Proportionality criteria for approval methods
There was some off topic discussion in this thread but I want to expand on it.
This paper discusses a lot of the approval proportionality criteria that have come up over the years. And there are a lot of them. For example, there's Justified Representation (JR), Fully Justified Representation (FJR), Extended Justified Representation (EJR), Proportional Justified Representation (PJR), Laminar Proportionality, Priceability, Stable Priceability, Perfect Representation (PR), Core Stability. And probably some I've missed.
That's a lot of proportionality criteria. But the question is whether we need that many and whether they're all useful. If I want to know if a particular approval method is "proportional", I don't want to have to check it against 10 different criteria and then weigh them all up.
Well actually, I don't really think any of them really capture the essence of proportionality very well. On page 56 in that paper, there's a chart showing which criteria imply which others. And as you can see, almost all of them imply lower quota. According to lower quota, under party voting, no party can receive less than the proportion of seats they are due, rounded down to the nearest integer. However, as can be seen here, Sainte-Laguë/Webster fail lower quota, and are seen by many as the most accurately proportional methods. I don't think proportionality criteria that imply lower quota are fit for purpose.
Of the remaining criteria, I think Justified Representation is too weak to be worth anything. Perfect Representation, however, is too strong, but I think it makes a good base for a criterion. According to the wiki:
if there is a possible election result where candidates could each be assigned an equal number of voters where each voter has approved their assigned candidate and no voter is left without a candidate, then for a method to pass the perfect representation criterion, such a result must be the actual result.
I would say the main reason it is too strong is that it is incompatible with strong monotonicity. Consider the following ballots:
x voters: A, B, C
x voters: A, B, D
1 voter: C
1 voter: D
With 2 to elect, a method passing Perfect Representation must elect CD regardless of the value of x. There could be almost unanimous support for both A and B, but CD (with half the votes each) would still be elected.
As the number of elected candidates increases, then for v voters, in the limit each voter should be able to be uniquely assigned to ¹⁄ᵥ of the representation, approved by them, as long as it is possible from the ballot profile."
As I explained in the paper:
The common thread among proportionality criteria is the notion that a faction that comprises a particular proportion of the electorate should be able to dictate the make-up of that same proportion of the elected body. But this can be subject to rounding and there can be disagreement as to what is reasonable when some sort of rounding is necessary. However, taken to its logical conclusions, each voter individually can be seen as a faction of ¹⁄ᵥ of the electorate for v voters.
I also say that any deterministic method should obey Perfect Representation when Candidates Equals Voters (PR-CEV):
For a deterministic approval method where a fixed number of candidates are elected, a stronger proportionality criterion is Perfect Representation when Candidates Equals Voters (PR-CEV): if the number of elected candidates is equal to the number of voters (v), then it must be possible for each voter to be assigned to a unique candidate that they approved, as long as it is possible from the ballot profile. This is because no compromise due to rounding is necessary at that point.
One other thing I explained about PRIL, in case it is considered too weak for any reason:
One potential downside is that it does not define anything about the route to Perfect Representation, other than that it must be reached in the limit as the number of candidates increases. However, in that respect it has similarities with Independence of Clones, which is a well-established criterion. Candidates are only considered clones if they are approved on exactly the same ballots (or ranked consecutively for ranked-ballot methods). We would also want a method passing Independence of Clones to behave in a sensible manner with near clones, but it is generally trusted that unless a method has been heavily contrived then it would do this. Similarly, one would expect the route to Perfect Representation in a method passing PRIL to be a smooth and sensible one unless a method is heavily contrived, and none of the methods considered in this paper are contrived in such a manner.
So this is why I consider PRIL to be the standard proportionality criterion for approval methods. Any deterministic method should also pass PR-CEV.