Phragmén's voting rules are
multiwinner voting
Multiwinner voting, also called multiple-winner elections or committee voting or committee elections, is an electoral system in which multiple candidates are elected. The number of elected candidates is usually fixed in advance. For example, it can ...
methods that guarantee
proportional representation
Proportional representation (PR) refers to a type of electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to geographical (e.g. states, regions) and political divis ...
. They were published by
Lars Edvard Phragmén
Lars Edvard Phragmén (2 September 1863 Örebro – 13 March 1937) was a Swedish mathematician.
The son of a college professor, he studied at Uppsala then Stockholm, graduating from Uppsala in 1889. He became professor at Stockholm in 1892, afte ...
in French and Swedish between 1893 and 1899, and translated to English by
Svante Janson
Carl Svante Janson (born 21 May 1955) is a Swedish mathematician. A member of the Royal Swedish Academy of Sciences since 1994, Janson has been the chaired professor of mathematics at Uppsala University since 1987.
In mathematical analysis, Jans ...
in 2016.
There are two kinds of Phragmén rules: rules using
approval ballot An approval ballot, also called an unordered ballot, is a ballot in which a voter may vote for any number of candidates simultaneously, rather than for just one candidate. Candidates that are selected in a voter's ballot are said to be ''approved'' ...
s (that is,
multiwinner approval voting
Multiwinner approval voting, also called approval-based committee voting, is a multi-winner electoral system that uses approval ballots. Each voter may select ("approve") any number of candidates, and multiple candidates are elected. The number of ...
), and rules using
ranked ballots
The term ranked voting (also known as preferential voting or ranked choice voting) refers to any voting system in which voters rank their candidates (or options) in a sequence of first or second (or third, etc.) on their respective ballots. Ran ...
(that is, multiwinner
ranked voting
The term ranked voting (also known as preferential voting or ranked choice voting) refers to any voting system in which voters rank their candidates (or options) in a sequence of first or second (or third, etc.) on their respective ballots. Ra ...
).
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
Proportional representation (PR) refers to a type of electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to geographical (e.g. states, regions) and political divis ...
. 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. But 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).
Phragmén's method for approval ballots
Phragmén's method for unordered (approval) ballots can be presented in several equivalent ways.
Here is one description.
Each voter has identical voting-power, denoted by ''t''. Each candidate needs a total voting-power of 1 in order to be elected. For each elected candidate ''j'', the required voting-power of 1 is deducted equally from all voters who approved ''j''. The winners are elected one by one: the next winner is the one who requires the smallest ''t'' in order to attain a total voting-power of 1.
Example
There are 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.
Properties
Homogeneity
For each possible ballot ''b'', let ''v
b'' be the number of voters who voted exactly ''b'' (for example: approved exactly the same set of candidates). Let ''p
b'' 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
House monotonicity (also called house-size monotonicity) is a property of apportionment methods and multiwinner voting systems. These are methods for allocating seats in a parliament among federal states (or among political party). The property s ...
property: when more seats are added, the set of winners increases (no winner loses a seat).
They also satisfy the following
monotonicity criterion
The monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. A ranked voting system is monotonic if it is neither possible to prevent the election of a candidate by ranking them h ...
:
* 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).
* 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
A voting system is consistent if, whenever the electorate is divided (arbitrarily) into several parts and elections in those parts garner the same result, then an election of the entire electorate also garners that result. Smith calls this property ...
. 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
Plurality voting refers to electoral systems in which a candidate, or candidates, who poll more than any other counterpart (that is, receive a plurality), are elected. In systems based on single-member districts, it elects just one member pe ...
- it always selects the candidate ranked first by the largest number of voters.
Further reading
More information on Phragmén's methods is available at.
Implementations
* Some of Phragmén's voting rules are implemented in the Python packag
''abcvoting''
Both the simple and complicated versions are used in the substrate of the
cryptocurrency Polkadot.
References
{{reflist
Approval voting
Multi-winner electoral systems
Preferential electoral systems