In the 1903 Swedish commission report that I mentioned earlier in this
thread, Cassel describes a method designed by Thiele, in all probability
Thorvald Thiele (1838-1910), professor of astronomy at Copenhagen
University. In this method, the value of a ballot is 1/n for the nth
candidate elected that the ballot elects. The ballot is not preferential.
The method is intended to be a generalization of d'Hondt's rule.

Phragmén gives the reference "Om flerfoldsvalg" [On multiple election],
Overs. over d. K. Danske Vidensk. Selskabs Forh. 1895.

Oversigt over Det Kongelige danske videnskabernes selskabs forhandlinger=
Bulletin de L'Académie royale des sciences et des lettres de Danemark,
Kongelige Danske Videnskabernes Selskab = Royal Danish Academy of Sciences
and Letters

Sites on Thorvald Thiele:
A book by Lauritzen

On this list, Thiele's method is known as sequential Proportional Approval
Voting (PAV). There's also non-sequential PAV but you need a computer for
it because it checks all the possible election results.

The main reference to sequential PAV:

It's interesting that PAV and Thiele's method have been described with
almost the same words.
Michael Welford:
"The key to proportional approval voting (PAV) as I conceptualize it is to
assign to each voter a kind imputed utility that I call fair satisfaction,
and to maximize that sum of all voters satisfaction scores."

"The n-th candidate selected by that voter adds 1/n to that voters
satisfaction score. With a computer it's easy to find the set of
cnandidates that maximizes the sum of fair satisfaction scores over all the

Thiele (as quoted by Phragmén):
"The sum of satisfaction shall be the maximum for the winning combination
of candidates"

Both Cassel and Phragmén criticize Thiele's method. It's easy to implement,
but it is a false generalization of d'Hondt's rule. Cassel gives the
following example with 9 seats:

Thiele's method elects ABCDEFGHI, while Phragmén's method elects ABCDEFGJK.
Phragmén's examples are similar but shorter. He finds it bad that the
smaller group is penalized for voting for common candidates. The smaller
party would be tempted not to vote for a common candidate, and in that case
even the larger party might prefer not to vote for one, especially if the
difference between the sizes of the parties was smaller. Both parties might
be willing to accept a common candidate as chair of a committee but under
Thiele's method it might happen that neither party would want to vote for
him or her.

Phragmén finds justification for his method from the idea of branching
lists, which have common candidates only at the top of the lists.


With such a branching list Phragmén thinks that the whole combination
should be given as many seats as the total vote entitles. Three of those
seats should go to the common candidates, while the rest should be
apportioned between the branches according to d'Hondt's rule, in the order
D 4200, E 2100, J 1710, F 1400, G 1050, K 855. Phragmén's method gives the
same result.

If I've understood the method correctly and if there's no mistake in the
following calculations, ordinary non-sequential PAV with d'Hondt's rule
gives the same result as Thiele's method. The two outcomes above get the
following satisfaction scores:



4200*(1+1/2+1/3+...+1/7)+1710*(1+1/2+1/3) is the same in both, so to
determine which is greater we only need to count

4200*(1/8+1/9)=991 2/3

1710*(1/4+1/5)=769 1/2

So ABCDEFGHI would be preferred. I haven't calculated the other
possibilities, but if the order of a party's candidates is irrelevant there
aren't many of them.

Another way of comparing the methods is to check where the votes are at the
end of the election. Let's imagine the following election, with 3 seats to
be filled:
12 ABC
12 ADE

In Thiele's method, each ballot is equally divided between the candidates.
A common candidate needs more votes than other candidates. It is as if A
were two persons, one on the first list, another on the second.

        A       B       D
12 ABC  6       6
12 ADE  6               6
Sum    12       6       6

In Phragmén's method, the burden of one seat is shared between all ballots
that contain the name of the candidate. Both groups elect only half of A.
The next priority number is 12/(0.5+1)=8 for both groups.

        A       B       D
12 ABC  4       8
12 ADE  4               8
Sum     8       8       8

The result is the same with both methods in this case, but the votes are
distributed differently, which leads to differences in results in other

Thiele's method was used in Sweden for a while after 1909 (but only within
party lists), together with another rule. If more than n/(n+1) of the
voters placed the same n candidates in the same order at the head of the
list, these were given the first n seats. If this rule could not be
applied, recourse was taken to Thiele's method. This was replaced by
preferential d'Hondt-Phragmén's method, probably in the 1920s. One of the
problems with the 1909 Elections Act was the combination of a preferential
rule with a non-preferential one, which together with the problems that
Thiele's mehods has, could lead to unacceptable outcomes.

Olli Salmi

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