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Let's say we are electing a single district of twenty representatives using the single transferable vote. This is an unusually large district, but that's what it is.

We would also like to select five committees, each of four representatives, such that every representative in the district is on one committee. We would like these committees to be as proportional as possible.

Is there a established way to do this using the ballots cast for the original election? If not, is there an obvious way to do it?

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    @divibisan I could phrase it more generically. Would that be useful? Personally, i find specific examples like this easier to grasp. – Tom Anderson Oct 9 '19 at 18:54
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    Seriously? Who was obnoxious enough to downvote this question and why? If this type of practical question is not only on topic but also excellent, we should be shutting down Politics SE. Nearly everything else that gets asked here belongs in a forum. – Denis de Bernardy Oct 9 '19 at 19:06
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    Is en.wikipedia.org/wiki/Matrix_vote relevant? – endolith Oct 9 '19 at 19:14
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    @Roger If the district has ten Big-Endians and ten Little-Endians, and two of the committees have four Big-Endians, two have four Little-Endians, and one has two of each, then that's not proportional, because there are four committees whose makeup does not reflect the division of opinion in the electorate. You would ideally have all five committees have two from each party. – Tom Anderson Oct 9 '19 at 22:23
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    @endolith That is a different approach to the overall problem, but yes, that might work. I need to have a think about that. – Tom Anderson Oct 11 '19 at 11:31
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You can use the ballots cast for the original election, with the caveat that some votes might end up not counting if you had a lot more candidates than elected representatives. (Consider doing a new election for this purpose if that was the case.)

In non-profits it's common (source: worked with an electronic voting company in the past) to elect several officials in a single pass using STV. The top candidate becomes President, the next becomes General Secretary, the next Treasurer, etc. until all roles are filled. You can optionally swap positions according to the top candidates' preferences, a bit like what happens in competitive exams (e.g. if #2 wants to be Treasurer, then let them do so and continue assigning roles based on the ranked order).

You could arguably adapt this for multiple committees. Have voters rank order candidates, tally the results, and assign committee positions based on candidate preferences and the results until all positions are filled.

It's reasonable to add a constraint in there to ensure that there is a degree of proportionality in each committee. Specifically, disallow that any committee have more seats held by candidates with a certain color than the proportion of votes that they got overall (round up). For instance, if one party has 6 seats, there are 20 seats in total, and 5 committees, then there are 4 seats per committee, and that party can have more than 4 * 6 / 20 rounded up = 2 seats per committee at most. (You could require a minimum of one seat per committee in this case by rounding down.)

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Assuming that the representatives are members of parties, for some reason, and that the proportionality of the committees should reflect the relative sizes of the parties, then the obvious way of assigning representatives is:

Start with the party with the most representatives. Pull out a representative and put it into a committee. Then pull out the next representative and put it into the next committee. Repeat until you run out of representatives.

Repeat with the next most populous party. Continue until all representatives are assigned.

That might not be clear, so an example:

20 Representatives: A B C D E F G H I J K L M N O P Q R S T

Five Committees of four members: () [] {} <> «»

Parties:

  • Party $ has 10 members: A B C D E F G H I J

  • Party & has 5 members: K L M N O

  • Party * has 3 members: P Q R

  • Party ~ has 2 members: S T

Round 1 of assignment -- starting with Party $:

  • (A)

  • [B]

  • {C}

  • < D >

  • «E»

Round 2 of assignment -- continuing with Party $:

  • (A F)

  • [B G]

  • {C H}

  • < D I >

  • «E J»

Round 3 of assignment -- finished with Party $; starting with Party &:

  • (A F K)

  • [B G L]

  • {C H M}

  • < D I N >

  • «E J O»

Round 4 of assignment -- finished with Party; starting with Party *:

  • (A F K P)

  • [B G L Q]

  • {C H M R}

  • < D I N >

  • «E J O»

Round 5 of assignment -- finished with Party *; starting with Party ~:

  • (A F K P)

  • [B G L Q]

  • {C H M R}

  • < D I N S >

  • «E J O T»

And then we're done.

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