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The excess method: a multiwinner approval voting procedure to allocate wasted votes

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In using approval voting to elect multiple winners to a committee or council, it is desirable that excess votes—approvals beyond those that a candidate, especially a shoo-in, needs to win a seat—not be wasted. Common voting procedures such as sequential proportional approval voting (seqPAV) elect candidates in sequence and depreciate the voting weights of voters who have gotten one or more of their approved candidates elected. As a consequence, a voter is “punished” for approving a candidate when his or her approval was not necessary for that candidate to be elected. To alleviate this problem, we propose a modification of seqPAV that we call the excess method. In each round, the method determines the excess votes of the winning candidate and transfers them to other candidates approved of by the winner’s supporters. In doing so, the excess method transforms wasted votes into support of a voter’s other approved candidates, who in later rounds deserve, and may benefit from, receiving this additional support. In parliamentary systems with party lists, the excess method is equivalent to seqPAV and thus ensures the approximate proportional representation of political parties. When voters are restricted to voting for a single party list, both methods coincide with the well-known apportionment method due to Jefferson/D’Hondt.

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  1. Under approval voting, voters do not rank candidates or parties but simply express their approval for those they like or find acceptable. We assume that the decision of voters about where to draw the line between acceptable and nonacceptable candidates or parties is a subjective judgment they make, but it may follow certain guidelines (Brams and Fishburn 2007, Chapter 5).

  2. On a committee or council, diversity may be more important than proportional representation. As a case in point, one would not want all members of a university grievance committee to come from the same department or be of the same gender. Ratliff (2014) and Bredereck et al. (2018) analyze constraints of this kind, and Brams (1990, and 2008, Chapter 4) analyzes constraints for providing proportional representation to two different constituencies (e.g., geographic and party-based).

  3. Whereas Peters (2018) focussed on resolute voting rules, Kluiving et al. (2020) have generalized the impossibility result to irresolute rules.

  4. What we call seqPAV is sometimes referred to in the literature as reweighted approval voting (RAV); a nonsequential version of proportional approval voting is referred to as PAV. Aziz et al. (2017) show that PAV, but not seqPAV, satisfies “justified representation” and “extended justified representation.”

  5. Whereas calculating excess votes and updated profiles gets tedious quickly (even for small numbers of rounds), it is straightforward to implement these operations with a computer in an efficient way.

  6. If B did not have a big excess (say, only 1 more vote than C), then A and C would not be significantly helped, enabling D to win a seat under Excess in a modified Example 3.

  7. We could make this “support transfer” explicit by informing voters that if there are candidates whom they have not approved of after all their approved candidate(s) have been elected—as occurred with the A voters in Example 1—Excess will attribute approval in the transfer process proportional to the approval of A voters who approved of other candidates. But simply informing voters that their unexpressed approval will be delegated in this way still violates their sovereignty. It’s as if they were told: “Even if you don’t have an opinion, trust other A voters who do.”

  8. In Example 3, as changed to 25 rather than 5 A voters, \(\frac{25}{45} = \frac{5}{9}\) of the \(45 - 15 = 30\) excess A votes, or \(16\frac{2}{3}\) votes, would be fictitiously transferred to the 25 A voters.

  9. The total number of approvals of elected candidates can be considered as the social welfare of an elected committee, as it is equal to the sum, over all voters, of the number of approved candidates in the committee. If the goal is to maximize social welfare, which is equivalent to minimizing the first type of wasted votes, then simply selecting the k candidates with highest approval scores would be optimal. However, this method is far from being proportional (e.g., Aziz et al. 2017).

  10. To be precise, whenever Excess and seqPAV Excess elect the same candidates in the same order, Excess never wastes more votes than seqPAV. When Excess and seqPAV elect different committees, however, there are examples in which Excess wastes more votes than seqPAV.

  11. The possibility of expressing and aggregating approval votes over parties has been studied by several scholars. For example, Speroni di Fenizio and Gewurz (2019) suggest transforming such votes into a standard party-list election by assigning each voter to one of her approved parties. For a general exposition of approval-based apportionment, see Brill et al. (2020). Crosson and Tsebelis (2021) recommend the use of approval voting for parties and simulate its usage in four European countries, which they argue would lead to greater parliamentary representation of centrist parties. Not combining approval voting with an apportionment method, however, may upset the proportional representation of parties, except as measured by their total approval.


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Markus Brill’s work was supported by the Deutsche Forschungsgemeinschaft under grant BR 4744/2-1. We thank the editor and two reviewers for their insightful comments and suggestions, which substantially improved the present version of the article. We also thank Sean Horan, Karen Long Jusko, D. Marc Kilgour, Jean-François Laslier, and Marcus Pivato for helpful comments, and Dung Thuy Pham for help with formatting.

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Correspondence to Steven J. Brams.

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Brams, S.J., Brill, M. & George, AM. The excess method: a multiwinner approval voting procedure to allocate wasted votes. Soc Choice Welf 58, 283–300 (2022).

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