A Mathematical Analysis of an Election System Proposed by Gottlob Frege
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We provide a mathematical analysis of an election system proposed by the eminent logician Gottlob Frege (1848--1925). His proposal was written presumably in 1918, was (re)discovered around the turn of the millennium, and published for the first time in the original German in 2000. A remarkable feature of Frege's proposal is its concern for the representation of minorities and its sensitivity to past election results. Frege's proposal is based on some highly original and relevant ideas; his core idea is that the votes of unelected candidates are carried over to the next election. All candidates thus accumulate votes over time and eventually each candidate is elected at some point. We provide a mathematical formulation of Frege's election system and investigate how well it achieves its aim of a fair representation of all political opinions in a community over time. We can prove that this goal is fulfilled remarkably well. However, we also show that, in other aspects, it falls short of Frege's high ambition that no voter's vote be lost. We propose a slight modification of his voting rule, the modified Frege method, that remedies these shortcomings. We analyse both methods from the perspective of modern social choice and apportionment theory, and can show that they are novel contributions with noteworthy proportionality properties over time.
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