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. It is a special case of Thiele's voting rules, proposed by Danish statistician Thorvald N. Thiele in the early 1900s.E. Phragmén (1899): "Till frågan om en proportionell valmetod." Statsvetenskaplig tidskrifts Vol. 2, No. 2: pp 87-95 It was used in Sweden from 1909 to 1921, when it was replaced by a "party-list" style system, and is still used for some local elections.

Description Sequential proportional approval voting uses approval voting ballots to elect multiple winners on a round-by-round basis. With approval voting ballots, each voter may support any number of candidates on their ballot as they see fit. For tabulation, each ballot is weighted according to a formula, the candidate with the most support is elected, and the process is repeated until there are no more seats to fill.Steven J. Brams, D. Marc Kilgour (2009): "Satisfaction Approval Voting": p4

The aforementioned formula for each ballot in a given round is as follows: where is the number of candidates approved on that ballot who were elected in the previous rounds, and is its weight. For the first round, is naturally 0, and so each ballot has a weight of 1.

Example

As a clarifying example, consider an election for a committee with three winners. There are six candidates, representing two main parties: A, B, and C from one party, and X, Y, and Z from another party. Each voter casts their vote by selecting all the candidates they support. The following table shows the results of the votes. Each row represents a possible candidate support combination and the first column indicates how many ballots were cast with that combination. The bottom row shows the number of votes each candidate received.

+ Votes from 200 voters # of votes Candidate A Candidate B Candidate C Candidate X Candidate Y Candidate Z 112 ✓ ✓ ✓ 6 ✓ ✓ 4 ✓ ✓ ✓ ✓ 73 ✓ ✓ ✓ 4 ✓ ✓ ✓ ✓ 1 ✓ ✓ Total Votes 116 122 126 82 78 77

Because Candidate C has the most support, they are the first winner, , and they cannot win any subsequent rounds. In the second round, for any ballot where Candidate C is approved, becomes 1 in the formula, meaning the ballot's new weight equals . Below is the chart for round 2. A column has been added to indicate the weight of each set of ballots.

+ Second Round Results # of votes Weight of Vote Candidate A Candidate B Candidate C Candidate X Candidate Y Candidate Z 112 1/2 ✓ ✓ ✓ 6 1/2 ✓ ✓ 4 1/2 ✓ ✓ ✓ ✓ 73 1 ✓ ✓ ✓ 4 1/2 ✓ ✓ ✓ ✓ 1 1 ✓ ✓ Weighted Votes 58 61 78 76 75

Despite Candidates A and B having so many votes in the first round, Candidate X is the second winner (), because all ballots that support A and B also support C and thus already have representation on the council. In round 3, ballots that voted for both and have their vote weighted by one third. Any ballot that supports only one of the two winners will be weighted by one half. Ballots that indicate support for neither winner remain at full weight. Below is a table representing that information.

+ Third Round Results # of votes Weight of Vote Candidate A Candidate B Candidate C Candidate X Candidate Y Candidate Z 112 1/2 ✓ ✓ ✓ 6 1/2 ✓ ✓ 4 1/3 ✓ ✓ ✓ ✓ 73 1/2 ✓ ✓ ✓ 4 1/3 ✓ ✓ ✓ ✓ 1 1/2 ✓ ✓ Weighted Votes 57 1/3 60 1/3 38 1/3 37 5/6

Candidate B is the third and final winner, . The final result has two winners from the party that received about two thirds of the votes, and one winner from the party that received about one third of the votes. If ordinary approval voting had been used instead, the final committee would have all three candidates from the first party, as they had the highest three vote totals in the first round.

Properties SPAV 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.

Pareto efficiencyCommittee monotonicitySupport monotonicity with additional votersSupport monotonicity without additional votersConsistencyinclusion-strategyproofnessComputational complexityApproval votingstrong✓✓✓✓✓PProportional approval votingstrong×✓cand✓×NP-hardSequential proportional approval voting×✓candcand××P

There is a small incentive towards tactical voting where a voter may withhold approval from candidates who are likely to be elected, just like there is with cumulative voting and the single non-transferable vote. SPAV is a much computationally simpler algorithm than harmonic proportional approval voting and other proportional methods, permitting votes to be counted either by hand, rather than requiring a computer to determine the outcome.

When comparing sequential proportional approval to single transferable vote (STV), SPAV is more likely to elect candidates that individually represent the average voter, where STV is more likely to elect a range of candidates that match the distribution of the voters. The larger the number of candidates elected, the smaller the practical difference.

See also Proportional approval voting Satisfaction approval voting Reweighted range voting Approval voting Single transferable vote Sainte-Laguë method D'Hondt method

References