[EM] A very simple quota method based on Bucklin

[Raph Frank]

On 8/26/08, Kristofer Munsterhjelm <km-elmet at broadpark.no> wrote:
>  Inputs are ranked ballots. Each voter starts with a weight of one. The
> quota is Droop (Hare does much worse).

Can a voter skip ranks and also is there a limited number of ranks?

If you allow rank skipping, then a voter can distinguish between

A>B>>C
and
A>>B>C

E.g.
A:1
B:2
C:10

and

A:1
B:9
C:10

In the second case, the voter will only compromise and vote for B if A
can't get elected even after 9 rounds.

In fact, the notation could include the number of skipped ranks

A>>>B

This means
A:1
B:4

i.e.

A>(empty)>(empty)>B

>  For all of those voters that voted for the winner, reweight their weights
> by (new weight = old weight * (votes for winner - quota)/(votes for
> winner)).
>  Don't alter the quota, but in all other respects, restart the election with
> the winner removed from all ballots, as if he never entered. Keep on doing
> this until enough candidates have been elected.

It might be worth recalculating the quota based on exhausted ballots.
Otherwise, your method might end without electing enough candidates.

You can just recalculate the Droop quota using the new seat total and
the reweighted number of votes.

For example, assuming 100 voters and 4 seats

Q = 100/(4+1) = 20

After round 1, your reweigting will decrease the effective number of
ballots by 20 and seats to 3.  This has no effect on the quota
(assuming no exhausted ballots)

Q = 80/(3+1) = 20

This means that you can just keep recalculating the quota to take
account of exhausted ballots.

Another option for weightings is to weight each ballot at

w = 1/(candidates elected + 1)

If the ballot was voting for a candidate who gets elected, its
'candidates elected' count goes up by 1.

This also achieves proportionality.  It works like proportional approval voting.

>  That's it. For the single-winner case, the method reduces to Bucklin, which
> is monotonic. I'm not sure if the method is monotonic in the multiwinner
> case as well, but I think so.
>
>  According to my simulation, the method isn't as proportional as STV.

What does this mean?  It looks like the method meets Droop
proportionality, so should be proportional.

If sims are showing non-proportional effects, it probably means that
votes are 'bleeding' over into other parties.

If I vote

A1>A2>B1

I could end up helping party B get seats instead of my favourite
party.  A better vote (from my point of view) would be

A1>A2

This means none of my vote bleeds over into party B.

it is likely that people could start bullet voting in order to prevent
it.  Allowing people to skip ranks could help here.

I could vote

A1>A2>>>>>>>>B1