Phragmen's voting rules

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. A translated and revised into modern terminology definition is as follows

1. The ballots are Approval voting 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 "cost.") All ballots initially have cost=0.
3. Seats are elected sequentially. Perform steps 4-5 until all seats are filled:
4. As soon as any candidate is elected, the N ballots share the cost of 1 from that candidate in such a way that their ballots each have equal total costs. This is to keep the maximum cost on any one voter at a minimum. (Note: at any moment, the sum of all the ballot costs is equal to the number of seats filled so far. This fact can help with checking one's calculations.)
5. The candidate who wins the next seat is the one whose N supporters' ballots will each have the lowest total cost. (So, for example, the first winner is simply the most-approved candidate, because if he is approved by N voters the cost per approving-ballot is 1/N, which is minimal because N is maximal.)
6. Once a candidate has been elected and the cost distributed among the voters, this cost is fixed and cannot be redistributed once later candidates are elected. If this redistribution were allowed, then the result would be further optimized and closer to the non-sequential max-Phragmén method.

In the case of a single candidate to be elected, Phragmén's method reduces to Approval voting, because the candidate resulting in the minimal load on each voter is the one with most voters to share the load. When all votes are in party list order, it reduces to the D'Hondt method.

Variations and implementations

As described above, Phragmén's is a method that attempts to ensure that the winners' support are as widely distributed as possible. This can be done in different ways, and thus there exist different methods that follow the same broad principle. Brill et al. identify the following variants:

• seq-Phragmén: Phragmén's original method, described above. It attempts to minimize the maximal load by sequentially electing candidates.
• max-Phragmén: The non-sequential variant of seq-Phragmen: the objective is the same (minimize the maximal load), but it's treated as a global optimization problem. The load from each candidate does not have to be evenly spead across their voters, but is done so optimally to minimise the total maximum load (from all candidates) on any one voter.
• var-Phragmén: This variant minimizes the variance of the load distribution. As with max-Phragmén, the load from each candidate can be spread unevenly across their voters.
• Ebert's method: As var-Phragmén but where a candidate's load is evenly spread across their voters.

In addition, these variants are also described on Electowiki:

• Sequential Ebert: The sequential variant of Ebert's method.
• PAMSAC: An Ebert variant, developed by Toby Pereira, that restores monotonicity.

Further reading

• https://www.rangevoting.org/Phragmen.html