Sequentially Shrinking Quota
Sequentially Shrinking Quota (SSQ) is a sequential Multi-Winner Cardinal voting system built on Score voting ballots. It is a modification to Sequentially Spent Score designed to limit cases of free-riding. [1]
Definition
All ballots initially start as having weight 1, and the quota is initially set at the Hare quota.
The score each ballot contributes to a candidate in a particular round is equal to the weight times the score given for the "weighted" variant, and min{Max Score*weight, score given} for the capped variant. If there is a candidate (or multiple candidates) who exceeds the quota, then the candidate with the highest score in this round is elected, and the reweighting step is applied. If no candidate meets the quota, then a smaller quota size is chosen so that if all of the previous reweighting steps are recalculated for this smaller quota size (maintaining the prior order of election), then one candidate will be able to meet the new quota. This candidate is selected next.
Reweighting Step
Each round, a ballot loses weight equal to (Score contributed to the candidate elected this round / Max score) * (Quota / Total Score for that candidate this round).
Notes
SSQ decays into D'Hondt in the party list case. This is because it essentially searches for the highest divisor that each party can pay for its seats. 2-seat party list example:
30 A 14 B
The Hare quota is 22 ((30+14=44)/2). A has the most votes and gets the first seat, and then is left with 30-22=8 votes. So it becomes
8 A 14 B
Since no party can get 22 votes now, we look for the largest quota that anyone can pay now if earlier winners (the one A candidate) had paid that same amount. We end up finding that if the quota is shrunken to 15:
15 A (8 original votes + (22 (old quota) - 15 (shrunken quota) = 7 restored votes)) 14 B
A has the most votes, so they take the second seat, resulting in a winner set of (2 A) i.e. (A, A). This is a notable contrast to most Hare quota-based methods that reduce to largest remainder methods, such as regular Sequentially Spent Score, which would've stopped at the second round, seen that B had 14 votes to A's 8, and thus elected (A, B).