Sequential proportional approval voting
Sequential proportional approval voting (SPAV) or reweighted approval voting (RAV) is an electoral system that extends the concept of approval voting to a multiple winner election. It is a simplified version of proportional approval voting. Proposed by Danish statistician Thorvald N. Thiele in the early 1900s, it was used (with adaptations for party lists) in Sweden for a short period after 1909. Description. This system converts Approval Voting into a multi-round rule, selecting a candidate in each round and then reweighing the approvals for the subsequent rounds. The first candidate elected is the Approval winner  ("w"). The value of all ballots that approve of "w" are reduced in value from 1 to 1/2 and the approval scores recalculated. Next, the unelected candidate who has the highest approval score is elected ("w"). Then the value of ballots that approve of both "w" and "w" are reduced in value to 1/3, and the value of all ballots that approve of either "w" or "w" but not both are reduced in value to 1/2. At each stage, the unelected candidate with the highest approval score is elected. Then the value of each voter’s ballot is set at  1/(1+m) where "m" is the number of candidates approved on that ballot who were already elected, until the required number of candidates is elected. This reweighting is based on the D'Hondt method (Jefferson method). The system disadvantages minority groups who share some preferences with the majority. In terms of tactical voting, it is therefore highly desirable to withhold approval from candidates who are likely to be elected in any case, as with cumulative voting and the single non-transferable vote. It is however a much computationally simpler algorithm than proportional approval voting, permitting votes to be counted either by hand  or by computer, rather than requiring a computer to determine the outcome of all but the simplest elections. Example. For this example, there is an election for a committee with 3 winners. There are six candidates from two main parties: A, B, and C from one party, and X, Y, and Z from another party. About 2/3 of the voters support the first party, and the other roughly 1/3 of the voters support the second party. Each voter casts their vote by selecting the candidates they support. The following table shows the results of the votes. Each row starts by  saying how many voters voted in that way and marks each candidate that group of voters supported. The bottom row shows the number of votes each candidate received. Because Candidate C has the most support, they are the first winner, "w", and their vote is not counted in later rounds. For the second round, anyone who voted for Candidate C has their vote counted as only 1/2. Below is the chart for round 2. A second column on the left has been added to indicate the weight of each ballot. Despite Candidates A and B having so many votes in  the first round, Candidate X is the second winner, "w", because not as many of the votes for Candidate X were halved. In round 3, anyone who voted for either Candidates C or X has their vote count 1/2, and anyone who voted for both has their vote count 1/3. If anyone had voted for neither, their vote would remain at 1. Below is that table. Candidate B is the third and final winner, "w". The final result has 2/3 winners from the party that had about 2/3 of the votes, and 1/3 winner from the party that had about  1/3 of the votes. If approval voting had been used instead, the final committee would be all three candidates from the first party, as they had the highest three vote totals without scaling. Properties. Sequential-PAV satisfies the fairness property called justified representation whenever the committee size is at most 5, but might violate it when the committee size is at least 6.