[EM] PR-STV with approval based elimination

[Dan Bishop]

Kristofer Munsterhjelm wrote:
> The problem in using any other method than Plurality for counting the 
> votes in an STV method is that of elimination. If someone votes for A 
> with strength 1 and B with strength 0.5, and A wins, then how much do 
> you downweight that voter? The A voters should be reweighted so that 
> the surplus is distributed evenly, then A is eliminated, but if this 
> causes score changes, then the effective voting power is no longer 
> equal to the surplus.
>
> I ran into that while trying to generalize STV to work with Borda (and 
> more generally, any sort of weighted positional system). Approval 
> without rescaling might dodge it (since the voting power doesn't 
> change by removing a candidate), but Range with rescaling wouldn't. 
You could use Plurality (with vote-splitting between equally ranked 
candidates) to determine surpluses and a different method to determine 
eliminations.  For example,

3 seats, Hare quota (100 votes)

100: Escher>Andre=Bush=Gore=Nader
110: Andre>Nader>Gore>Bush=Escher
 18: Nader>Gore>Andre=Bush=Escher
 21: Gore>Nader>Andre=Bush=Escher
  6: Gore>Bush>Andre=Escher=Nader
 45: Bush>Gore>Andre=Escher=Nader

After dealing with surpluses, we have:

28: Nader>Gore>Bush
21: Gore>Nader>Bush
  6: Gore>Bush>Nader
45: Bush>Gore>Nader

Using Plurality for elimination would eliminate Gore.  BUT if we instead 
used the Borda count, the scores are:

Bush: 96
Gore: 127
Nader: 77

So *Nader* gets eliminated.  In the final round, we have

Bush: 45
Gore: 55

So the winning set is {Andre, Escher, Gore}.  Coincidentally, the same 
as the CPO-STV result.