[EM] A very simple quota method based on Bucklin

[Kristofer Munsterhjelm]

Raph Frank wrote:
> 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

That would cause exhaustion. Here's an example with six candidates, 
single winner.

10: A>>>D
10: B>>>E
10: C>>>F

The quota is 50% + 1, or 16. However, none of the candidates get more 
than 10 votes.

If the ballots are fully specified, then by pigeonhole, once all ranks 
have been included, each candidate must have got one vote per voter. 
Thus some candidate will be above quota.

One could perhaps fix this by equal-ranking all remaining candidates 
last, below any specified rank.

>>  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.

It should have no effect on ballots that aren't exhausted, since the 
reweighting reduces the nominator by a quota, and the election of a seat 
reduces the denominator by one, thus canceling out.

Say there are k votes for the winner, and all weights are 1. The quota 
is Q < k. Then the sum of the new weights is k * (k - Q) / k. Cancel out 
the  factor of k and we get (k-Q). Call the number of those who didn't 
vote for the winner r. Then the quota was (r + k)/(numseats + 1). 
Afterwards, we have

(r + k - Q) / (numseats),

which has reduced the numerator by a quota, and the denominator by one, 
which was what we wanted.

> 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.

Does that pass Droop proportionality? It looks like D'Hondt.

>>  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.

It means that if voters and candidates have binary opinion profiles and 
vote in order of Hamming distance (number of opinions where they 
disagree) to each candidate, ranking those with greater Hamming distance 
lower and breaking ties randomly, then the difference of the proportion 
that hold the "yes" stance on some issue or issues in the assembly 
differ more from that proportion in the population, on average, than 
would be the case for an assembly elected using STV.

The simulation shows different scores even among methods that satisfy 
Droop proportionality. QPQ does best, then STV, then this.

> 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.

That makes sense. Bucklin passes Later-no-help while failing 
Later-no-harm, thus producing an incentive to truncate ballots. IRV (and 
thus STV) passes both, but pays for it by being nonmonotonic.