Do cardinal PR methods pass Droop PR when Droop quotas max-score as well as bullet vote?

In the single-winner case, any cardinal PR method which reduces to Score passes Droop PR when a Droop Quota (majority) bullet votes (and in fact, guarantees the Droop Quota representation if they max-score a set of candidates and min-score all others, so basically Droop PSC i.e. mutual majority), so I’m wondering why this wouldn’t be the case in the multi-winner case, if it isn’t. If it is, why does everyone prefer to only say that they pass Hare PR under “max-score bullet voting”; it’d seem that Droop PR is an even stronger criterion and builds connections between ranked and rated PR methods (namely, that Droop Quotas are guaranteed their preference in part because a) these methods treat all voters equally and b) ranked PR methods generally treat voters’ highest preferences as maximally preferred while rated PR methods do the same if the voters themselves vote that way)

Also, can the Hare Quota Criterion be generalized and modified to be " Whenever more than a (k) Hare Quota(s) of the voters gives max support to a single candidate (set of candidates) and min support to every other candidate, that (k of that set of) candidate(s) is (are) guaranteed to win regardless of how any of the other voters vote." ?

I think proportionality for solid coalitions is a more general thing than Droop proportionality criterion.

As for how you would define the Hare proportionality criterion, I think what I posted here might be right.

Just to finish the thought on this, under Sainte-Laguë/Hare proportionality, for a party/faction to guarantee themselves s seats, then I think they would need s-1 Hare quotas and then a Droop quota for their last seat, but it would be a Droop quota considering only the remaining seats and voters rather than all of them.

For example, let’s say there are five seats.

To guarantee one seat, a party would need 0 Hare quotas and one Droop, so 1/6 of the total vote.

To guarantee two seats, a party would need 1 Hare quota and one Droop quota of four seats, so 1/5 + 4/5*1/5 = 9/25 or 0.36 of the vote

To guarantee three seats, they would need 2 Hare quotas and one Droop quota of three seats, so 2/5 + 3/5*1/4 = 11/20 or 0.55 of the vote.

To guarantee four seats, they’d need 3 Hare quotas and one Droop of two seats, so 3/5 + 2/5*1/3 = 11/15 or 0.73 of the vote.

To guarantee five seats, they’d need 4 Hare quotas and one Droop of one seats, so 4/5 + 1/5*1/2 = 9/10 = 0.9 of the vote.

I think this is correct now. According to the Wikipedia https://en.wikipedia.org/wiki/Proportionality_for_Solid_Coalitions the Hare version of proportionality for solid coalitions requires a full Hare quota for each seat, but I think this is a more sophisticated definition.

As for which is better, it depends on your viewpoint. Take the following approval ballots with 2 to elect:

6: A1, A2
2: B

Under Droop, A gets both seats. But under Hare, it would be a tie because both parties have exactly half the support for the final seat. Which seems the better result? Obviously the A faction can split themselves to guarantee both seats though:

3: A1
3: A2
2: B

Party B needs a third of the vote (well just over) to guarantee a seat under Droop, so 2/3 of the seat’s proportion of the total number of seats. Under Hare it’s just a quarter of the total vote (with just the two parties anyway), so half the seat’s proportion.

But Droop arguably looks more unfair to the smaller party with more seats. If there are 10 seats, the smaller part of the two would need 10/11 (over 90%) of the seat’s proportion to be guaranteed to get it. On the other hand the larger party only needs 10/11 of the total vote, so that’s 9 full seats’ worth plus just 1/11 towards the final seat. This is quite a big difference - the smaller party needs 5% of the total vote under Hare (assuming two parties), and 9.1% under Droop.

Really it depends on whether you see a party as having to “use up” a whole seat once they take it, or just a Droop quota of it and still have the rest of the vote for use elsewhere.

So this might be the reason cardinal PR methods with Droop Quotas acting maximally strategic pass Droop PR in the single-winner case but only Hare PR in the multi-winner case, if that’s the case; in the single-winner case, a strategic Droop Quota would need 0 Hare Quotas and 1 DQ there, which is just a majority of votes (but really, if we consider an HB quota, half of the votes, that guarantees at least a tie).

It seems you are suggesting that if one wishes to use Droop Quotas with cardinal PR methods, the selection of winners should be based on Hare Quotas but the spending based on Droop Quotas. In other words, with something like Sequential Monroe Voting a candidate’s worthiness to be elected is based on their most-supporting Hare Quota, but once elected, a Droop Quota’s worth of that Hare Quota is spent instead, correct?

Well, I wasn’t really suggesting anything in particular, just putting a few ideas out there.