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, Copenhague Kongelige Danske Videnskabernes Selskab = Royal Danish Academy of Sciences and Letters Sites on Thorvald Thiele: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Thiele.html A book by Lauritzen http://www.oup.co.uk/isbn/0-19-850972-3 Symposium http://www.math.ku.dk/~michael/thiele/ 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: http://groups.yahoo.com/group/election-methods-list/message/8744 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 voters." http://groups.yahoo.com/group/election-methods-list/message/6391 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: 4200 ABCDEFGHI 1710 ABCJKLMNO 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. DEFGHI / ABC \ JKLMNO 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: ABCDEFGHI 4200*(1+1/2+1/3+...+1/9)+1710*(1+1/2+1/3) ABCDEFGJK 4200*(1+1/2+1/3+...+1/7)+1710*(1+1/2+...+1/5) 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 ABCDEFGHI 4200*(1/8+1/9)=991 2/3 ABCDEFGJK 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 cases. 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 ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em