[EM] The 6 paper 4W1CD problem solved.

[Craig Carey]

**
At 08:03 03.10.00 +1000 Tuesday, David Catchpole wrote:
 >Problem is though that the US Presidential election system distorts the
 >vote by its "winner takes all" selection of delegates from each
 >State. Meaning, say, a huge number of voters in a huge State like
 >California have their vote float away into insignificance 'cause they
 >didn't support the plurality winner in that State.
 >
The Electoral College debate is a debate over small errors (difference
between figures from a weighted sum and the sum resulting from counting
votes of each counted voter). There is rough aim to keep the errors
about minimal, I guess. The URL omitted:

"Electoral College calculator" page:
http://www.avagara.com/e_c/ec_calc.htm

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I wrote that truncation resistance is more important than monontonicity
but that might be wrong: it is best to get both. STV has a face that
tilted too much, and a long vector the length of the face could pierce
it from the wrong side (according to P1). Monotonicity is similar
to Truncation Resistance: a prohibition on winner changes along lines
parallel to lines between two edges (or more than two edges in the case
of truncation resistance).


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At 01:52 02.10.00 -0400 Monday, DEMOREP1 at aol.com wrote:
...

I read the comments.

So you want the list to gets its act together and You also note merit in
a modified Approval Vote (for use with an ignorant public that has had
rotten schooling). A list with its act together may be one that has
principles that rule out your YES/NO methods.

If you do have a very lean set of basic principles, why don't you kick off
by stating them. It is a possibility. It think it is better to call it the
list's year of the weed.


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At 15:00 02.10.00 +1000 Monday, David Catchpole wrote:
 >Hell no, FPP is _not_ monotonic. If one ranks one's real preference
 >first rather than one's "lesser evil" [sic], a repugnant candidate can
 >win. Tell me that's not nonmonotonicity.
 >

Monotonicity has a definition and your comments are distant from that.
I'd say it would be this (note that I have not found a trustable source
so this is just a definition I believe that is right and agreeable):

Monotonicity

    For each candidate and any/som/all papers, if a preference naming a
    candidate is added and/or shifted to the left, and it is nowhere the
    case that another change occurs or that a preference of the
    candidate is either removed or shifted to the right [i.e. away from
    the first], then the candidate's will not change from being a winner
    into a loser.

The number of winners is not stated so the definition is for any number of
winners.

FPTP/FPP will pass that definition won't it?.

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At 04:35 02.10.00 +0000 Monday, MIKE OSSIPOFF wrote:

>Craig asked whether CVD is hiding IRV's problems, or whether CVD
>is merely ignorant of them. I'd say it's a little of both.


People might guess that they are idealists.
But it is a touch impossible to comprehend that they lack idealism.

It is able to be simple: the CVD leader gets an A4 sheet and writes
on it that the CVD does not promote methods that negate the direction
of people's votes. It can frame it up and hand it in the hallway of
their building. How much will power does it take from the CVD staff
to pay the rent on their buildings.
Maybe the problem is not with a lack of will or idealism at the CVD,
but a shortage of paper. Has anybody joined the CVD. Maybe the fees
keep well stocked their inner library wax museum (check for fake
books behind the door). Big or small, the CVD can learn-em them all.
I need to be careful to not threaten the capacity to receive overblown
donations for their work, or I could be suddenly banned from the list
buy Rob.
Mr Ossipoffs' Approval Vote is monotonic, but it is not truncation
resistant, and Condorcet variant methods have the same problem.
Demorep is off side too. I am in the clear. Mr Schulze believes in
SSD which fails lots of good rules. Would you _please_ tell me which
major rules passing my meta-rules, your considered-best method, fails?,
Markus. This is an implementation of Demorep's call for a better list.



The following is a copy of a message I sent to my
Politicians and Polytopes list.

This next result was first obtained today.

__________________________________________________________________________

  /-------------------------------------------------------------------
  |
  |  VBCD =
  |      A    a0
  |      AB   ab
  |      AC   ac
  |      AD   ad
  |      B    b
  |      C    c
  |      D    d
  |
  |  bWCD = (a<b) . (a+c<b+d or a+c<2*b) . (a+d<b+c or a+d<2*b) .
  |            (Tacb or (2c<a+b or 2c<a+d)(c+d<a+b or b+c<a+d)) .
  |            (Tadb or (c+d<a+b or b+d<a+c)(2d<a+b or 2d<a+c))
  |
  |  aWCD = (b<a)(c<a)(d<a) = aW = (A wins VCD in IFPP)
  |
  |  Tacb = (c+ac<b+ab)
  |  Tadb = (d+ad<b+ab)                       :    a = a0+ab+ac+ad
  |
  |  bWCD = (B wins VCD in 1 winner IFPP) = bWBCD
  |
  \-------------------------------------------------------------------
bWCD is not checked. I presume it is unlikely to contain an error.

  /-------------------------------------------------------------------
  |
  |  V =
  |      A    a0
  |      AB   ab
  |      AC   ac
  |      B    b
  |      C    c
  |
  |  bW  = (c+a<2b)(Tacb or (2c<b+a))(Tcab or (2a<b+c))
  |
  |  Tacb = (c+ac<b+ab)
  |  Tcab = (a+ca<b+cb)
  |
  |  bW  = (c+a<2b)((c+ac<b+ab) or (2c<b+a))((a+ca<b+cb) or (2a<b+c))
  |
  |  These bW equations also give the solution to the fully general 3
  |  candidate 1 winner problem having the 15 papers:
  |      A,AB,AC,B,BC,BA,C,CA,CB, ABC,ACB, BCA,BAC, CAB,CBA.
  |  This is 1 winner IFPP.
  |  Formula are on http://www.ijs.co.nz/ifpp.htm
  |
  \-------------------------------------------------------------------

...

Conclusion.

This is not a good prompter to lead to understanding and a wait for
the 10 paper 4W1CD solution may be needed. Once that is available
then the whole solution is out. The persistence of the (a+c<b+d) may
be under threat.

Keywords: polytopes, quantifier elimination, logic.