Is the quota rule broken?

The quota rule sounds reasonable at first glance, but the more you examine it, the more unreasonable it appears. First, the quota rule is known to be incompatible with two criteria:
 Population monotonicity
 Independence of irrelevant ballots
The latter criterion is most revealing, because the quota rule requires voting methods to try to represent "unrepresentable" voters, and punishes the largest factions when unrepresentable voters are present. By unrepresentable voters, I mean voters who vote for very small parties that aren't big enough to win even 1 seat. The quota rule counts these voters as part of the denominator, meaning that the larger parties have lower quotas, get fewer seats, and some of the smaller parties need to be bumped up from 0 to 1 seat to compensate. Note that the unrepresentable voters aren't voting for these smaller parties that are getting seats, they're voting for even smaller parties and just warping the entire election by changing the denominator of all the quotas. If there are a whole lot of unrepresentable voters, the quota rule tends towards ChamberlinCourantlike behavior, where the top n factions all get 1 seat, because their quotas are so low that 1 is the upper bound for all of them.
I don't think the quota rule should be placed on a pedestal, and I don't think methods should be bashed for violating the quota rule if they pass independence of irrelevant ballots and population monotonicity. PAV and SeqPAV do this, as well as D'Hondt party lists  they basically ignore the unrepresentable voters because of how the math works.

@isocratia I'd agree that a method that obeys the quota rule that a party or faction must get their "correct" number of seats rounded up or down. So if a party is due 9.4 it must get exactly 9 or 10. I prefer methods that reduce to highest averages in the partylist case, so D'Hondt or SainteLaguë. So there certainly should not be an upper quota limit. SainteLaguë can break lower quota in some cases as well.

@tobypereira
Does the method Sequentially Shrinking Quota obey independence of irrelevant ballots? I just found out about it yesterday, it's based on a quota but the article says it reduces to the D'Hondt method with partybloc voters. The shrinking quota might correct for irrelevant ballots, but I'm not sure. 
@isocratia said in Is the quota rule broken?:
Does the method Sequentially Shrinking Quota obey independence of irrelevant ballots?
nope

@andydienes said in Is the quota rule broken?:
@isocratia said in Is the quota rule broken?:
Does the method Sequentially Shrinking Quota obey independence of irrelevant ballots?
nope
Interesting. But I'd say there are two main ways to fail IIB. One is with "full" ballots and one with "empty" ballots. Some methods won't be affected by a load of ballots approving noone or just approving unelectable candidates. But the harder one to pass is by adding ballots that approve all the candidates. Does SSQ fail both do you know?

@tobypereira I don't have examples on hand but I imagine it fails both. When you add ballots you change the amount that the first block of voters get reweighted by (but the winner is the same). shouldn't be too hard to construct scenarios where that has downstream effects.

I just came up with an alternative quota rule which might avoid these problems with irrelevant ballots and also be passed by methods that satisfy independence of irrelevant ballots. There are 2 new concepts:
 The quota denominator starts at 1 and increases until it reaches a stopping point
 There is an "excess", i.e. the current denominator can require a seat allocation that is higher than the number of seats you are trying to fill
At the start, the denominator is 1 so the excess will be massive. As the denominator increases, the excess decreases until it eventually reaches 0  that is, you find quotas that are possible to allocate seats for. Then you stop increasing the denominator. Those are the quotas.
Notice that the irrelevant ballot problem is avoided because the denominator has been abstracted away from the total number of ballots.
I also think that this new quota rule doesn't need upper bounds because there are implicit upper bounds created by the lower bounds of all the other parties.

@isocratia Can you give an example of how this would work with a particular set of ballots?

@tobypereira
Well I'd have to write a program to generate examples because I did a trivial example with 10 seats, 60 voters for Party A, 30 voters for Party B, and 10 voters for Party C, and even that was tedious to do by hand, even though it seemed to work correctly.