Phragm\'{e}n's Voting Methods and Justified Representation

In the late 19th century, Swedish mathematician Lars Edvard Phragmen 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 maximal load and one minimizing the variance of loads -- and a sequential variant. We study Phragmen'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 Phragmen's methods and provide mixed-integer programming based algorithms for computing them.