[EM] Sequential Best Assigment (multiwinner method)

[Andy Jennings]

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
manner:
  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
formula:

  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
representative.

(**) 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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