Phragmén’s voting methods and justified representation

In the late 19th century, 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, ´ focussing on justified representation and related properties that have recently been introduced by Aziz et al. (2015a) and Sanchez-Fern ´ andez et al. (2017). We show that the sequential ´ variant satisfies proportional justified representation, making it the first known polynomial-time computable method with this property. 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.