[EM] Holding byelections with PR-STV

Raph Frank raphfrk at gmail.com
Wed Sep 16 05:22:53 PDT 2009


On Wed, Sep 16, 2009 at 11:45 AM, James Gilmour
<jgilmour at globalnet.co.uk> wrote:
>> In Meek, elected means that you have at least a Droop quota
>> (and can have any "keep value" between zero and one) .
>
> This is confusingly expressed.  It is impossible to be elected with a keep value of zero.

The rule is that they can have a keep value in the range from zero to 1.

Eliminated candidates have a keep value of zero and remaining/not
elected/running candidates have a keep value of 1.

For each ballot, a voting weight of 1 is given to the first choice.
That candidate keeps a fraction of the voting weight received equal to
his keep value and passes the rest on to the next choice and so on.

What is nice is that if you multiply a candidate's keep value by a
constant, then his vote total will also scale by the amount.

This is because every piece of voting weight he receives is multiplied
by his keep value.

Thus if a candidate has more than a quota you can multiply his keep
value by (Quota)/(His vote total) and he will automatically end up
with exactly a quota, after that step.

The vote weight that he lost is shared between all the other
candidates.  Thus the candidate's vote total drops to the quota, but
all other candidates' totals increase or stay the same (if they were
ranked higher on a ballot, their vote is not affected, and if they are
lower on a ballot, they receive more vote weight from that ballot).
Thus this operation will never move a candidate who has more than a
quota to a total that is below the quota.

If you do this over and over for all elected candidates, they will
converge to a quota each (as each step decreases their keep value).
There is a theorem that shows that this will always converge to the
same final set of keep values.

Once another candidate goes above the quota, he can also be designated
as elected and have his keep value reduced too.

If after the convergence completes, all the seats aren't filled, then
the candidate with the fewest votes is eliminated and has his keep
value set to zero.

A more accurate rule is that if
- the total surplus of all the elected candidates is
-- greater than the difference between the 2 weakest candidates, and
-- greater than the difference between the strongest unelected
candidate and the quota
then eliminate the lowest candidates

Effectively, that means the surplus isn't enough to elect another
candidate even if it all went to the strongest unelected candidate and
it isn't enough to change the relative order of the bottom 2, then you
can eliminate and there is no point in further convergence processing.

The bottom 2 rule could also be changed so that the bottom N
candidates can be eliminated if the sum of their votes and the surplus
is less than that (N+1)th worst candidate (unless that would result in
insufficient remaining candidates to fill all the seats).

In New Zealand, convergence is consided to have happened when all
elected candidates are within 10E-8 of a vote of the quota, rather
than defining it in terms of it being impossible for any other
candidate to be elected.



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