In the late 19th century, Swedish mathematician Lars Edvard Phragm\'{e}n 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 Phragm\'{e}n'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 Phragm\'{e}n's methods and provide mixed-integer programming based algorithms for computing them.

中文翻译：

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弗拉门的投票方法和合理的表述

在19世纪后期，瑞典数学家Lars Edvard Phragm'{e} n提出了一种基于批准投票来选择委员会的负载平衡方法。我们考虑了这种方法产生的三项委员会投票规则：两个优化变体（一个最小化最大负载，一个最小化负载变化）和一个顺序变体。我们从公理的角度研究Phragm'{e} n的方法，重点是捕获比例表示的属性。我们表明，顺序变体满足比例合理的表示形式，这是委员会单调方法的罕见属性。此外，我们证明了优化变体可以满足完美的表示形式。我们还分析了Phragm \'{e} n'的计​​算复杂度