Best cardinal PR method(s)?
I thought I might revisit this general topic to see where people's thoughts are.
After a lot of deliberation Allocated Score was chosen as the PR method for STAR voting by the Equal Vote committee. I happened to be looking at the EndFPTP Reddit and found this post about a potentially better version of it. By the way, I was involved in some of the discussions that led to Allocated Score being selected, but I was not really central to the discussions and largely took a back seat.
But anyway, my thinking since then is that all methods that elect a candidate and then remove a quota of votes are pretty crude, and there are better methods. I discussed the problems with quotas in this thread. As I argued in that thread, removing quotas in this way is just a way of approximating what Phragmén-based methods do. So really they are all inferior to Phragmén. You then just have to decide how to convert approval Phragmén to score, e.g. by using the KP-transformation. Phragmén can be done sequentially to make it computationally realistic.
But then Phragmén-based methods have their own problems, like being only weakly monotonic. Thiele Proportional Approval Voting is strongly monotonic and also has the advantage of passing Independence of Irrelevant Ballots (IIB) but it's not exactly proportional, and not just because of rounding making an exact result impossible. It also fails the Universally Liked Candidate criterion (ULC).
Then you have COWPEA Lottery. As its name suggests, it is not deterministic, but this can be seen as good or bad depending on your perspective. One advantage of this is that it gives better proportionality across all the constituencies as whole rather than being limited by the constituency size of e.g. 5 or 6 seats. It also has the advantage of being strongly monotonic and it passes IIB and ULC. It is also much simpler to calculate results than the other methods. Determinism aside, it easily has the best criterion compliance.
In any case, I see no reason to advocate for sequential quota-dumping methods any more.
A lot of tumbleweed floating about here. I didn't just come here to post that though. I meant to mention Monroe as well. But I see it as a lesser version of Phragmén. Fairly similar, except that Monroe reduces to Hamilton for party voting, whereas Phragmén reduces to D'Hondt or Sainte-Laguë, so Monroe has a worse IIB failure.
There's also a discussion of different PR philosophies on the wiki. It doesn't really cover all the ground though, as COWPEA or COWPEA Lottery wouldn't be included, for example. But perhaps non-deterministic methods deserve their own entry on there.
Optimised PAV Lottery is another non-deterministic method. In this you work out the optimum amount of weight each candidate should have (by e.g. infinitely cloning all candidates and running an election with a very large number of seats), and then elect candidates probabilistically according to these weights. (Though you would have to work out the distribution again every time a candidate is elected). Unlike deterministic PAV, it is thought (though not known) to be proportional by passing the Perfect Representation In the Limit criterion.
I also think that sequential methods generally fail participation (for a suitable multi-winner definition), whereas optimal elect-all-at-once methods are computationally infeasible. However, I think non-deterministic sequential methods can get around this failure. Optimised PAV Lottery is computationally infeasible anyway, but COWPEA Lottery is easily runnable.