[EM] Power of votes: Min(Sum[max-Vi],Sum[Vi-min])

[Craig Carey]

The usual definitions of the Borda method have the weights wrong,
 and there is a serious problem with truncations of preferences
 altering the power of a vote.

Before that can be fixed, a definition of power (for a vector which
 is the sums for candidates), is needed.

Here is a definition (devised today):

-----------------------------------------------------------------------
             Definition of the Power of a list of numbers

 Votes = V
  (V may be the counts of all voting papers, or a vector of the weights
   in a method like Borda.).

 Power = Min( (Sum i)[max(V)-V(i)], (Sum i)[V(i)-min(V)] ).

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

The definition has the property that the power is independent of t if
 V is replaced  with V+(t,t,t,t,...). It also performs OK when some
 votes are negative.
The definition correctly calculates the power of (a,b) as |a-b|.

Taking the sum of the absolute values around a value that minimizes that
 sum is not a good definition because it finds that the power P of
 (0,x,1-x), 0<=x<=0.5, to be as follows:

  P = (Min u)[|0-u|+|x-u|+|1-x-u|] = (Min u)[u+|x-u|+1-x-u]
    = (Min u)[|x-u|+1-x] = 1-x.

 The correct power is P=1.
 The definition above calculates the power as:
  P = Min( [(1-x)-0]+[(1-x)-x], [x-0]+[(1-x)-0] )
    = Min(2-3x,1) = if x<1/3 then 1 else 2-3x
 The power is minimal when x=0.5, and the vector = (1/2, 1/2, 0), which
  should have the same power as (0, 0, -1/2), which clearly ought have
  a power equalling 1/2.

 My previous suggestion of calculating the power of a voter as the sum
 of the absolute values of the weights of the voter's votes is a bad
 idea. (This could be applied to an enhanced-STV election with negative
 ballot counts).

I am not aware of a derivation of the definition of Power, here.

Here is an example, of the Power definition applied to the Approval Voting
 method, with a finding of the "hump-shaped-curve" that voters under that
 election would want to know something about.

 If there are n candidates and the voter casts k sub-votes/approvals,
  [each having weight 1], then:
   V(i) = (1 if i<=k else 0), k=0,1,2,3,...,n.  V=(1,1,...,1,0,0,,...)
   Power = Min( (Sum i=1..n)[1-V(i)], (Sum i=1..n)[V(i)-0]) )
         = Min(n-k, k)

If anybody has another suggestion of power, something for fixing the
 problems with Borda weights, then please note it.


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At 07:25 04.03.00 , Bart Ingles wrote:
>Craig Carey wrote:
>> >Bart Ingles:
...
>> When there are 4 candidates, then specifying the last preference can alter
>>  the set of winners.
>
>
>Using that argument, what system doesn't fail?  With approval voting,
>voting for all candidates is a form of abstention.  If you abstain under
>STV, can't that alter the set of winners?
...

If there are n candidates, P2({n-1}) prohibits that odd behaviour with the
 very last preference. It is a problem absent in the Borda method?.
 Certianly STV/AV & SNTV/FPTP select the same winners when the very last
 preference is entered. If there is a problem there, there is more likely
 to be problems throughout the voting method.