@lime Cardinal PR and Holy Grail criteria is my main area of interest when it comes to voting methods, so any mention of them and I'm back in the thread.
Your criteria aren't the same that I would pick. There was also some discussion in this thread, but Sainte-Laguë/Webster does not pass core stability or priceability. I (along with many other people) would consider it to be the most accurate party PR method, so if a cardinal method reduces to Sainte-Laguë/Webster where there is party voting, it should not be disqualified.
As I understand it, D'Hondt is the only divisor method that satisfies lower quota, so therefore the only one in the running to pass core stability. As I also understand it, non-divisor methods all fail Independence of Irrelevant Ballots, so I think we would be forced into a method that reduces to D'Hondt under party voting if we also wanted to pass that, which isn't satisfactory.
It is an inconvenient truth actually that most PR criteria that have popped up in recent years for cardinal methods are failed by Sainte-Laguë/Webster, so I would argue that they are not really fit for purpose.
Single-candidate Pareto efficiency is pretty non-controversial. If candidate A is approved on all the ballots that B is and at least one other, then if B is elected so should A.
However, the multiple-candidate criterion, while intuitive, is at least debatable. Basically: for the winning set of candidates, there should not be another set for which every voter has approved at least as many elected candidates as they have in the winning set, and at least one voter has approved more. Take the following example with 500 voters and 2 to elect:
150 voters: AC
100 voters: AD
140 voters: BC
110 voters: BD
AB and CD are the only two electable sets for a PR method. In either case every voter will have one elected candidate. PAV and welfarist methods in general would regard them as the same because they just look at number of candidates elected for a voter.
However, under AB each elected candidate is has been approved by 250 voters. Under CD, it's 290 for C and 210 for D. AB is therefore more proportional than CD, and certainly the result I would prefer. But then you could have:
150 voters: AC
100 voters: AD
140 voters: BC
110 voters: BD
1 voter: C
1 voter: D
So unless AB's win over CD in the previous example was just of a tie-break nature, AB should still win here. But under the Pareto criterion above, CD must win. I regard this as unsatisfactory.
The four main criteria I look for in the Holy Grail are:
Perfect Representation in the Limit (the primary PR criterion I use, which does not disqualify Sainte-Laguë/Webster)
Strong Monotonicity (so an extra approval should count in favour of a candidate rather than merely not against them and not just as a tie-break measure)*
Independence of Irrelevant Ballots
Independence of Universally Approved Candidates
*Phragmén, while monotonic, fails strong monotonicity. E.g. two to elect:
1: AB
1: AC
It regards all results (AB, AC, BC) as equally good other than possibly in a tie-break (despite A's universal support). This makes a difference in the following example:
99: AB
99: AC
1: B
1: C
Phragmén prefers BC in this case, which does not seem right. Electing sequentially avoids this obviously, but examples can easily be found where electing sequentially does not save it from a bad result.
Anyway, my criteria may be incompatible, in a deterministic method at least. COWPEA Lottery passes them all but is non-deterministic. Optimised PAV Lottery might also pass as well, but this is unproven as far as I am aware.
Pretty much everything I have discussed here is discussed in detail in my non-peer-reviewed arXiv paper on COWPEA and COWPEA Lottery. I think it's at least worth a look. Arguably the most exciting part is the section on COWPEA v Optimised PAV as the ultimate in cardinal PR for cases where there are no limits to the number and weight of elected candidates.