Problems with quotas
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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.
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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.
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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.
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In addition to the others I've mentioned, I made a topic on Reddit about quotas, basically adding to this. I'll copy it out here but I can't be bothered to re-link the links (you can go to the post to click on them):
The crude tool that is quota-removal proportional representation
I'll be talking specifically about proportional approval methods here, but the problems exist with ranked methods too. But alternatives are easier to come by with approval methods, so there's less excuse for quota-removal methods with them.
Electing the most approved candidate, removing a quota of votes (e.g. Hare, Droop), and then electing the most approved candidate on the modified ballots (and so on) has intuitive appeal, but I think that's where the advantages end.
First of all the quota size is essentially arbitrary, particularly with cardinal or approval ballots where any number of candidates can be top-rated, and 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.
Also adding voters that don't approve any of the candidates that have a chance of being elected can change the result, giving quite a bad failure of Independence of Irrelevant Ballots (IIB), which I'd call an IIB failure with "empty" ballots. Adding ballots that approve all of the candidates in contention and changing the result is a failure of IIB with "full" ballots, but this is harder for a method to pass and not as bad anyway. It is not that hard to pass with empty ballots, but quota-removal methods do fail. I'll give an exaggerated case of where quotas can go badly wrong:
3 voters: A1; A2; A3
1 voter: B1
1 voter: B2
1 voter: B3
6 voters: Assorted other candidates, none of which get enough votes to be elected
4 candidates are to be elected. There are two main parties, A and B, but the B voters have strategically split themselves into three groups. We'll use the Hare quota, but it doesn't really matter. This example could be made to work with any quota.
With 12 voters, a Hare quota is 3 votes. 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. If the 6 "empty" ballots weren't present there would be a 50:50 A:B split.
If you have a fixed quota like this, the voters that get their candidates elected 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".
What you really want to do is look for a quota that distributes the cost more evenly, and that's essentially what Phragmén methods do. They distribute the load or cost across the voters as evenly as it can. So really quota-removal methods are just a crude approximation to Phragmén. Phragmén passes the empty ballot form of IIB and generally would give more reasonable results than quota-removal methods.
Also Thiele's Proportional Approval Voting (PAV) passes all forms of IIB, and has better monotonicity properties than Phragmén, but it is really only semi-proportional, as I discussed here, except where there are unlimited clones, or for party voting.