A Mathematical Analysis of an Election System Proposed by Gottlob Frege

TL;DR

This paper provides a mathematical analysis of Gottlob Frege's rediscovered election proposal, demonstrating its proportional representation guarantees and proposing a modified version that enhances these properties and offers a new solution to the apportionment problem.

Contribution

The paper introduces a formal analysis of Frege's election method, proving its proportionality properties and proposing a modified version with improved guarantees and a novel apportionment approach.

Findings

Frege's method guarantees long-term proportional representation.

A slight modification enhances proportionality and addresses the apportionment problem.

The modified method offers a new approach distinct from existing apportionment methods.

Abstract

In 1998 a long-lost proposal for an election law by Gottlob Frege (1848--1925) was rediscovered in the Th\"uringer Universit\"ats- und Landesbibliothek in Jena, Germany. The method that Frege proposed 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 elected candidates incur a cost of winning. We prove that this sensitivity to past elections guarantees a proportional representation of political opinions in the long run. We find that through a slight modification of Frege's original method even stronger proportionality guarantees can be achieved. This modified version of Frege's method also provides a novel solution to the apportionment problem, which is distinct from all of the best-known apportionment methods, while still possessing noteworthy proportionality properties.