# Phragmén's Method

**Phragmén's Method** is one of the three common interpretations of non-Partisan Proportional representation. It is named after the inventor Lars Edvard Phragmén. It was devised as a solution to a flaw he found in Thiele's method.

Phragmén describes his method on page 88 of his original work ^{[1]}. A translated and revised into modern terminology definition is as follows

- The ballots are Approval voting i.e. each ballot lists the set of candidates that voter "approves."
- 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.
- Seats are elected sequentially. Now perform steps 4-6 until all seats are filled:
- 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.)
- [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.
- 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.)

In the case of 1 candidate to be elected, Phragméns method degenerates to Approval voting (because the candidate resulting in the minimal load on each voter is the one with most voters to share the load).

Expanding on the original method there are several implementations.