## Phragmén's proportional representation multiwinner voting method

Warren D. Smith. Nov 2015.

Once upon a time, in 2004, I invented what seemed to be a nice new proportional representation multiwinner voting method, which I called "reweighted range voting" (RRV). But what I did not know was, the same basic idea had already been invented in 1895 (albeit for approval-style, rather than rating-style, ballots, and only in the "d'Hondt" case) by Thorvald N. Thiele (1838-1910) chief astronomer at the University of Copenhagen. His paper was not exactly well known in the non-Danish speaking voting theory world, but nowadays you can read it online, thanks to the joys of "google scanned books":

Thorvald N. Thiele: Om flerfoldsvalg [On multiple election], 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 3,4 (1895-6) 415-441.

Thiele was a mathematician of some lasting repute; he invented Thiele's rational-function interpolation formula employing continued fractions. But then a Swedish mathematician, Lars Edvard Phragmén (1863-1937), stepped into the fray. Phragmen was a professor of mathematics at Stockholm university 1892-1904, and later became an insurance company executive, e.g. was president of national life insurance co 1908-1930. Phragmén is nowadays remembered for the Phragmén-Lindelöf principle in complex analysis. Phragmén was particularly noted (or feared) for his ability to find flaws in the work of others.

Lars Edvard Phragmén: Till frågan om en proportionell valmetod [On the Question of a Proportional Election Method], Statsvetenskaplig Tidskrift 2,2 (1899) 87-95. Phragmén's method was first described in "Proportionella val" [Proportional Elections], Svenska spörsmâl 25 (1895) Stockholm.

And, sure enough, Phragmén found some interesting flaws (or anyway, worrying features) of Thiele's method. Further criticism of Thiele was by

Nore B. Tenow: Felaktigheter i de Thieleska valmetoderna [problems with Thiele's methods], Statsvetenskaplig Tidskrift [Swedish state scientific journal published by Fahlbeckska Foundation of Political Science in Lund] (1912) 145-165.

And then Phragmén proposed his own voting method, which also could be proved to enjoy proportional representation, but seemed not to suffer from those flaws. Both methods, among others, then were reviewed and criticized by the economist Gustav Cassel (1866-1945) in a book length 1903 Swedish government report titled Proportionella val.

Phragmén describes his method on page 88. Unfortunately, his description is pretty incomprehensible even if you do read Swedish. Here is my version of his description, which I've translated into English then reworded heavily to cause it to make sense and employ modern terminology.

1. The ballots are "approval style" i.e. each ballot lists the set of candidates that voter "approves."
2. Later on, we shall associate a "cost" with each ballot. (Phragmén used the Swedish word "belastning." Other people often prefer to translate this into the English word "load" rather than my "cost.") All ballots initially have cost=0.
3. Seats are elected sequentially. Now perform steps 4-6 until all seats are filled:
4. As soon as any candidate is elected, the N ballots that approved him have 1/N added to each of their costs. (Note: at any moment, the sum of all the ballot costs, equals the number of seats filled so far. This fact can help with checking one's calculations.)
5. [This step is really peculiar, and perhaps things would be better if it were omitted.] Immediately after a candidate is elected, we then redistribute the costs among his approvers, to make their ballots each have equal costs.
6. The candidate who wins the next seat is the one whose N supporters' ballots will have the least average cost. (So, for example, the first winner is simply the most-approved candidate, because if he is approved by N voters the average cost per approving-ballot is 1/N, which is minimal because N is maximal.)

Phragmén's election example on page 90:

```         approved
#voters  canddts
1034   ABC
519   PQR
90   ABQ
47   APQ
```

A wins the first seat since he is approved by the most voters, 1171; hence those approving him have the least average cost, 1/1171. We now give those 1171 ballots each cost=1/1171 and continue:

```         approved  summed
#voters  canddts   cost
1034   ABC     1034/1171
519   PQR         0
90   ABQ      90/1171
47   APQ      47/1171
```
Q wins the second seat because his 656 supporters have the lowest total cost. Because: If Q is elected, his 656=519+90+47 supporters would have average cost (1+(90+47)/1171)/656=327/192044≈0.00170273. If instead B were elected, his 1124=1034+90 supporters would have slightly greater average cost (1+1124/1171)/1124=2295/1316204≈0.00174365. Therefore Q wins. We now add 1/656 to the costs on each of his supporters' ballots:

```         approved  summed
#voters  canddts   cost
1034   ABC      1034/1171
519   PQR       519/656
90   ABQ   90/1171+90/656
47   APQ   47/1171+47/656
```

and then we redistribute those costs so that each Q-approving ballot has cost=327/192044:

```         approved  summed
#voters  canddts   cost
1034   ABC      1034/1171
519   PQR   519×327/192044
90   ABQ   90×327/192044
47   APQ   47×327/192044
```

Now B wins the third seat. That is because his 1124=1034+90 approvers will have least average cost (1+1034/1171+90×327/192044)/1124=195525/107928728≈0.00181161. P does not win, because his 566=519+47 approvers would have had greater average cost (1+519×327/192044+47×327/192044)/566=188563/54348452≈0.00346952.

If we had not redistributed according to rule 5, then B still would have won the third seat, because his 1124=1034+90 approvers still would have had least average cost (1+1034/1171+90/1171+90/656)/1124=805455/431714912≈0.00186571. P would not have won, because his 566=519+47 approvers would have had greater average cost (1+519/656+57/1171+47/656)/566=734177/217393808≈0.00337718.

Proportionality theorem: Actually, Phragmén did not state one. But it is the same as the RRV proportionality theorem, and it holds regardless of whether we use or omit Phragmén's peculiar rule 5.

RRV and Phragmén behave almost exactly the same if all voters are "total racists" i.e. always approve candidates of their own "color" and never approve those with different color – in that case both Phragmén and RRV guarantee the color composition of the parliament will duplicate that of the voters (up to small round-to-integer errors, and provided enough candidates of each color run; obviously there cannot be 3 Red winners if only 2 Reds run).

The differences between Phragmén and RRV behaviors only arise when the voters are not all "extreme racists."

One difference is as follows. Suppose there is some universally-approved candidate A. He, of course, wins the first seat. We now proceed to elect further seats. With Phragmén (and this is true whether or not we include the peculiar rule 5), the entire subsequent process proceeds in the same manner as if A had not existed. But with RRV, the further seat-elections B,C,D,... can (and often will) be altered (starting with C) as a consequence of A's existence.

So if the voters behave as "total racists" about all candidates except for universally-approved candidates, then Phragmén (with or without rule 5) will yield color-proportionality in all the seats not occupied by universally-approved candidates. But RRV usually will fail to deliver that kind of proportionality.

In this kind of example, which method behaves better – Phragmén or RRV? That depends on your view about the universally-approved candidates U. If you regard U as "uncolored" then Phragmén will elect a parliament whose color-composition duplicates that of the electorate. If, on the other hand, we regard U as "multicolored," then RRV will elect a parliament whose color-composition duplicates that of the electorate, provided we get to choose the "internal color composition" of U after the election and do it in just the right way (there will always be a suitable way). But Phragmén also enjoys this latter property. Therefore Phragmén seems superior to RRV for proportional approval-style voting in situations involving both universally-approved candidates and "racism." (Actually a more accurate word than "racism" for describing what is going on here, might be "partisanship.")

Indeed, we can go further. Suppose there is some "gray" candidate who is approved by fraction X of all voters, regardless of their color. (E.g, each voter independently flips a coin with bias X, to decide whether to approve him.) With Phragmén (either with or without the peculiar redistribution rule 5) if there only are gray candidates (possibly each with different grayness levels X) and colored candidates, then the parliament seats occupied by non-gray candidates automatically will have color composition that duplicates the electorates'.

And we can go even further. Suppose a "blue & red" candidate exists, who is approved by fraction X of blue voters, and by same-fraction X of the red voters, but is not approved by any other color (green etc) of voter. Then the relative proportion of seats occupied by pure-blue and pure-red candidates, will duplicate the blue and red fractions of the electorate, regardless of how many "blue & red" candidates get elected.

So Phragmén really enjoys a quite sophisticated kind of proportionality.

However, Phragmen's method is not perfect. It suffers "non-monotonic" phenomena:

Anti-Phragmen election example by Jan Lanke (mentioned in an unpublished book by Svante Janson): 2 seats to fill, 44 votes:

15 A&B, 12 B&X, 14 C&Y, 3 Z&W.

The first seat goes to A and the second to C. Suppose now that the three voters who approved Z&W change to instead vote A&C. One might think that both A and C have been strengthened, and that it everything should proceed at least as happily for them after the change. But now, paradoxically, A&B win.