Phragmén's Voting Methods and Justified Representation
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In the late 19th century, Swedish mathematician Lars 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 maximal 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.
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