Phragmen's voting rules

They were published by Lars Edvard Phragmén in French and Swedish between 1893 and 1899,[1] and translated to English by Svante Janson in 2016.

The goal is to find a committee for which the load can be divided among the voters in the most "balanced" way.

Seq-Phragmen can alternatively be described as the following continuous process: The following simple example resembles party-list voting.

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(b2) time, the total run-time is O(k3 m n log2 n).

[2]: Sec.9 In the adapted version, in each round, each voter effectively votes only for the highest-ranked remaining candidate.

[4] The Seq-Phragmen rule was adapted to the more general setting of combinatorial participatory budgeting.

[5] Jaworski and Skowron[6] constructed a class of rules that generalise seq-Phragmen for degressive and regressive proportionality.

Intuitively: 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.

Motivated by online Q&A applications,[8] they assume that some candidates were already chosen, and use this information in computing the ranking.

[2]: Sec.6  This reduces one incentive for strategic manipulation: adding "dummy" candidates to attract votes.

[2]: Ex.13.16 The Sequential Phragmen rule satisfies an axiom known as Proportional Justified Representation (PJR).

One example is given here:[3] Another example is given here (for the setting of parties):[9] Seq-Phragmen also fails a different, incompatible axiom called Perfect Representation (PER).

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 When there is a single seat (k=1): Motamed, Soeteman, Rey and Endriss[16] present a sequential load balancing mechanism, that generalizes Phragmen's rule to participatory budgeting with multiple resources.