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 give s 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." ?