Quota Borda system

QBS is a multiwinner generalization of the Borda count designed to pass PSC. It works by assigning seats based on how many quotas (usually Droop) candidates or groups of candidates have of top preferences, and elects among them according to their Borda scores.

Notes

QBS can deny representation to minority groups that support irrelevant alternatives. 3-winner example:

1: c1, c2, c3, e1, e2, e3, e4, d1

1: c2, c3, c1, e1, e2, e3, e4, d1

1: c3, c1, d1, c2, e1, e2, e3, e4

6: e1, e2, e3, e4, c1, c2, c3, d1

4 : e1, e2, e3, e4, c1, c2, c3, d1

5 : e1, e2, e3, e4, c1, c2, c3, d1

6 : e1, e2, e3, e4, c1, c2, c3, d1

7 : e1, e2, e3, e4, c1, c2, c3, d1

8 : e1, e2, e3, e4, c1, c2, c3, d1

9 : e1, e2, e3, e4, c1, c2, c3, d1

Had d1 not run, there would've been a Droop solid coalition for c1-3, guaranteeing one of them a seat. But instead, e1-3 all win. This is an example where the Expanding Approvals Rule and STV, two other common PSC-compliant methods, would elect one of c1-3.