With the current 13 ballots, pure approval picks the top 4 without incident, but then hits a 5-way tie for the 5th seat. The 4 alternatives I mentioned again all return a unique winning set of 5, and - incidentally - Satisfaction Approval Voting (SAV) also returns that set.

I’ll briefly explain what those alternative are. Contrast, e.g., with the brain-bustingly “huh?” description of CIVS’s proportional Condorcet schemes.

The approval schemes I’ll name PAV-NUMBER and SPAV-NUMBER.

PAV is Proportional Approval Voting and SPAV Sequential Proportional Approval Voting. They’re all deeply related.

SPAV is probably easier to understand at first:

```
while a seat remains to be filled:
pick a highest-scoring unpicked candidate
```

The score of a candidate is the sum of that candidate’s current approvals, but from weighted ballots. When a ballot has `w`

approvals that have already won, the ballot’s weight falls to `1/(1+h*w)`

where `h`

is a global constant. So all ballots start at weight 1 (`w=0`

at first), and the first winner picked is the same as the top scorer under pure approval voting. The weights of the ballots that approved the first winner fall to `1/(1+h)`

, and the other ballots remain at weight 1. The more winners a voter gets, the less influence that voter’s ballot has on remaining choices. That’s all there is to it.

`h=0`

is the same as pure approval voting (ballot weights remain 1 forever).

SPAV-1 picks `h=1`

(“Jefferson”) and SPAV-2 `h=2`

(“Webster”). Weights fall faster under Webster:

Jefferson: 1, 1/2, 1/3, 1/4, …

Webster: 1, 1/3, 1/5, 1/7, …

A problem with sequential schemes is that the way ties are broken along the way can have major effects on the outcome. For example, if one voter approves of A & B, and another of C & D, none have been picked yet, and A & C tie for highest score, what to do? If A is picked, B’s chances fall (because the ballot’s weight falls), while if C is picked, D’s chances fall.

So PAV-1 and PAV-2 are not sequential, but are computationally much more demanding. Instead each possible set of winners is given a score, and a set with maximal score is picked. The score of a set of winners is the sum over all ballots of the sum of the ballot’s weights “as if” each candidate the ballot approved from the winning set had been picked one at a time.

For example, if a ballot approved 3 winners, it would contribute `1 + 1/(1+h) + 1/(1+2*h)`

to the set’s score.

SPAV-n can viewed as a computationally efficient “greedy” approximation to PAV-n.

An interesting thing about the current 13 ballots: PAV-h and SPAV-h pick the same set for *any* `h`

greater than 0 (note `h`

is a non-negative real number here, not necessarily an integer). So “the math” detects a faction clear enough that the slightest concession to proportionality resolves pure-approval’s 5-way tie in favor of a unique winner.