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A Mathematical Analysis of an Election System Proposed by Gottlob Frege

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  • Paul Harrenstein
  • Marie-Louise Lackner
  • Martin Lackner

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.

Suggested Citation

  • Paul Harrenstein & Marie-Louise Lackner & Martin Lackner, 2019. "A Mathematical Analysis of an Election System Proposed by Gottlob Frege," Papers 1907.03643, arXiv.org, revised Sep 2020.
    • Handle: RePEc:arx:papers:1907.03643
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    References listed on IDEAS

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    1. James Green-Armytage, 2015. "Direct voting and proxy voting," Constitutional Political Economy, Springer, vol. 26(2), pages 190-220, June.
    2. Moulin, Herve, 1988. "Condorcet's principle implies the no show paradox," Journal of Economic Theory, Elsevier, vol. 45(1), pages 53-64, June.
    3. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    4. Rae, Douglas W., 1969. "Decision-Rules and Individual Values in Constitutional Choice," American Political Science Review, Cambridge University Press, vol. 63(1), pages 40-56, March.
    5. Dan Alger, 2006. "Voting by proxy," Public Choice, Springer, vol. 126(1), pages 1-26, January.
    6. Federica Ricca & Andrea Scozzari & Bruno Simeone, 2013. "Political Districting: from classical models to recent approaches," Annals of Operations Research, Springer, vol. 204(1), pages 271-299, April.
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