[EM] Approval PR as a primary

[Ted Stern]

Approval PR is a very simple method:

For M multiwinner seats to be allotted, use the Hagenbach-Bischoff quota
("Droop") of Q = Number of Votes / (M + 1)

Voters may approve of as many candidates as they wish.

Ballots start with a weight of 1 each.

Repeat until M candidates have been seated:

   - Seat the candidate with highest weighted approval, then remove that
   seat winner from the list of candidates.
   - If that candidates total weighted approval is greater than Q, then
   reweight all ballots by multiplying by the factor
      - F = 1 - Q / Approval
   - Re-tabulate approvals for all remaining candidates.

When used as a primary for an M-seat position, I propose winnowing to 2*M +
1 candidates.

What are the consequences of using Droop quota and 2*M+1 candidates?

For a single winner election, the quota is 25% and 3 primary winners are
chosen.  If one party commands more than 75% of the vote, they will have 3
candidates in the general election. If a party has less than 75% but more
than 50%, they will have 2 candidates in the general. If a party has less
than 50% but more than 25%, they will have 1 candidate in the general.

For most such single winner elections, this would all but eliminate the
single-party runoff problem associated with top-two runoff (of any form).

I am aware that there are many potential failures of such a method. Why
would one bother using it?

   - First off, I would not have an APR primary if the number of candidates
   is less than 2*M+1. The main purpose is to winnow the field to a manageable
   number.
   - 2*M + 1 means that you have at least double the candidates as the
   number of seats, and a Droop quota ensures that the majority party gets at
   least M + 1 candidates in the runoff.
   - Approval is simple to implement and has good properties for each seat.
   - For a single-winner election, a 3 candidate general is actually
   feasible to tabulate summably for a ranked method such as Approval Sorted
   Margins.

My goal, however, is to eliminate gerrymandering completely by having as
many representatives as possible elected in large multiwinner elections.

For example, for a state senate with 49 seats (e.g. Washington state), you
could have 7 districts of 7 seats each.  I would allow as many as 12 seats
per district. If subdivision of the region is necessary to stay under 12
seats per district, the districts should be subdivided to no smaller than 6
seats.

If you had a district with 11 seats, and more than 23 candidates, you would
hold a primary to reduce to 23 candidates.
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