Phragmén's voting methods and justified representation.

In the late 19th century, Swedish mathematician Edvard Phragmén proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants-one minimizing the maximum load and one minimizing the variance of loads-and a sequential variant. We study Phragmén 's methods from an axiomatic point of view, focusing on properties capturing proportional representation. We show that the sequential variant satisfies proportional justified representation , which is a rare property for committee monotonic methods. Moreover, we show that the optimization variants satisfy perfect representation . We also analyze the computational complexity of Phragmén 's methods and provide mixed-integer programming based algorithms for computing them.