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.
Each of these variants has two sub-variants:
- 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.
Realistic
Here is a more 'realistic' example. There are k=3 seats and 6 candidates, denoted by A, B, C, P, Q, R. The ballots are: 1034 vote for ABC, 519 vote for PQR, 90 vote for ABQ, 47 vote for APQ. The winners are elected sequentially as follows: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 | |
| P | |
| Q | |
| R |
We select the candidate with the lowest score, which is A.
Round 2
Now, candidate scores are increased by the product of 1/1171 and the proportion of approvers who also approve of A.| Candidate | Derivation | Score |
| B | 1/1124+1 | |
| C | 1/1034+1 | |
| P | 1/566+ | |
| Q | 1/656+ | |
| R | 1/519+0 |
We select the candidate with the lowest score, which is Q.
Round 3
Now, candidate scores are increased by the sum of the product of 1/1171 and the proportion of approvers whose 'most recent' approval was A, and the product of 327/192044 and the proportion of approvers whose 'most recent' approval was Q.| Candidate | Derivation | Score |
| B | 1/1124++ | |
| C | 1/1034+1+0 | |
| P | 1/566+0+1 | |
| R | 1/519+0+1 |
We select the candidate with the lowest score, which is B.
Computation
Var-Phragmen and Leximax-Phragmen are NP-hard to compute, even when each agent approves 2 candidates and each candidate is approved by 3 voters. The proof is by reduction from the maximum independent set on cubic graphs.Leximax-Phragmen can be computed by a sequence of at most 2n mixed-integer linear programs with O variables each ; see Lexicographic max-min optimization.
Var-Phragmen can be computed by solving one mixed-integer quadratic program with O variables.
Seq-Phragmen can be computed in polynomial time. A naive computation shows that the run-time is O: there are k steps ; in each step, we have to check all candidates to see which of them can be funded; and for each candidate, we have to check all voters to see which of them can fund it. However, to be accurate, we need to work with rational numbers, and their magnitude grow up to k log n. Since computations in b bits may require O time, the total run-time is O.
Phragmén's rules for Ranked ballots
Phragmén rules are commonly used with approval ballots, but they have variants using ranked ballots. An adaptation for Seq-Phragmen was proposed in 1913 by a Royal Commission on the Proportional Election Method. The method has been used in Swedish elections for the distribution of seats within parties since 1921.In the adapted version, in each round, each voter effectively votes only for the highest-ranked remaining candidate. Again, when a candidate is elected, his "load" of 1 unit should be distributed among the candidates who vote for him ; the load division should minimize the maximum load of a voter.
Variants
Party voting
It is possible to use Phragmen's method for parties. Each voter can approve one or more parties. The procedure is the same as before, except that now, each party can be selected several times - between 0 and the total number of candidates in the party.Participatory budgeting
The Seq-Phragmen rule was adapted to the more general setting of combinatorial participatory budgeting.Degressive and regressive proportionality
Jaworski and Skowron constructed a class of rules that generalise seq-Phragmen for degressive and regressive proportionality. Intuitively:- Degressive proportionality is obtained by assuming that the voters who already have more representatives earn money at a slower rate than those that have fewer;
- Regressive proportionality is implemented by assuming that the candidates who are approved by more voters cost less than those that garnered fewer approvals.
Using Phragmen's method to rank alternatives
The sequential Phragmen method can be used not only to select a subset, but also to create a ranking of alternatives, according to the order by which they are chosen. Brill and Israel extend this method to dynamic rankings. Motivated by online Q&A applications, they assume that some candidates were already chosen, and use this information in computing the ranking. They suggest two adaptations of Phragmen's rule:- Dynamic Phragmen: at each step, loop over the sequence of already-elected candidates, and divide their "cost" among their supporters. This creates, for each user, a potential "debt" - negative balance. Computing the debts can be done in time O, where m is the number of candidates and n the number of users. Then, users start accruing money as usual, where a user can start buying new candidates only after having paid its "debt". Users buy candidates sequentially, until the new ranking is computed. The new ranking is proportional. Computing the new sequence can be done in time O.
- Myopic Phragmen: the "debt" of each user is computed as in Dynamic Phragmen. Then, instead of creating a complete ranking by running Sequential Phragmen, the candidates are ranked by the amount of "debt" they will create to the users. That is: the candidates are ranked by their suitability to be elected next. The resulting ranking is not necessarily proportional. Computing the new sequence can be done in time O.
Properties
Homogeneity
For each possible ballot b, let vb be the number of voters who voted exactly b. Let pb be fraction of voters who voted exactly b. A voting method is called homogeneous if it depends only on the fractions pb. So if the numbers of votes are all multiplied by the same constant, the method returns the same outcome. Phragmén's methods are homogeneous in that sense.Independence of unelected candidates
If any number of candidates is added to a ballot, but none of them are elected, then the outcome does not change. This reduces one incentive for strategic manipulation: adding "dummy" candidates to attract votes.Monotonicity
Seq-Phragmén assign seats one-by-one, so it satisfies the committee monotonicity property: when more seats are added, the set of winners increases.They also satisfy several other monotonicity criteria.
For Phragmén's approval-ballot method: if some candidate C is elected, and then candidate C earns some approvals either from new voters who vote for C, or from existing voters who add C to their ballots, and no other changes occur, then C is still elected. However, this monotonicity does not hold for pairs of candidates, even if they always appear together. For example, it is possible that candidates C, D appear together in all ballots and get two seats, but if another ballot is added for C, D, then they get together only one seat. Similarly, monotonicity does not hold in the variant with parties: a party can get more approvals but still get fewer seats. For example:
- Suppose there are k=3 seats and 3 candidates: a,b,c. The ballots are: 4 for a, 7 for b, 1 for a+b, 16 for a+c, 4 for b+c. Then the elected committee is. But, if one of the b voters approves a too, then the elected committee is. So party a won an approval but lost a seat.
Justified representation
The Sequential Phragmen rule satisfies an axiom known as Proportional Justified Representation. This makes it one of the only methods satisfying both PJR and monotonicity.However, it fails a stronger axiom known as Extended Justified Representation. One example is given here:
- There are 14 candidates: a, b, c1,..., c12. There are 12 seats to fill.
- There are 24 voters: two voters approve ; two voters approve ; 6 voters approve ; 5 voters approve
Consistency
Phragmén's methods do not satisfy the consistency criterion. Moreover, they do not ignore full ballots: adding voters who vote for all candidates might affect the outcome.[Consistency criterion|]Special cases
When there is a single seat :- Phragmén's approval-ballot method reduces to approval voting - it always selects the candidate with the largest number of approvals.
- Phragmén's ranked-ballot method reduces to plurality voting - it always selects the candidate ranked first by the largest number of voters.
Implementations and demonstrations
- Some of Phragmén's voting rules are implemented in the Python package .
- Some of Phragmén's voting rules can be tried online on the https://pref.tools/abcvoting/ pref.tools website.
- Both the simple and complicated versions are used in the substrate of the cryptocurrency Polkadot.