[EM] STV and weighted positional methods

[Kristofer Munsterhjelm]

Kathy Dopp wrote:
> James,
> 
> Thanks but I don't need to read any references, the arithmetic is obvious.
> 
> In Borda there can be more than one candidate with majority approval
> and the candidate with the *most* majority approval may not be the
> plurality majority winner like it would be in a first round majority
> winner in IRV.
> 
> Apparently your definition of majority winner, only includes first
> choice winners, and that's OK, but then an obvious consequence of your
> definition is that it compels you to admit that IRV finds "majority"
> winners far *less* often than a primary/general election or esp. lots
> less often than top-two runoff.
> 
> I mean *let's get real* and start telling it like it is.

I think what he's talking about is that Borda can fail to discover a 
true majority. Consider this Wikipedia example, for instance:

51: A > C > B > D
  5: C > B > D > A
23: B > C > D > A
21: D > C > B > A

100 voters, so A has a majority outright. But the Borda scores are (when 
0-based):
A: 51 * 3 = 153
B: 51 + 5 * 2 + 23 * 3 + 21 = 151
C: 51 * 2 + 5 * 3+ 23 * 2 + 21 * 2 = 205
D: 5 + 23 + 21*3 = 91

so C wins.

Borda can still be interesting, because its single winner elimination 
versions (Nanson and Baldwin) pick from the Mutual majority set. The 
multiwinner equivalent of mutual majority is Droop proportionality, so 
it would be of interest to see if Borda-"STV" would be proportional. I 
don't think it would be (given my Left-Right-Center example in another 
post), but how far does it fall from the mark, and why does plain old 
STV pass it in that case, given that Plurality itself doesn't pass 
mutual majority?

Of course, if I could find a reweighting scheme that works for any WPS, 
the generalized/altered STV doesn't have to use Borda; it can use any 
weighted positional system, and even some that are not (like Range).