[EM] Semiproportional Bucklin method

[Raph Frank]

On Sat, Apr 25, 2009 at 1:26 PM, Kristofer Munsterhjelm
<km-elmet at broadpark.no> wrote:
> Start as Bucklin. Call each step a "round". A set of q candidates has a
> support equal to the support of that candidate in the set that has the least
> support. So, for instance, the set {A, B, C} has support 20 if the
> individual support of each candidate is A: 20, B: 30, C: 98.
> At round p, a candidate has support equal to the number of voters who placed
> that candidate in rank 1..p inclusive, where rank 1 is first preference,
> rank 2 second preference, etc.
>
> At round p, if there's some set of cardinality p that's supported by a Droop
> quota more than the number of members we've already elected from it, pick a
> candidate from that set (use any method you want to decide which).
> Otherwise, go on to round p+1.

It seems like it would meet Droop to me.

For example,

Assume you have a Party with a Droop quota of supporters and is
running N candidates.  All supporters rank all the party's candidates
and rank them before any other party's candidates.

In round N, all of the party's candidates will have a Droop quota of
support.  You can therefore create a group containing all of the
party's candidates and it will be size N and have a Droop quota worth
of support.

Thus, unless the party has already had one of its candidates elected
before then, one of the members of the party will be elected in round
N.

Also, since the party's supporters didn't rank any other candidates
outside the party, the other parties cannot get more than (seats-1)
Droop quotas between them and so can only fill (seats-1) seats, and so
the count will definitely reach round N (unless a party member is
elected before that).

This means a party with a Droop quotas worth of support is guaranteed
to get at least one candidate elected.

The only problem I can think of is if only 1 candidate can be elected
per round (I am not clear if  that is a requirement).  You could end
up with 2 parties with a Droop quota each "colliding" in round N, if
they both ran N candidates.

This is not a problem if more than 1 candidate can be elected if 2
distinct groups manage to met the election condition in the same
round.