Proportional approval voting

Proportional approval voting is a voting system that adapts approval voting to seek proportional representation in multi-winner election.
One form is as follows. Voters vote the same way as in normal approval voting. Define the satisfaction of each voter as (1 + 1/2 + 1/3 + ... + 1/n), where n is the number of candidates who are approved by the voter and elected. The election result is the combination of candidates that maximizes the sum of satisfactions.
Exact optimization requires computers, but an approximation can be achieved by hand. For each seat, elect the candidate with the most votes (approvals). Then re-weight each ballot, giving it a value of 1/(n+1), where n is the number of candidates already elected and approved on that ballot, and recount. Repeat until all seats are filled.

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