Problems with quotas
-
We've discussed a few problems with PR methods that use quotas before (e.g. here). They tend to fail Independence of Irrelevant Ballots as despite these ballots not expressing any preference between the sets that might win, adding these ballots changes the quota size, which has knock-on effects. Also the quota size can be seen as arbitrary, particularly with cardinal ballots where any number of candidates can be top-rated, as any number of candidates can reach a full quota of votes. This can be considerably more or less than the number of candidates to be elected.
Anyway, that's just the introduction. I have a specific example to discuss. In a subtractive quota method (where candidates are elected and a quota of votes is then removed), parties can gain more than their fair share of seats by getting each of their voters to vote for just one of their candidates, which wouldn't happen in an e.g. D'Hondt party list election or Proportional Approval Voting with D'Hondt divisors. With Sainte-Laguë party list or PAV, there can be a splitting incentive, but it's limited far more than with subtractive quota methods, where very extreme examples can be contrived.
In the following example there are 4 seats. It doesn't really matter which quota is used - it works for any fixed quota - so I'll just use Hare. There are 12n voters, which makes the quota 3n, and the votes are as follows:
3n: A1, A2, A3
n: B1
n: B2
n: B3
6n: Assorted other candidates, none of which get enough votes to be elected.Let's say A1 is elected first. That uses up the entire A vote. All the other seats then go to B candidates, so a 3:1 ratio despite there being a 50:50 split between A and B voters. This example can be made as extreme as you like in terms of the A:B seat ratio.
I think all subtractive fixed quota methods are vulnerable to this. This includes Allocated Score, Sequentially Spent Score, Threshold Equal Approval and the Method of Equal Shares.
This problem might be averted or at least mitigated using something like Sequentially Shrinking Quota rather than a fixed quota, but I'm not sure exactly how all the methods would interact with this.
-
Just to add to this - Sequentially Shrinking Quota doesn't fully optimise the quota because it always maintains the same order of election of the candidates. Whereas if you knew the "optimum" quota from the start, it might elect in a different order and end up with different candidates.
But finding the optimum quota and using it from the start means you just end up with Phragmén. That is to say that subtractive quota methods are all essentially just approximations of Phragmen's voting rules.
-
Just to clarify on the Phragmén thing:
If you just have a fixed quota then the voters that get their candidates early can get a bad deal because they pay a whole quota, whereas later on, the might not be a candidate with a whole quota of votes and yet you have to elect one anyway, so the voters of this candidate get their candidate more "cheaply".
So you might then look for a quota that distributes the cost more evenly, and that's all Phragmén really does. It distributes the load or cost across the voters as evenly as it can.
I see subtractive fixed-quota methods as a cheap hack really.