By Aaron Hamlin
Approval voting typically refers to the single-winner version. Choose as many as you want, and the candidate with the most votes wins. But there are multi-winner approaches, including proportional ones.
A multi-winner approval method can merely use an approval-style ballot, though often with more candidates (since you’re electing more seats) like this:
A proportional version of approval voting can behave quite differently than its single-winner counterpart. This difference in behavior is important to realize when comparing any single and multi-winner methods that look similar.
If you just tally up approval votes and elect multiple winners by whoever gets the most votes, then those winners will tend to look alike. They’ll be good, high-utility winners, but you’re not getting much diversity there. If your aim with a multi-seat body is getting a broad viewpoint for policy discussion, then you’ll want to use what’s called a proportional voting method.
Proportional methods are multi-winner systems that elect candidates to office roughly in proportion to the support they receive. If we’re talking independents, then ideologies receive proportional seats. For example, if there are three sides to a single big issue, then candidates would get elected in proportion to the support of each side of that issue. This also means that a slate of candidates on one side of an issue shouldn’t get more seats than their proportion of support.
You’ll see that proportional methods tend to get more proportional as they simultaneously fill a larger number of seats. It’s hard to get proportional outcomes with only two or three seats, for instance. Really, you want no fewer than five, if possible. Five is also enough to guard against gerrymandering where parties attempt to draw district lines to unfairly benefit themselves.
The reason for this effect is that as the number of seats increases (called district magnitude), the threshold needed to get someone elected decreases. Here’s a figure indicating this using a common threshold formula, the Droop Quota, that’s general enough for us to extrapolate.
Diving Into Proportional Approval Voting
There are several variations of multi-winner approval voting that are proportional. Technically, the variation we’ll talk about is called sequential proportional approval voting. There are some more sophisticated versions, but this version balances performance and complexity well.
Here’s how it works. First up, you have your ballot, as was shown earlier. Party affiliation is not necessary for any calculation element, but you can add it if you like for informational purposes. You might have 31 candidates and have to elect nine of them. Perhaps you only like one, or perhaps you like over a dozen candidates. This setup for the number elected might be used for a city council in a large city.
This proportional approval voting method works like many proportional methods. If a voter gets someone they want elected, then their votes for other candidates get reweighted. Those voters’ ballots are reweighted to allow other voters the chance to elect someone. If a voter doesn’t get anyone elected in a particular round, then that voter’s ballot weight remains the same as before.
Specifically, all ballots start with the same weight: one. This version of proportional approval voting will reweight someone’s ballot by ½ the first time they elect someone, then ⅓ for the second time, ¼ for the third time, and so on. Compared to other reweighting schemes, this approach—the Jefferson method—gives less weight to the votes of voters who have already elected one or more candidates. This is beneficial to minority groups by giving them a better chance to elect at least one person who represents them.
You can find an example at the end of this article where you can follow along.
Many proportional methods tend to be inherently more complicated than their single-winner counterparts. While there is some complexity with this version of proportional approval voting, the ballot remains super easy even with a long candidate list. And that’s fortunate because a long candidate list is commonplace with multi-winner elections.
Also, this method does allow geographic representation if you have multiple large districts. The districts just have to be larger, multi-member districts.
Want to try out proportional approval voting on your own? Upload some ballot data into this application developed by one of our awesome volunteers, Nuño Sempere.
A Walkthrough Example
The following example is adapted from Brams, Kilgour, and Potthoff.
There are 16 voters four candidates: A, B, C, and D. Three candidates will be elected.
Here are the raw votes:
3 A, B
4 A, B
5 A, B
6 A, B
7 A, B
8 A, C
9 A, C
10 A, C
11 B, C
12 B, C
For simplicity, we consolidate it to this (“# of this ballot type”: “candidates selected”):
Round 1 Totals:
A is elected in round 1.
Now we recalculate for R2 (all ballots including winning candidate A are reweighted by ½; those excluding candidate A keep their original weight):
2 (½) = 1: A
5 (½) = 2.5: AB
3 (½) = 1.5: AC
2 (1) = 2: BC
4 (1) = 4: D
Round 2 Totals:
B is elected in Round 2.
Now we recalculate for round 3 (all ballots including winning candidates A or B are reweighted by ½; all ballots including winning candidates both A and B are reweighted by ⅓; all ballots excluding A and B keep their original weight):
2 (½) = 1: A
5 (⅓) ~ 1.67: AB
3 (½) = 1.5: AC
2 (½) = 1: BC
4 (1) = 4: D
Round 3 Totals:
D is elected for round 3.
In all, A, B, and D are elected.
Aaron Hamlin is a co-founder of CES and served as its Executive Director until 2023.