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. There are two kinds of Phragmén rules: rules using approval ballots (that is, multiwinner approval voting), and rules using ranked ballots (that is, multiwinner ranked voting). Phragmen's rules were adapted to the more general setting of combinatorial participatory budgeting. 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 (where k may be, for example, the number of parliament members). 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 wanted to keep the vote for individual candidates, so that voters can approve candidates based on their personal merits. In the special case in which each voter approves all and only the candidates of a single party, Phragmén's methods give the same results as D'Hondt's method. However, Phragmén's method can handle more general situations, in which voters may vote for candidates from different parties (in fact, the method ignores the information on which candidate belongs to which party). Sequential Phragmén's method for approval ballots 
Phragmén's method for unordered (approval) ballots can be presented in several equivalent ways.One way is as a 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 (as they "used" their vitrual money to fund y). Voters keep earning virtual money and funding candidates until all k committee members are elected. Examples 
Let us start with a simple example, that resembles party-list voting. Suppose 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. Voters start earning money at a fixed rate of 1 per day. After 1/31 day (~0.0323 day), 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 day (~0.0476 day), 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 ~0.0646 day, the abc voters again have 0.0323 each, so they buy another one of their approved candidates, say b. After 1/11 day (~0.091 day) the ghi voters have 0.091 each, so together they can fund one of their approved candidates, say g (at this point, the abc voters have only 0.0264 each and the def voters have 0.0434 each, so none of them can by another candidate). After 0.0952 day, the def voters again have 0.0476 each, so they can buy another candidate, say e.
 After 0.0969 day, the abc voters again have 0.0323 each, so they can buy another candidate, say c.
The final committee is a,b,c; d,e; g. Note that each "party" is represented approximately in proportion to its size: 3 candidates for 31 voters, 2 candidates for 21 voters, and 1 candidate for 11 voters. Here is a more complex 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:
 First, we compute for each candidate the required value of t so that the candidate can get a total voting-power of 1. This value is 1/1171 for A (since A appears in 1171 ballots); 1/1124 for B; 1/1034 for C; 1/566 for P; 1/656 for Q; 1/519 for R. Thus, A is elected first. Now, we re-compute for each candidate the required value of t so that the candidate can get a total voting-power of 1, keeping in mind to deduct 1/1171 from each voter who approved A. The required value for B is 1/1124+1/1171, since there are 1124 voters who approve B, and all of them already approved A. Similarly, the required value for C is 1/1034+1/1171; for Q it is 1/656+(137/656)/1171, since 137 out of 656 voters for Q already voted for A; for P it is 1/566+(47/566)/1171; and for R it is 1/519. The value is smallest for Q, so it is elected as the second winner. Similarly, B is elected as the third winner. Variants 
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. Properties

Homogeneity 
For each possible ballot b, let vb be the number of voters who voted exactly b (for example: approved exactly the same set of candidates). Let pb be fraction of voters who voted exactly b (= vb / the total number of votes). 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 is elected, then the outcome does not change. This reduces the incentive for strategic voting. Monotonicity 
Since Phragmén's methods assign seats one-by-one, they satisfy the house monotonicity property: when more seats are added, the set of winners increases (no winner loses a seat). They also satisfy several 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 (so one of them loses a 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 {a,b,a}. But, if one of the b voters approves a too (so that the ballots are: 4 for a, 6 for b, 2 for a+b, 16 for a+c, 4 for b+c), then the elected committee is {a,c,b}. So party a won an approval but lost a seat. For Phragmén's ranked-ballot method: if some candidate C is elected, and then candidate C is promoted in some of the ballots, or earns some new votes, and no other changes occur, then C is still elected. However, if some other changes occur simultaneously, then C might lose his seat. For example, it is possible that some voters change their mind, and instead of voting for A and B, they vote for C and D, and this change causes C to lose his seat. Consistency 
Phragmén's methods do not satsify the consistency criterion. Moreover, they do not ignore full ballots: adding voters who vote for all candidates (and thus are totally indifferent) might affect the outcome. Special cases 
When there is a single seat (k=1):

 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. 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 adaptaions of Phragmen's rule:

 Dynamic Phragmen: at each step, they 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(m n2), where m is the number of candidates and n the number of users. Then, users start accuing money as before, where a user can start buying new candidates only after having paid its "debt". They buy candidates in order, until the new ranking is computed. The new ranking is proportional. Computing the new sequence can be done in time O(m2 n2). 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 (in particular, when the sequence is empty, Myopic Phragmen coincides with utilitarian approval voting). Computing the new sequence can be done in time O(m n2). They analyze the monotonicity and fairness properties of these adaptations, both theoretically and empirically. Further reading 

 More information on Phragmén's methods is available at. Mathematical properties of Phragmen's methods vs. Thiele's methods. The methods of Enestrom and Phragmen. Implementations and demonstrations 

 Some of Phragmén's voting rules are implemented in the Python package abcvoting. Some of Phragmén's voting rules can be tried online on the pref.tools website. Both the simple and complicated versions are used in the substrate of the cryptocurrency Polkadot. See also 

 Single transferable vote and the Method of Equal Shares both have the same goal: allowing to vote for individual candidates while guaranteeing proportional representation. References 

Approval voting
Multi-winner electoral systems
Preferential electoral systems