[EM] Sequential Best Assigment (multiwinner method)

Andy Jennings elections at jenningsstory.com
Mon Jul 24 21:58:28 PDT 2017

Here's a multiwinner system that's so simple that it should have a name,
but I don't think it does.  Let me know if it does.

It uses rated ballots.  The goal is to repeatedly find the candidate whose
top quota's-worth of grades are highest and elect that candidate, then
de-weight a quota's-worth of voters.  Some names worth considering:

Sequential Best Assignment
Sequential Constituent Matching
Sequential Quota Allocation

The method:

N = Number of voters
S = Number of seats

1. Every voter grades every candidate.  (I'd say 4 or 6 grades.)

2. Each voter starts with weight 1.

3. Choose quota Q = N / S. (*)

4. For each candidate, calculate the minimum of their top Q grades.  Let G
be the highest minimum.  Elect the candidate with that minimum.  (Break
ties as in GMJ: calculate for each candidate what fraction of their G
grades are in their top Q grades, and elect the candidate with the smallest
such fraction.  Break further ties by choosing the candidate with the least
number of G grades in their top Q grades.)

5. Deweight some voters to decrease the total voter weight by Q, in this
  a) any voter who gave the minimum grade to all remaining candidates is
deweighted to 0.
  b) for the voters not deweighted in (a) who gave this candidate a grade
of G or above, find the deweighting D such that when the deweighting

  W_new = max(W_old - D, 0)

is applied, the total voter weight in this round is decreased by Q. (**)

6. Repeat steps 4 and 5, applying voter weights when calculating the top Q
grades, until S seats are filled.

(*) With this quota, when you are filling say, 4 seats, then 25% of the
voting weight gets used up with each seat filled.  25% of the voting weight
will remain when choosing the last seat.  That last seat will be determined
by the tie-breaker rule, so it is essentially equivalent to approval
voting, with any above-bottom grade counting as approval.

The other common choice of quota, Q = N / (S + 1), could also be
considered.  When filling 4 seats, then, 20% of the voting weight gets used
up with each seat filled.  40% of the voting weight remains to choose the
last seat, so the last seat is essentially filled with a median-based
method (GMJ).  20% of the voters' opinions are, by design, left without a

(**) I thought about another step (a') where anyone who gave a grade
strictly above G was deweighted completely, but I think it gives the voters
too much incentive to down-weight candidates who they think can get elected
without their help.

I also considered another step (a'') where anyone who graded the chosen
candidates strictly above all other candidates was deweighted completely,
but I don't think there's much benefit for the added complexity.

Any thoughts on which quota is better or on the right name?

~ Andy Jennings
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