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

[Kristofer Munsterhjelm]

On 06/29/2013 01:27 AM, Vidar Wahlberg wrote:
> On Fri, Jun 28, 2013 at 03:04:13PM +0200, Vidar Wahlberg wrote:
>> This gave me an idea.
>> We seem to agree that it's notably the exclusion part that may end up
>> excluding a party that is preferred by many, but just isn't their first
>> preference.
>> I'm sticking to quota election because I don't fully grasp how to apply
>> other methods (Sainte-Laguë, for instance) to determine when to start
>> excluding parties.
>> 1. Give seats to parties exceeding the quota (seats = votes / quota)
>> 2. Create an ordered list using Ranked Pairs/Beatpath, exclude the least
>>     preferred party and redistribute its votes. Repeat.
>
> Chris, Kristofer.
> Spending the rest of the day on this, I think I finally understood what
> you meant with "best formula for apportioning seats in List PR". Or at
> least I eventually came up with a very simple method, even though it
> does not meet my concerns about excluding a second preference party that
> is far more popular than a party that have some more first preference
> voters.
> For larger parties who are very likely to get a seat there's neither any
> reason to create an ordered list, as those parties who do receive one or
> more seats will never have any votes transfered.
>
> Basically, this is what I do:
> 1. Distribute seats using Sainte-Laguë.
> 2. If any parties received no seats, exclude the party with least votes
>     and redistribute votes to 2nd preference.
> 3. Repeat 1-2 until all non-excluded parties got at least 1 seat.
>
> Although as noted a party that is a popular as second preference (but
> less popular as first preference) will easily be excluded, even though
> more voters would prefer this party over another party.

I think that when the number of seats is large enough, you could combine 
the two methods. That is, by combining them, you handle the problem 
arising from voters not having an influence, but the problem arising 
from the method not becoming Condorcet-like when there are few seats 
remains.

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.

For a similar reason, it is not perfect: if the second preference party 
has widespread support but is hidden behind many parties that get one 
seat each, then the council will fill up with the smaller parties and 
the second preference party never gets a shot. But in a sense, that is 
proportional: every voter is represented. The question is how much 
second preferences should override first preferences. I think that an 
answer to that, and implementation thereof, would also fix the L-C-R 
problem, because they're two aspects of the same thing.

(And good luck explaining the purpose of the elimination order, and why 
it should be determined by Condorcet, to the average voter!)