % votes and their subtotal weightings:
5: % (A) : (1,0,0)
5: % (AB) : (p,q,r)
5:
5: % a is parwise comparing matrix, for an A before B pairwise comparing sum.
5:
5: a := mat( %
5: ( 1, 1, 0), % A
5: ( 1, 1, 1), % AB
5: ( 1, 1,1), % AC
5: (1, 0, 1), % B
5: (1, 1, 1), % BA
5: (1,1, 1), % BC
5: ( 0,1,1), % C
5: ( 1,1,1), % CA
5: (1,1,1)); % CB

 [1 1 0 ]
 [ ]
 [1 1 1 ]
 [ ]
 [1 1 1]
 [ ]
 [1 0 1 ]
 [ ]
a := [1 1 1 ]
 [ ]
 [1 1 1 ]
 [ ]
 [0 1 1]
 [ ]
 [1 1 1]
 [ ]
 [1 1 1]


6:
6: a;

[1 1 0 ]
[ ]
[1 1 1 ]
[ ]
[1 1 1]
[ ]
[1 0 1 ]
[ ]
[1 1 1 ]
[ ]
[1 1 1 ]
[ ]
[0 1 1]
[ ]
[1 1 1]
[ ]
[1 1 1]


7: tp a;

[1 1 1 1 1 1 0 1 1]
[ ]
[1 1 1 0 1 1 1 1 1]
[ ]
[0 1 1 1 1 1 1 1 1]


8: % the nullspace is the set of perpendicular vectors
8:
8: nullspace a;

{}

9:
9: n := nullspace tp a;

n := {

 [2]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [1]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [1]

 ,


 [0]
 [ ]
 [1]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [1]

 ,


 [0]
 [ ]
 [0]
 [ ]
 [1]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [1]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]

 ,


 [0 ]
 [ ]
 [0 ]
 [ ]
 [0 ]
 [ ]
 [2 ]
 [ ]
 [0 ]
 [ ]
 [1]
 [ ]
 [0 ]
 [ ]
 [1 ]
 [ ]
 [0 ]

 ,


 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [1]
 [ ]
 [0]
 [ ]
 [0]
 [ ]
 [1]
 [ ]
 [0]

 ,


 [0 ]
 [ ]
 [0 ]
 [ ]
 [0 ]
 [ ]
 [0 ]
 [ ]
 [0 ]
 [ ]
 [0 ]
 [ ]
 [2 ]
 [ ]
 [1]
 [ ]
 [1]

 }

10:
10: h := for k:=1:length(n) do write (0 = b * part(n,k));

0 =

[  p  q + 2*r + 2]


0 =

[p  2*q + r]


0 =

[0]


0 =

[p + q  2*r  2]


0 =

[  p + 2*q  r]


0 =

[0]


11: f := mat ((2, 1, 1, 2), (0, 1, 2, 1));

 [2 1 1 2]
f := [ ]
 [0 1 2 1]


12: nullspace f;

{

 [3 ]
 [ ]
 [0 ]
 [ ]
 [2]
 [ ]
 [4]

 ,

 [0]
 [ ]
 [1]
 [ ]
 [1]
 [ ]
 [1]

 }

13: nullspace tp f;

{}

14: rlqea rlex ((0 =  p  q + 2*r + 2) and
14: (0 = p  2*q + r) and
14: (0 = p + q  2*r  2) and
14: (0 =  p + 2*q  r));

{{true,

 3*p  2 3*p  4
 {q = ,r = }}}
 3 3

15:

It would seem that there is a Borda method that can computed using only the
3 pairwise comparing numbers of 3 candidate elections.
Craig Carey
http://www.ijs.co.nz/quota13.htm (Notes on origin of some voting axioms)

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