A Mathematical Analysis of an Election System Proposed by Gottlob Frege

In 1998 a long-lost proposal for an election law by Gottlob Frege (1848--1925) was rediscovered in the Thuringer Universitats- und Landesbibliothek in Jena, Germany. Frege's method for the election of representatives of a constituency features a remarkable concern for the representation of minorities. Its core idea is that votes cast for unelected candidates are carried over to the next election, while the elected candidates incur a cost of winning. We prove that this sensitivity to past elections guarantees a proportional representation of political opinions over time. We find that through a slight modification of the original voting method even stronger forms of proportionality can be achieved. This modified version of Frege's method can also be seen as providing a novel solution to the apportionment problem. We prove that it is distinct from all of the best-known apportionment methods, while it still possesses noteworthy proportionality properties.

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