The 3 main cardinal PR proposals (SSS, SMV, RRV) all fail Hare-PSC, a criterion for PR methods stating that a group of voters comprising k Hare quotas who prefer a set of candidates above all others should always elect at least k of those candidates. Hare-PSC seems like an elephant in the room when discussing which cardinal PR proposals are best.
Something to consider is that if the Hare quota is 10 and some voters vote:
This is not technically a solid coalition, yet arguably A deserves a guarantee of winning just as much as they would if D wasn’t running.
Some ideas for implementing Hare-PSC:
Before running a cardinal PR method, first identify all solid coalitions, then use combinatorics to decide which candidates must win. Use the cardinal PR method’s selection method to decide which candidate within the PSC-guaranteed set is best, elect them, reweight their supporters’ ballots, then recalculate the PSC set of candidates and repeat. Once all PSC guarantees are fulfilled, only then run the method as normal.
When running a cardinal PR method, at each stage eliminate all candidates whose election would force violation of PSC. (I think this doesn’t cause problems with cardinal methods since they pass IIA with unchanged ballots, but not sure.)