[EM] Droop fails the Markus Schulze Rule

[Craig Carey]

I reply to some messages of Mr Catchpole lower down.


At 18:43 26.10.99 , Bart Ingles wrote:
>
>Craig Carey wrote:
>
>> Example: Some method will find six winners:
>> 
>> Case 1: paper=(ABCDEFGH), winner={C,D,E,F,G,H}, Satisfaction=63/256
>> Case 2: paper=(ABCDEFGH), winner={B,I,J,K,L,M}, Satisfaction=1/4
>> 
>> So the 2nd alternative is the alternative that the satisfaction value of
>>  the paper would prefer.
>
>I see that if you assume that the utility of each successive ranked
>choice declines by a factor of two, then the utility of a given choice
>
>will always be greater then the sum of utilities of all lower choices.
>
>What I was questioning was the use of base-2 satisfaction itself.  Is
>someone actually using this as an approximation of reality, or is it
>just a basis for a certain class of examples?
>
>--Bart

It wouldn't make a difference if the base were greater than 2.
The base can't be less than 1.6. The 'satisfaction' numbers are a
 convenient way to represent Boolean expressions.
 Two of those expressions are given below:

paper  : Satisfaction of 'paper' is higher in V than in U

  A.   : ga = not uwa and vwa
  AB   : gab = (not uwa and vwa) or
            ((uwa and vwa or not uwa and not vwa) and (not uwb and vwb))
  uwa = (A in Winners(U))
  vwb = (B in Winners(V)), etc.

Alternatively:
  ga  = (Satisfaction(W(U),(A.....)) <  Satisfaction(W(V),(A.....))
  gab = (Satisfaction(W(U),(AB....)) <  Satisfaction(W(V),(AB....))

--

Not "examples" as far as I know, but something that can be a metric for
 existing methods, and that can be strictly held in some new methods.

It may be possible to enhance an existing method by getting it to be
 more resistant to strategic voting of a type indicated appropriate
 by these satisfaction values.

------------------------------------------------------------------------

At 14:11 27.10.99 , David Catchpole wrote:
>On Tue, 26 Oct 1999, Craig Carey wrote:
...
>Pi(A,B,V) _is not_ Condorcet pairwise comparison! It's a proposition
>regarding whether a voter i expresses a preference for candidate A over
>candidate B in scenario V!

That last sentence is lengthy given the information it holds.
A "proposition". Two types of preferences, one on paper known to the
 formula, and another in the thoughts of voters. There is still no
 definition of "Pi". There is no presumption in me at all that you have
 ever defined Pi. You did write that you had thoroughly tested the
 rule. No one else can: for example, what if a Catchpole-"voter" actually
 voted strategically in a 150 candidate election, but did not "express"
 a "preference". I want to prove your rule is too strong or too weak, if
 possible, but you have not stated what the entire rule is. Does rule
 reject FPTP with 3,000 candidates?. I don't know.
 

...
About the 2nd rule

>> The First Past the Post Method, satisfies monotonicity but it is not
>>  Condorcet, so the statement is false. The term representing neutrality
>>  is on the left hand side of an implication. Mr Catchpole defended his
>>  position by saying it was clearly true ([r]elettering is possible).
>
>What's the problem with neutrality being on the "left hand side" of the
>implication? That I remember, that's where universal assumptions go.
...
This:  (All m)(True and y .=>z) = (All m)(y=>z),
 where (y=>z) = 'if method m is monotonic then the method is Condorcet'.
 Consider the method FPTP...

>> How could FPTP not be a preferential voting method when it is the method
>>  that results from modifying STV so that all subsequent preferences are
>>  discarded during transfers and during elimination (approximately).
>
>It is a preferential voting method according to _that_ definition.

FPTP is a monotonic preferential voting method, and it isn't Condorcet.
So therefore the and rule is finally rejected.

--------------------------------
At 14:39 27.10.99 , David Catchpole wrote:
>On Wed, 27 Oct 1999, Craig Carey wrote:
...
>
>Um... what do you mean by "Condorcet picks the wrong number of winners?"
>I've been talking about single-winner election systems, SF-dammit!
>
>Again, Pi is not Condorcet pairwise comparison. It's an individual
>comparison with respect to voter i, not an aggregate comparison. ...

Saying what Pi is not, does not say what it is.
I have a symbolic algebra program, and on the line describing "Pi",
 I had to enter a best guess of Pi. Mr Catchpole said I got that guess
 wrong.
 This is part of the idea of "running" with an idea.
If you define Pi, then can then be embedded into an expression holding
 an equation for a method, and sent into "rlqe" (REDLOG
    <http://www.fmi.uni-passau.de/~redlog/htmldoc/redlog_35.html> ).

--------
Also, Condorcet does picks the wrong number of winners.

 A.  za
 AB  zab
 AC  zac
 B.  zb
 BC  zbc
 BA  zba
 C.  zc
 CA  zca
 CB  zcb

 a = za+zab+zac,  b = zb+zbc+zba,  c = zc+zca+zcb

This is where Condorcet gets the wrong number of winners:

rlcnf not wok;   % Condorcet 3 candidates, 1 winner, wok=(1='#W')

       (a-c+zba-zbc >= 0 or b-c+zab-zac >= 0)
   and (a-b+zca-zcb >= 0 or b-c+zab-zac <= 0)
   and (a-b+zca-zcb <= 0 or a-c+zba-zbc <= 0)

rldnf not wok;   % Condorcet 3 candidates, 2 winners, wok=(2='#W')

      (a-b+zca-zcb > 0  and a-c+zba-zbc < 0  and b-c+zab-zac > 0)
   or (a-b+zca-zcb = 0  and a-c+zba-zbc <= 0 and b-c+zab-zac <= 0)
   or (a-b+zca-zcb < 0  and a-c+zba-zbc > 0  and b-c+zab-zac < 0)
   or (a-b+zca-zcb <= 0 and a-c+zba-zbc = 0  and b-c+zab-zac >= 0)
   or (a-b+zca-zcb >= 0 and a-c+zba-zbc >= 0 and b-c+zab-zac = 0)
   or (a-b+zca-zcb = 0  and b-c+zab-zac = 0)

Both regions are not empty (shown with "rlqe rlex not wok")




G. A. Craig Carey, Avondale, Auckland. 
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Mr G. A. 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
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