[EM] Preferential voting system where a candidate may win multiple seats

[Kristofer Munsterhjelm]

On 06/30/2013 03:02 AM, Chris Benham wrote:
> **
> Kristofer Munsterhjelm**wrote (29 June 2013):
>> "The combined method would go like this:
>>
>> 1. Run the ballots through RP (or Schulze, etc). Reverse the outcome
>> ordering (or the ballots; these systems are reversal symmetric so it
>> doesn't matter). Call the result the elimination order.
>> 2. Distribute seats using Sainte-Laguë.
>> 3. Call parties that receive no seats "unrepresented". If there are
>> unrepresented parties, remove the unrepresented party that is listed
>> first in the elimination order.
>> 4. Go to 2 until no party is unrepresented.
>>
>> This should help preserve parties that are popular as second preferences
>> but not as first preferences, because the elimination order will remove
>> parties that hide the second preferences before it removes the party
>> that is being hidden, thus letting the second-preference party grow in
>> support before it is at risk of being eliminated.
>>
>> Note that this doesn't solve the small-council problem. If we have:
>>
>> 46: L > C > R
>> 44: R > C > L
>> 10: C > R > L
>>
>> 1 seat,
>>
>> then the first seat goes to L just like in Plurality. The elimination
>> order never enters the picture."

> Kristopher,
> I don't see this. Your elimination order is obviously L, R,C.  R and C
> are "unrepresented" so we eliminate R.
> Then we have
> 46: L
> 54: C
> Then we redistribute the seat to C and then eliminate L and confirm the
> final redistribution.

Ah, right. I erred there; good call.

> But I'm not on board with the spirit of this method, because it seems to
> give a say to voters who are efficiently represented a say in which
> party/candidate will represent other voters.

Well, in one sense the problem is that the parties that have no seats 
are fragmented. So replacing each of the fragmented parties with a 
larger party that is closer to the center would help with that.

I can see your objection, though, because there is an element of 
majority rule. If the majority prefers left-wing to right-wing, and 
there are right-wing voters who are unrepresented and vote for 
right-wing parties from most extreme to least, then those voters would 
prefer to get a party as right-wing as possible into the assembly, but 
the elimination order will preferentially preserve left-wing parties.

Would you suggest that the elimination ordering only be calculated based 
on the votes of those who currently don't get any representation?

I suppose that could work, although that may also introduce some path 
dependence. But it doesn't handle the example very well anymore:

46: L > C > R
44: R > C > L
10: C > R > L

As you noted, R and C are unrepresented. So the RP ordering among the R- 
and C-voters is R > C > L (by Majority), hence the elimination ordering 
is L > C > R.

Now we have two options. Either we can eliminate L - which will give the 
right result but override all the L-voters - or we can eliminate "of the 
unrepresented, the one first in the elimination ranking", which is C. If 
we go by the second path, then we have

46: L
54: R

so L is eliminated anyway and R wins, which seems to be a more IRV-like 
outcome.

The problem here seems to be that the L-voters *become* unrepresented. 
What we really have are competing desires, and a sequential elimination 
process will favor those desires in a particular order. If I'm right, 
then we have a situation kind of like solving simultaneous equations. 
Thus it could be solved by an iterative method - in this case, noticing 
L would be eliminated anyway and thus eliminating L ahead of time to 
make C win - or by something that doesn't use eliminations at all, such 
as some analog of Schulze STV.

But either suggestion will make the method a lot more complex. It seems 
that PR methods get really complex really quickly as one places 
additional demands on them.