This is my idea how approval ballots could be converted into proportionally allocated seats.
- Define influence of candicate x as the expected value of number of winning candidates for which a randomly chosen voter that approves the winning candidate also approves x.
- Elect a candidate with the highest number of approvals per unit of influence, where influence is calculated as if that candidate was already elected.
- Repeat until all the seats are filled.
The formula for the quotient between approval ratio and influence can be written as
where P stands for probability and Ax is the event that a randomly chosen voter approves x. In each turn, the candidate with the highest quotient is elected.
This method is equivalent to SPAV when voters vote along party lines, but doesn’t disadvantage overlapping voter groups.
It’s also possible to use it with score ballots. In the score-based variant, voters would be able to state a probability with which they approve each candidate.