Phragmen's voting rules
Phragmén's voting rules are rules for multiwinner voting. They allow voters to vote for individual candidates rather than parties, but still guarantee proportional representation. They were published by Lars Edvard Phragmén in French and Swedish between 1893 and 1899, and translated to English by Svante Janson in 2016.
Background
In multiwinner approval voting, each voter can vote for one or more candidates, and the goal is to select a fixed number k of winners. The question is how to determine the set of winners?- The simplest method is multiple non-transferable vote, in which the k candidates with the largest number of approvals are elected. But this method tends to select k candidates of the largest party, leaving the smaller parties with no representation at all.
- In the 19th century, there was much discussion regarding election systems that could guarantee proportional representation. One solution, advocated for example by D'Hondt in 1878, was to vote for party-lists rather than individual candidates. This solution is still very common today.
Phragmén's rules for approval ballots
Phragmén's method for unordered ballots can be presented in several equivalent ways.Load balancing
Each elected candidate creates a "load" of 1 unit. The load of a candidate must be born by voters who support him. The goal is to find a committee for which the load can be divided among the voters in the most "balanced" way.Depending on the exact definition of "balanced" several rules are possible:
- Leximax-Phragmen: Minimizing the maximum load, and subject to that the second-maximum load, etc..
- Leximin-Phragmen: Maximizing the minimum load, and subject to that the second-minimum load, etc..
- var-Phragmen or Ebert's method: Minimizing the variance of the load.
- A global optimization variant, which is usually NP-hard to compute;
- A sequential variant, in which candidates are selected sequentially, and in each turn, the next elected candidate is the one who attains the optimal measure among all candidates.
In practice, the rules that have the best axiomatic guarantees in the global-optimization category are leximax-Phragmen and var-Phragmen. Among the sequential variants, the best guarantees are given by Seq-Phragmen.
Phragmen illustrated his method by representing each voter as a vessel. The already-elected candidates are represented by water in the vessels. To elect another candidate, 1 liter of water has to be poured into the vessels corresponding to voters who voter for that candidate. The water should be distributed such that the maximum height of the water is as small as possible.
Virtual money
Seq-Phragmen can alternatively be described as the following continuous process:- Each voter starts with 0 virtual money, and receives money in a constant rate of 1 per day.
- At each time t, we define a not-yet-elected candidate x as affordable if the total money held by voters who approve x is at least 1.
- At the first time in which some candidate is affordable, we choose one affordable candidate y arbitrarily. We add y to the committee, and reset the virtual money of voters who approve y.
- Voters keep earning virtual money and funding candidates until all k committee members are elected.
Examples
Party-list
The following simple example resembles party-list voting. There are k=6 seats and 9 candidates, denoted a,b,c,d,e,f,g,h,i. There are 63 voters with the following preferences: 31 voters approve a,b,c; 21 voters approve d,e,f; and 11 voters approve g,h,i.Round 1
Let candidates be given a score equal to the reciprocal of the number of approvers. We get the following:| Candidate | Score |
| a | |
| b | |
| c | |
| d | |
| e | |
| f | |
| g | |
| h | |
| i |
We select the candidate with the lowest score. In this case, it's tied between a, b, and c. Let's suppose that a was selected.
Round 2
Now, candidate scores are increased by the product of 1/31 and the proportion of approvers who also approve of a.| Candidate | Derivation | Score |
| b | 1/31+1 | |
| c | 1/31+1 | |
| d | 1/21+0 | |
| e | 1/21+0 | |
| f | 1/21+0 | |
| g | 1/11+0 | |
| h | 1/11+0 | |
| i | 1/11+0 |
We select the candidate with the lowest score. In this case, it's tied between d, e, and f. Let's suppose that d was selected.
Round 3
Now, candidate scores are increased by the sum of the product of 1/31 and the proportion of approvers whose 'most recent' approval was a, and the product of 1/21 and the proportion of approvers whose 'most recent' approval was d.| Candidate | Derivation | Score |
| b | 1/31+1+0 | |
| c | 1/31+1+0 | |
| e | 1/21+0+1 | |
| f | 1/21+0+1 | |
| g | 1/11+0+0 | |
| h | 1/11+0+0 | |
| i | 1/11+0+0 |
We select the candidate with the lowest score. In this case, it's tied between b and c. Let's suppose that b was selected.
Round 4
Now, candidate scores are increased by the sum of the product of 1/31 and the proportion of approvers whose 'most recent' approval was a, and the sum of the product of 1/21 and the proportion of approvers whose 'most recent' approval was d, and the product of 2/31 and the proportion of approvers whose 'most recent' approval was b.| Candidate | Derivation | Score |
| c | 1/31+0+0+1 | |
| e | 1/21+0+1+0 | |
| f | 1/21+0+1+0 | |
| g | 1/11+0+0+0 | |
| h | 1/11+0+0+0 | |
| i | 1/11+0+0+0 |
We select the candidate with the lowest score. In this case, it's tied between g, h, and i. Let's suppose that g was selected.
Round 5
We select the candidate with the lowest score. In this case, it's tied between e and f. Let's suppose that e was selected.Round 6
We select the candidate with the lowest score, which is c.- Voters start earning money at a fixed rate of 1 per day. After 1/31 or ~0.0323 days, the 31 abc voters have 0.0323 each, so together they can fund one of their approved candidates. One of a,b,c is chosen arbitrarily; suppose it is a.
- After 1/21 or ~0.0476 days, the 31 abc voters have only ~0.015 each, but the 21 def voters have 0.0476 each, so together they can fund one of their approved candidates. One of d,e,f is chosen arbitrarily; suppose it is d.
- After ~0.0645 days, the abc voters again have 0.0323 each, so they buy another one of their approved candidates, say b.
- After 1/11 or ~0.0909 days, the ghi voters have 0.0909 each, so together they can fund one of their approved candidates, say g.
- After 0.0952 days, the def voters again have 0.0476 each, so they can buy another candidate, say e.
- After 0.0968 days, the abc voters again have 0.0323 each, so they can buy another candidate, say c.
Small, non-{party-list}
As an example without a party structure, consider the following instance with 4 candidates, denoted by a,b,c,d, and 5 voters with approval sets 1: a; 2: b; 3: b and c; 4: a,b, and c; 5: d.Round 1
Again, let candidates be given a score equal to the reciprocal of the number of approvers. We get the following:| Candidate | Score |
| a | |
| b | |
| c | |
| d |
We select the candidate with the lowest score, which is b.
Round 2
Now, candidate scores are increased by the product of 1/3 and the proportion of approvers who also approve of b.| Candidate | Derivation | Score |
| a | 1/2+ | |
| c | 1/2+1 | |
| d | 1/1+0 |
We select the candidate with the lowest score, which is a.
Round 3
Now, candidate scores are increased by the sum of the product of 1/3 and the proportion of approvers whose 'most recent' approval was b, and the product of 2/3 and the proportion of approvers whose 'most recent' approval was a.| Candidate | Derivation | Score |
| c | 1/2++ | 1 |
| d | 1/1+0+0 | 1 |
We select the candidate with the lowest score. In this case, it's tied between c and d.
- Voters again start earning money at a fixed rate of 1 per day. After 1/3 days, the approvers of b have enough to buy b, resetting their money to 0, leading to a money distribution of.
- After 2/3 days, the money distribution is. Thus, the approvers of a can buy the candidate, with their money afterward being reset to 0 leading to a distribution of.
- Finally, after 1 day, the money distribution is. Thus, either c or d can be bought according to the tie-breaking used.