[EM] D'Hondt without lists

[Olli Salmi]

At 07:44 +0300 22.8.2002, Olli Salmi wrote:
>If we
>transfer votes with the value 1/n for the nth candidate if a candidate has
>the Droop quota, and eliminate like STV does, we get a system that works
>much the way I want.

I was wrong. This simple thing has defects and some more bookkeeping is
needed. I hope I'm not wrong again.

Let's see first how the transfers are done. Imagine the following transfer:

36 ABCD, quota=10

Transfer in STV:
36
10 26
10 10 10 6
ABC elected

Transfer in d'Hondt:
36
18 18
12 12 12
9 9 9 9
ABC elected, with the excess of 6 shared between them.

Transfer in Sainte-Laguë:
36
24 12
14.4 14.4 7.2
10.29 10.29 10.29 5.14

When the votes are transferred, the vote is divided into an odd number of
parts. The last candidate gets one part and the others two, because you are
elected if you have half the quota.

I don't know what the quota should be for Sainte-Laguë. If you use whole
number divisors, the Hare quota divided by two is often but not always
good. I'm not even sure that the method works with Sainte-Laguë.

It matters who holds the excess. In the d'Hondt example above, candidate D
might get votes from another source, and candidates ABC would end up being
elected with 9 votes only. In the following example with three seats the
seats would go to ABD if we gave 1/n votes to the nth preference. Quota>9.

13 AD
12 BD
11 CE

D gets 13/2+12/2 = 12 1/2 after B has been elected and the seat goes to D.
A has been elected with 6 1/2 votes and B with 6, which is not right. CE
exceeds the quota and C should get the seat. In STV, D has only the excess
5 votes which is not enough for election.

To remedy this we have to keep track of the combined number of seats that a
candidate has helped to elect. D's votes have already elected A and B so
the correct priority number for this candidate is (13+12)/3 = 8 1/3, the
sum of the votes divided by one more than the sum of the number of seats. C
is elected.

Such calculations are taken care of by the method described in Chapter 20,
Section 4 of the Swedish Elections Act. An English translation of the Act
can be found at http://www.val.se/att_rosta/international.html, together
with other information on Swedish elections in several languages. This
system is different from the Swedish method I've previously mentioned, the
one used within local councils, which is a simplified and not so accurate
version. Both methods are used to order candidates within party lists.

If the method is used without lists, it has to be supplemented with the use
of the Droop quota. You don't need the quota to transfer, you need it to
protect a solid coalition which is entitled to one seat but has its first
preferences spread evenly. The coalition has to have a chance to transfer
their preferences.

The basic thought behind the method in the Swedish Elections Act is that
groups with the same first preference are treated as lists. For each group
you count the number of votes, the number of seats it has helped to elect,
and a priority number, which represents the average number of votes that is
held by every candidate that the group has elected if it gets the next seat.

Here's the formula for the priority number:

p = v/(n+1)
p=priority number
v=the group's votes
n=the number of seats that the group has elected.

If k groups have the same first preference, the formula is like this:

p=(v1+v2+...+vk)/[(n1+n2+...+nk)+1)]

If a group breaks up into smaller groups when the votes are transferred,
the number of seats that a group has helped to elect has to be divided
between the smaller groups in proportion to the votes of the groups. Again
with three seats:

13 AD
4  BD
8  BE
11 CE

When B is elected with 12 votes, the group divides into two. The seat that
these ballots helped to fill is divided between the groups as well. The
Swedish Elections Act gives a handy method for it: you get the number of
seats elected by dividing the number of the votes that the group has by the
previous priority number, in this case 12, the one that elected B. (B)D has
4 votes. Divide that with twelve and you get 4/12=1/3 as the number of the
seats that this group has helped to elect. For (B)E it's 8/12=2/3. Here's
the formula:

n=v/p (this is the previous p; then a new p is calculated for each next
preference)

If p is the result of a combination of several groups, seats won by the
groups are divided in proportion to the votes of the groups, and the seat
in question is divided in proportion to the remainders. Let's take an
example:

32 ABCDE
28 FGDH

When D is elected the bigger group already has 3 seats and the smaller
group 2. There are 60 voters supporting D with 10 votes per seat. 30 votes
are needed to elect ABC and 20 votes to elect FG. To elect D, the bigger
group has 2 votes to spare and the smaller group has 8. D's seat is divided
between the groups in this proportion, so the number of seats for E is 3.2
and the number of seats for H is 2.8. This proportion is actually changed
if only one of the parties gains a further seat. This is perhaps a small
flaw in the method.

To illustrate the formulas, D's priority number is (32+28)/[(3+2)+1]=10.
The n for E is v/p=32/10=3.2 and for H it's 28/10=2.8. In Sweden they
truncate after two decimals.

These numbers have to follow the groups, and unfortunately they cannot be
calculated on the basis of the ballot alone, which doesn't make this method
much easier than STV, which is what I was hoping for. Although they don't
do it in Sweden, it's possible -- and perhaps better, to make strategic
voting more difficult -- to transfer votes to candidates already elected,
in which case you'd have to calculate the number of seats and the priority
number for the group before you do further transfers. Unfortunately this
doesn't affect the number of seats figure for the group that originally
elected the candidate. This may be a flaw.

Adjusting the quota when votes become non-transferable is tricky, because
the excess is evenly distributed between the elected candidates. I'm not
sure how necessary the adjustment is.

Exclusions do not appear to be problematic. If a group breaks up into
smaller groups according to next preferences, ni for group i has to be
counted differently:

ni= n*vi/v

I had to do a bit of sleuthing to find out about the origin of this method.
I found a reference to "the d'Hondt-Phragmén count and the group based
concept of lists". The method was probably invented by Edvard Phragmén,
professor of mathematics at Stockholm University. Swedish libraries contain
only two short articles on elections by Phragmén, but he was a member of
the commission on electoral reform which published its report in 1903 (I've
ordered the report so maybe I'll go on with this monologue). Sweden
switched over to proportional elections in 1909 after a reform bill had
been twice rejected.

Phragmén, Lars Edvard 1863-1937
http://www.math.uio.no/abel/1902/Phragmen.html

In Sweden this section of the Elections Act is a dead letter because voters
can't change the preferences on the ballot if they vote for registered
parties. It's only used if parties nominate two different lists, to get
better geographical proportionalty, for instance. It generates a priority
list from which the candidates are elected. No quota is used, because the
method is meant only for allocating seats within parties. Since 1997 it has
been supplemented with a system of personal votes. Before 1997 it was
possible to strike out and write in names, as long as the two top names
were official candidates of the party.

Olli Salmi

PS. I ordered the 1903 commission report and got it today. The method
sketched above hadn't been invented. It seems to be a further development
of Phragmén's method of branching lists. Phragmén's first method is a kind
of fractional approval STV (but elimination was unknown) and his second
method is sequential Proportional Approval Voting but with reduction of the
value of a ballot in proportion to the number of votes of a group, a group
being a parcel of ballots with the same set of candidates. These are only
first impressions, I've only had the book for a few hours. Appendix I
describes about 15 PR methods.


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