In multiwinner approval voting, each voter can vote for one or more candidates, and the goal is to select a fixed numberk of winners (wherek may be, for example, the number of parliament members). The question is how to determine the set of winners?
The simplest method ismultiple non-transferable vote, in which thek candidates with the largest number of approvals are elected. But this method tends to selectk 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 guaranteeproportional representation. One solution, advocated for example byD'Hondt in 1878, was to vote forparty-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.[2]: Sec.11 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).
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:[3]
Leximax-Phragmen: Minimizing the maximum load, and subject to that the second-maximum load, etc. (usinglexicographic max-min optimization).
Leximin-Phragmen: Maximizing the minimum load, and subject to that the second-minimum load, etc..
var-Phragmen orEbert's method: Minimizing thevariance of the load.
Each of these variants has two sub-variants:
Aglobal optimization variant, which is usually NP-hard to compute;
Asequential 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 (i.e., agreedy algorithm).
Phragmen's original method is the sequential method that minimizes the maximum load, which is currently known asSeq-Phragmen.[3]
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.
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 timet, we define a not-yet-elected candidatex asaffordable if the total money held by voters who approvex is at least 1.
At the first time in which some candidate is affordable, we choose one affordable candidatey arbitrarily. We addy to the committee, and reset the virtual money of voters who approvey (as they have now "used" their virtual money to fundy).
Voters keep earning virtual money and funding candidates until allk committee members are elected.
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.
Voters start earning money at a fixed rate of 1 per day. After 1/31 or ~0.0323 days, the 31abc 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 31abc voters have only ~0.015 each, but the 21def 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, theabc voters again have 0.0323 each, so they buy another one of their approved candidates, say b.
After 1/11 or ~0.0909 days, theghi voters have 0.0909 each, so together they can fund one of their approved candidates, say g (at this point, theabc voters have only 0.0264 each and thedef voters have 0.0434 each, so none of them can by another candidate).
After 0.0952 days, thedef 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.
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.
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.[3]
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 (1/3, 0, 0, 0, 1/3).
After 2/3 days, the money distribution is (2/3, 1/3, 1/3, 1/3, 2/3). Thus, the approvers of a can buy the candidate, with their money afterward being reset to 0 leading to a distribution of (0, 1/3, 1/3, 0, 2/3).
Finally, after 1 day, the money distribution is (1/3, 2/3, 2/3, 1/3, 1). Thus, either c or d can be bought according to the tie-breaking used.
Hence, for the committee size k = 3 both {a,b,c} and {a,b,d} are valid seq-Phragmén committees.
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 fromMaximum independent set oncubic graphs.[3]
Var-Phragmen can be computed by solving one mixed-integerquadratic program with O(n m) variables.
Seq-Phragmen can be computed in polynomial time. A naive computation shows that the run-time is O(k m n): there arek steps (one for each elected candidate); 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 tok logn. Since computations inb bits may require O(b2) time, the total run-time is O(k3 m n log2 n).
Phragmén rules are commonly used withapproval ballots (that is,multiwinner approval voting), but they have variants usingranked ballots (that is, multiwinnerranked voting). 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.[2]: Sec.9
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 (i.e., rank him first); the load division should minimize the maximum load of a voter.
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.[4]
Jaworski and Skowron[6] constructed a class of rules that generalise seq-Phragmen fordegressive 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.
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[7] extend this method todynamic rankings. Motivated by online Q&A applications,[8] 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(m n2), wherem is the number of candidates andn 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(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 utilitarianapproval 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.
For each possible ballotb, letvb be the number of voters who voted exactlyb (for example: approved exactly the same set of candidates). Letpb be fraction of voters who voted exactlyb (=vb / the total number of votes). A voting method is calledhomogeneous if it depends only on the fractionspb. 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.[2]: Rem.2.1
If any number of candidates is added to a ballot, but none of them are elected (even if some of them are voted for), then the outcome does not change.[2]: Sec.6 This reduces one incentive for strategic manipulation: adding "dummy" candidates to attract votes.
Seq-Phragmén assign seats one-by-one, so it satisfies thecommittee monotonicity property: when more seats are added, the set of winners increases (no winner loses a seat).[2]: Sec.5
For Phragmén's approval-ballot method: if some candidateC is elected, and then candidateC earns some approvals either from new voters who vote forC, or from existing voters who addC to their ballots, and no other changes occur, thenC is still elected. However, this monotonicity doesnot 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).[2]: Ex.14.4, 14.5 Similarly, monotonicity doesnot hold in the variant with parties: a party can get more approvals but still get fewer seats. For example:[4]
Suppose there arek=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 candidateC is elected, and then candidateC is promoted in some of the ballots, or earns some new votes, and no other changes occur, thenC is still elected. However, if some other changes occur simultaneously, thenC 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.[2]: Ex.13.16
The Sequential Phragmen rule satisfies an axiom known asProportional Justified Representation (PJR).[3] This makes it one of the only methods satisfying both PJR and monotonicity.
There are 14 candidates: a, b, c1, ..., c12. There are 12 seats to fill.
There are 24 voters: two voters approve {a,b,c1}; two voters approve {a,b,c2}; 6 voters approve {c1,c2,...,c12}; 5 voters approve {c2,c3,...,c12); 9 voters approve {c3,c4,...,c12}.
Seq-Phragmen selects c1,...,c12. It violates EJR for the four voters who approve {a,b,c1} and {a,b,c2}: this group has 2 quotas and it is 2-cohesive, but no member has 2 approved winners.
Another example is given here (for the setting of parties):[9]
There are 3 candidate-parties and 10 seats to fill.
There are 10 voters, with approval sets ab,ab,ab; ac,ac,ac,ac; bc,bc; b.
Seq-Phragmen chooses a (at time 1/7); then b; then a,b,a,b,a,b,a,b.
Voters 1,2,3 approve all 10 candidates, but voters 4,...,10 approve only 5 candidates. However, the group of voters 4,5,6,7,8,9 all agree on party c, so EJR requires that at least one of them should approve 6 candidates, so EJR is violated (note that PJR is not violated for that group, since all 10 candidates are approved by at least one member of the group).
Seq-Phragmen also fails a different, incompatible axiom calledPerfect Representation (PER).
Var-Phragmen satisies PER, but fails PJR and EJR (except for the case L=1).
Leximan-Phragmen satisfies both PJR and PER, but still fails EJR.
Phragmén's methods do not satisfy theconsistency criterion. Moreover, they do not ignore full ballots: adding voters who vote for all candidates (and thus are totally indifferent) might affect the outcome.[2]: Ex.15.4, 15.6, 15.8, 15.9
Motamed, Soeteman, Rey and Endriss[16] present asequential load balancing mechanism, that generalizes Phragmen's rule to participatory budgeting with multiple resources.
^1. "Om proportionella val." (Summary of a public lecture). Stockholms Dagblad, 14 March 1893.2. "Sur une m ́ethode nouvelle pour r ́ealiser, dansles ́elections, la repr ́esentation proportionelle des partis". ̈Ofversigt avKongl. Vetenskaps-Akademiens F ̈orhandlingar 1894, N:o 3, Stockholm,133–137.3. "Proportionella val. En valteknisk studie." Svenskasp ̈orsm ̊al 25, Lars H ̈okersbergs f ̈orlag, Stockholm, 1895.4. "Sur la th ́eorie des ́elections multiples", ̈Ofversigt avKongl. Vetenskaps-Akademiens F ̈orhandlingar 1896, N:o 3, Stockholm,181–191.5. "Till fr ̊agan om en proportionell valmetod." Statsvetenskaplig Tidskrift2(1899), nr 2, 297–305.http://cts.lub.lu.se/ojs/index.php/st/article/view/1949
^Janson, Svante; Öberg, Anders (2017). "A piecewise contractive dynamical system and election methods".arXiv:1709.06398 [math.DS].
^Camps, Rosa; Mora, Xavier; Saumell, Laia (2019). "The method of Eneström and Phragmén for parliamentary elections by means of approval voting".arXiv:1907.10590 [econ.TH].
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