[EM] D'Hondt without lists

Thu Nov 14 06:45:23 PST 2002

Some time ago I posted about a way to use the preferential method in the
Swedish Elections Act with a quota.
http://groups.yahoo.com/group/election-methods-list/message/10068

I have now got hold of an article by the inventor of the method, E.
Phragmén: "Till frågan om en proportionell valmetod" [On the Question of a
Proportional Election Method], Statsvetenskaplig Tidskrift (StvT,) 1899, 2
(7), p. 87-95. He describes the method which is called "Phragmén's second
method" in the 1903 Swedish commission report, in which appendix I on
election systems is written by G. Cassel, possibly Gustav Cassel, dubbed on
one Web page as the most famous economist before Keynes. Phragmén mentions
that his method was first described  in "Proportionella val" [Proportional
Elections], Svenska spörsmål, 25, Stockholm, 1895. He is rather impatient
because he cannot understand why people think that his method is hard to
understand. However, his tabulation is easier to understand than Cassel's
more complicated way.

Phragmén's method is an non-preferential method,i.e. the order of names on
the ballot is not relevant. It's sequential, the seats are allocated one at
a time. It's a generalization of d'Hondt, a highest average method. Each
seat is given to the group of voters with the highest average number of
votes per seat.

Phragmén uses the concept of load (or burden, strain, load on machinery,
'belastning' in Swedish, 'Belastung' in German). If 1000 voters elect one
candidate, each has a load of 1/1000 of a representative. For each
candidate we check what the load on individual ballot papers will be in
case the candidate is elected. The seat is given to the candidate with the
lightest load on the individual voter.

The d'Hondt formula can be described in these terms as well. If a party
has, say, 36 voters, each voter has a load of 1/36 if one candidate is
elected, 2/36 if two candidates are elected, 3/36 if three candidates are
elected, etc. The smallest load is given preference. This is equivalent to
giving seats according to d'Hondt's quotients 36/1, 36/2, 36/3, etc.,
giving preference to the highest quotient. The smallest load is the inverse
of the highest average.

These are Phragmén's rules in a somewhat streamlined translation:
Rule one: The candidates are not ordered on the ballot.
Rule two: Seats are assigned sequentially.
Rule three: To see which candidate gets the seat, the ballot papers for
each candidate will receive a load in addition to the previous load in such
a way that
1) the total load on these ballots is incremented by one unit of load,
2) the total load on these ballots is divided equally between the ballot
papers.
The candidate whose individual ballot papers have the smallest load is
elected, the load is adjusted and retained in subsequent counts.

There are actually several kinds of load: load on a vote, load on a
candidate and load on a group. In theory it's the load on a vote that is
saved with the vote, and the loads and votes of a candidate are added up to
check what the smallest load per ballot is. In practice the loads are saved
with the groups of ballots with the same set of candidates.

Phragmén's tabulation is easy, because he only keeps track of the load.
Ballots with the same set of candidates are grouped into bundles (groups)
and the number of ballot papers in the bundles are counted. Here's
Phragmén's example.

1034 ABC
 519 PQR
  90 ABQ
  47 APQ

Then the votes for the individual candidates are counted:
A 1171
B 1124
C 1034
Q  656
P  566
R  519

Candidate A places the smallest load on the voter (1/1171) so he is
elected. Now the load of one seat will be apportioned between the groups in
proportion to their votes, or in other words each individual ballot paper
is assigned the load 1/1171. The load is measured in seats. In a futile
attempt to make the method easier to understand, Phragmén uses the word
ounce (the name of a small measure of weight, he explains) for a thousandth
of a seat. It is actually easier to leave out the decimal points but I'm
not going to do it here. Phragmén rounds the results off.

So we assign the load 1034/1171=0.883 to the group ABC, 90/1171=0.077 to
ABQ and 47/1171=0.040 to APQ, to the groups that have voted for A.

                A
1034 ABC        0.883
 519 PQR
  90 ABQ        0.077
  47 APQ        0.040
	_______________________________
Sum             1

The sum of the loads equals the number of seats filled so far. This offers
a nice way of checking your calculations.

Now we figure out who is elected next. It's not necessary to calculate the
clearly unsuccessful cases. B has a stronger claim than C -- he's on two
lists while C is on only one of them -- and Q has a stronger claim than P
or R.

We pick up B's votes, 1034+90=1124 (or look them up in the list we made),
and their loads, 0.883+0.077=0.960. If these voters elect a further
candidate, the load has to be increased by one: 0.960+1=1.960. The load on
each of B's votes will be 1.960/1124=0.00174.

Similarly, Q has 519+90+47=656 votes with the load 0.077+0.040=0.117 before
Q is elected. The load on each vote after Q's election would be
1.117/656=0.00170, so he's elected. The total load 1.117 is apportioned
between Q's voters in proportion to the votes of the groups.

                A       Q
1034 ABC        0.883   0.883
 519 PQR                0.884
  90 ABQ        0.077   0.153
  47 APQ        0.040   0.080
        _______________________________
Sum             1       2

For the next seat, Phragmén checks only B's and P's claims on the next seat
(B is on one more list than C). The loads and votes are as follows:

B       2.036/1124=0.00181
P       1.964/556=0.00347

B is elected but C comes close:

C       1.883/1034=0.00182

Here's the final tabulation:

                A       Q       B
1034 ABC        0.883   0.883   1.873
 519 PQR                0.884   0.884
  90 ABQ        0.077   0.153   0.163
  47 APQ        0.040   0.080   0.080
	_______________________________
Sum             1       2       3

The ideal load on a vote is 3/1690
1.836
0.921
0.160
0.083

Cassel uses a more complicated tabulation which I found harder to
understand. The terminology has also changed. Cassel uses the term load
only in the introduction. In the examples he uses the inverse of load,
which he calls "computed vote". So instead of

B	2.036/1124=0.00181

he uses

B	1124/2.036=552.062.

This is what I called in my previous posting the priority number (p), which
represents the average number of votes per seat. The candidate with the
smallest load has the highest priority number. I think it's easier to use
the priority number, because you can avoid the small fractions and the
number has a more concrete meaning.

The load or number of seats n is also counted differently.
Phragmén:
n=v*[(n1+n2+...+nk)+1]/(v1+v2+...+vk)

Elections Act and Cassel:
n=v/p
Since p=(v1+v2+...+vk)/[(n1+n2+...+nk)+1], the value of n is the same.
p=priority number
v=the group's votes
n=the load, the number of seats that the group has elected

Cassel picks up the original votes and priority numbers to calculate the
loads, which perhaps enables him to avoid some rounding errors. He says in
a footnote that a considerably easier tabulation can be used in practical
elections. This may be Phragmén's original one.

In Cassel's description in 1903 there's a further feature suggested by
Phragmén. Because the voter can express no preference for candidates, the
method is prone to what Cassel calls "decapitation", which means that the
top candidates of a party can fail to be elected because the order of
election of a party's candidates can be determined by the voters of other
parties.

Phragmén's solution to this problem is the line (streck, German Strich).
The voters list their candidates and draw a line after the ones they want
elected in the first place. The candidates after the line are taken into
account only after the ones before the line have all been elected. If the
voter marks ABC|D on the ballot, the ballot is only counted for ABC until
they have all been elected.

It seems to me that the preferential method used in parliamentary elections
in Sweden is derived from this variant. Having a line between every
candidate makes the ballot preferential, while the underlying mathematics
is the same. In my previous posting I suggested that the preferential
method can be used with the Droop quota and elimination to protect a solid
coalition that has spread it's first preferences.

Douglas Woodall has suggested some refinements. If ballots become
non-transferable, the quota can be adjusted. You just subtract the number
of the non-transferable votes from the total number of votes and their
total load from the total number of seats to be filled and recalculate the
quota.

Douglas Woodall has also pointed out that after eliminations the votes for
a newly revealed candidate may help a candidate they were not intended for.
To remedy this, he proposes that after elimination the count should be
restarted from scratch.

All in all, this is a very versatile method. It is a generalization of the
d'Hondt formula, which can be used for both proportional non-preferential
or approval elections and proportional preferential elections with or
without a quota, in addition to normal list PR.

As far as I know, the method is known only in Sweden. I haven't found out
yet when it was introduced, but committee reports in 1913 and 1921
recommended it, so it may have been in the twenties.

I'm going to send a further posting on Phragmén's and Thiele's methods
(sequential PAV).

Olli Salmi

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