What's fair in a multiple-winner election?

Consider an group of voters divided solely on some linear issue (e.g. how spicy to make tonight's dinner). Assume the breakdown is as follows:

Wants more45%
Wants same20%
Wants less35%

Pretend you’re a voting system and imagine the ballots these voters cast. How do you aggregate these votes to best represent their preferences? If you only need one winner, that's fairly straightforward. The majority don't want it more spicy, neither do the majority want it less. It's a little less clear with two or more winners.

Proportional sets

If we know the number of required winners in advance, I'd argue the following results are fair.

MoreSameLess
1 winner1 candidate
2 winners1 candidate1 candidate
3 winners1 candidate1 candidate1 candidate
4 winners2 candidates1 candidate1 candidate
5 winners2 candidates1 candidate2 candidate
6 winners3 candidates1 candidate2 candidate

Notice the oscillation between the 1 and 2 winner scenarios. Although the voters are best represented with a single Same winner, a More/Less pairing better represents the electorate than Same/Same. While this holds true when the number of winners is known and fixed, our notion of fair changes when the number of winners is variable as per the two scenarios below.

Proportional ordering

Consider the scenario where you have one representative, but might need a second, or a third. Since the first representative is fixed, you choose your second proportional to the electorate and the current winners. We’re looking at generating an ordered list of candidates.

First placeSame
Second placeMore
Third placeLess
Fourth placeMore
Fifth placeLess
Sixth placeMore

Note the difference here between a fixed two-winner election (where we chose More/Less) and the proportional ordering (which starts off Same/More). In this different problem space, fair has a different meaning and produces different results.

Further, by the third place proportional sets and proportional orderings are producing the same results. Issues that are less linear may take a little longer to match up.

Non-proportional ordering

Consider the scenario where you only need one winner, but you need a runner up in case the first winner is unable to serve. Unlike proportional ordering, the previous winners aren't taken into account when determining the remaining ordering.

First placeSame
Second placeSame
Third placeSame
Fourth placeSame

Where in the first example we provided More/Less as a fairer alternative to Same/Same, the problem space for this third example would dictate Same/Same over More/Less. Again, fair has a different meaning and produces different results.

Hopefully this highlights the need to understand the specifics of each multiple-winner voting situation. Choosing the wrong result type can drastically misrepresent your voters.

Examples

A book club wants to chose one book per month that everyone will read.

Separate elections per month with a single winner per election. The candidates (books) and voters (club members) may change between elections.

A book club wants to chose three books per month, each member likely reading only one of the three.

Separate elections per month with three proportional set winners. The options then cover the largest clusters of preference.

A city wants to elect their mayor along with a vice-mayor who serves in the mayors absence.

One election with non-proportional ordering and only two winners. The second place winner will be the most similar candidate.

A city wants to prioritize cultural event funding, without knowing in advance the available funds.

One election with proportional ordering, where each event is funded in the resulting order.

Creating a new election and still not sure which one to use?

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