[EM] "None of ..." -- Negative votes

Craig Carey research at ijs.co.nz
Wed Feb 23 18:59:38 PST 2000

At 06:34 24.02.00 , Craig Carey wrote:
>Voting against candidates using negative votes.
>At 23:05 23.02.00 , Donald E Davison wrote:

To some extent, any election method can be made to handle negative
 votes, just by preprocessing the data using P2. That would keep negative
 numbers out of the method. However for methods that are not so well
 designed (...), the sets of winners may be different.

Suppose the ballot counts are as follows (with the candidates being

A      a0
AB     ab0
ABC    abc0
ABCD   abcd
ABCE   abce
ABD    abd0
ABDC   abdc
ABDE   abde
ABE    abe0
ABEC   abec
ABED   abed
AC     ac0
ACB    acb0
...    ...

Let g1=min(a0,b0,c0,d0,e0)

Let g2=min(ab0,ac0,ad0,ae0, ba0,bc0,bd0,be0, ca0,cb0,cd0,ce0, ,da0,... ,...)

Let g3=min(abc0,abd0,abe0, acb0,acd0,ace0, adb0,adc0,ade0, aeb0,aec0,aed0,
           bac0,bad0,bae0, bca0,bcd0,bce0, bda0,bdc0,bde0,... ,... edc0)

Let g4=min(abcd,abce,abde, acbd,acbe,acde, adbc,adbe,adce, aebc,aebd,aecd,
           bacd,bace,bade, bcad,bcae,bcde, bdae,bdce,bdce,... ,... edba)

Then for each of {g1,g2,g3,g4}, a constant is added to very paper from which
 they were derived, just to make none of the ballot papers have a negative
 count. The example would be generalised.

An example. This election

  A.   2
  AB  -5
  B.   6
  C.   7

can be changed into this:

  A.  2
  AB  0
  AC  5
  B.  6
  BA  5
  BC  5
  C.  7
  CA  5
  CB  5

That could be changed into this (if allowed):

  A.   2
  AC   5
  B.  16
  C.  17


>A check of STV against P2 with votes allowed to be negative.
>  A.  a0-2t+u,              a = a0-x
>  AB  -x+t
>  AC  t
>  B.  b+u
>  C.  c+u
>"Quota" = Q = (a+b+c+3u)/K.     "Sum" = S = (a+b+c+3u).
>K=3 in "Droop-STV", and K=2 in "Hare-STV".
>Case (b+u<Q)(c+u<Q)(Q<a+u):
>      YBC = 0 < c - b + x(a+b+c+3u)/a
>          = 0 < a.c - a.b + x(a+b+c+3u)      (provided 0<a)
>      I much suspect that YBC (which decides whether the winners are
>      {A,B} or {A,C}) is not dependent on u, despite it containing u.

I guess that "is independent" statement ought be constrained by saying
 that all ballot counts need to be positive.

Mr. G. A. Craig Carey 

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