Consecutively Halved Positional Voting: A Special Case of Geometric Voting

Peter Charles Mendenhall; Hal M. Switkay

doi:doi:10.11648/j.ss.20231202.11

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Consecutively Halved Positional Voting: A Special Case of Geometric Voting

Peter Charles Mendenhall, Hal M. Switkay

Published in Social Sciences (Volume 12, Issue 2)

Received: 3 February 2023 Accepted: 21 February 2023 Published: 4 March 2023

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Abstract

The Borda count is a positional voting system that favors ‘consensual’ candidates with broad support while plurality is instead biased towards ‘polarizing’ ones with strong support. Our article focusses first on developing indices for quantifying system bias and then on vector analysis and design, while seeking to find an intermediate vector evenly balanced between consensus and polarization. The bias indices are based on the preference weightings of a normalized vector that represents a class of affine equivalent ones. The use of weightings that form a geometric progression evolves from this development. Such a ‘geometric voting’ vector can represent any positional voting vector with three preferences. With its common ratio as the sole variable, this vector can also span the whole spectrum of system bias continuously regardless of the number of preferences it employs; as demonstrated by our case study of the 1860 US presidential election with four candidates. Using this variable vector as an analytical tool, it establishes the ‘consecutively halved positional voting’ vector as the optimum one for balance. In our case study of the 2019 Nauru general election, this balanced vector is compared to its Dowdall rival that comprises a harmonic progression of weightings and several advantages are identified.

Published in	Social Sciences (Volume 12, Issue 2)
DOI	10.11648/j.ss.20231202.11
Page(s)	47-59
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Positional Voting, Consensus, Polarization, Geometric Progression, Borda Count, Plurality, Dowdall

References

[1]	Saari, D. G. (1995) Basic Geometry of Voting, Chapter IV: Positional Voting and the BC, pp. 101-106. Berlin: Springer-Verlag.
[2]	Fraenkel, J. and Grofman, B. (2014) The Borda Count and its real-world alternatives: Comparing scoring rules in Nauru and Slovenia. Australian Journal of Political Science, Vol 49, No 2, pp. 186-205. https://doi.org/10.1080/10361146.2014.900530
[3]	Smith, W. D. (2006) Descriptions of single-winner voting systems. Self-published. https://www.researchgate.net/publication/242756803_Descriptions_of_Single-winner_Voting_Systems. Accessed 2021 July 4.
[4]	Saari, D. G. (2001) Chaotic Elections! A Mathematician Looks at Voting. Providence: American Mathematical Society.
[5]	Tideman, N. (2006) Collective Decisions and Voting, Part 2, Chapter 13: Vote Processing Rules for Selecting One Option from Many When Votes Have Predetermined Weights: Alternatives to Plurality, p. 166 and p. 179. Farnham and Burlington: Ashgate.
[6]	Riker, W. H. (1982) Liberalism Against Populism. New York: W. H. Freeman.
[7]	Szpiro, G. G. (2010) Numbers Rule: The Vexing Mathematics of Democracy from Plato to the Present, pp. 71-72. Princeton: Princeton University Press.
[8]	Online Encyclopedia of Integer Sequences: sequences A000295 https://oeis.org/A000295, A125128 https://oeis.org/A125128, and A130103 https://oeis.org/A130103. Accessed 2021 May 23.
[9]	Nauru Electoral Commission (2019) Parliamentary Election Final Report https://election.com.nr/wp-content/uploads/2019/12/2019-Parliamentary-Election-Report.pdf. Accessed 2022 Feb 04.
[10]	Nauru Electoral Commission (2022) Parliamentary Election Results https://election.com.nr/election-results/. Accessed 2023 Jan 30.
[11]	Saari, D. G. (2008) Disposing Dictators, Demystifying Voting Paradoxes: Social Choice Analysis, Chapters 1 and 5. Cambridge: Cambridge University Press.
[12]	Saari, D. G. (2001) Decisions and Elections: Explaining the Unexpected, p23. Cambridge: Cambridge University Press.
[13]	Black, D. (1958) The Theory of Committees and Elections: Cambridge: Cambridge University Press.
[14]	Emerson, P. (2020) Indicative (SIC) Votes. Representation, Vol 56, No 2, pp. 273-283. https://doi.org/10.1080/00344893.2019.1696393
[15]	Reynolds, A., Reilly, B., and Ellis, A. (2008) Electoral System Design: The New International IDEA Handbook. Stockholm: Strömsborg.

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APA Style

Peter Charles Mendenhall, Hal M. Switkay. (2023). Consecutively Halved Positional Voting: A Special Case of Geometric Voting. Social Sciences, 12(2), 47-59. https://doi.org/10.11648/j.ss.20231202.11

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Peter Charles Mendenhall; Hal M. Switkay. Consecutively Halved Positional Voting: A Special Case of Geometric Voting. Soc. Sci. 2023, 12(2), 47-59. doi: 10.11648/j.ss.20231202.11

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AMA Style

Peter Charles Mendenhall, Hal M. Switkay. Consecutively Halved Positional Voting: A Special Case of Geometric Voting. Soc Sci. 2023;12(2):47-59. doi: 10.11648/j.ss.20231202.11

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@article{10.11648/j.ss.20231202.11,
  author = {Peter Charles Mendenhall and Hal M. Switkay},
  title = {Consecutively Halved Positional Voting: A Special Case of Geometric Voting},
  journal = {Social Sciences},
  volume = {12},
  number = {2},
  pages = {47-59},
  doi = {10.11648/j.ss.20231202.11},
  url = {https://doi.org/10.11648/j.ss.20231202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ss.20231202.11},
  abstract = {The Borda count is a positional voting system that favors ‘consensual’ candidates with broad support while plurality is instead biased towards ‘polarizing’ ones with strong support. Our article focusses first on developing indices for quantifying system bias and then on vector analysis and design, while seeking to find an intermediate vector evenly balanced between consensus and polarization. The bias indices are based on the preference weightings of a normalized vector that represents a class of affine equivalent ones. The use of weightings that form a geometric progression evolves from this development. Such a ‘geometric voting’ vector can represent any positional voting vector with three preferences. With its common ratio as the sole variable, this vector can also span the whole spectrum of system bias continuously regardless of the number of preferences it employs; as demonstrated by our case study of the 1860 US presidential election with four candidates. Using this variable vector as an analytical tool, it establishes the ‘consecutively halved positional voting’ vector as the optimum one for balance. In our case study of the 2019 Nauru general election, this balanced vector is compared to its Dowdall rival that comprises a harmonic progression of weightings and several advantages are identified.},
 year = {2023}
}

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TY - JOUR
T1 - Consecutively Halved Positional Voting: A Special Case of Geometric Voting
AU - Peter Charles Mendenhall
AU - Hal M. Switkay
Y1 - 2023/03/04
PY - 2023
N1 - https://doi.org/10.11648/j.ss.20231202.11
DO - 10.11648/j.ss.20231202.11
T2 - Social Sciences
JF - Social Sciences
JO - Social Sciences
SP - 47
EP - 59
PB - Science Publishing Group
SN - 2326-988X
UR - https://doi.org/10.11648/j.ss.20231202.11
AB - The Borda count is a positional voting system that favors ‘consensual’ candidates with broad support while plurality is instead biased towards ‘polarizing’ ones with strong support. Our article focusses first on developing indices for quantifying system bias and then on vector analysis and design, while seeking to find an intermediate vector evenly balanced between consensus and polarization. The bias indices are based on the preference weightings of a normalized vector that represents a class of affine equivalent ones. The use of weightings that form a geometric progression evolves from this development. Such a ‘geometric voting’ vector can represent any positional voting vector with three preferences. With its common ratio as the sole variable, this vector can also span the whole spectrum of system bias continuously regardless of the number of preferences it employs; as demonstrated by our case study of the 1860 US presidential election with four candidates. Using this variable vector as an analytical tool, it establishes the ‘consecutively halved positional voting’ vector as the optimum one for balance. In our case study of the 2019 Nauru general election, this balanced vector is compared to its Dowdall rival that comprises a harmonic progression of weightings and several advantages are identified.
VL - 12
IS - 2
ER -

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Author Information

Peter Charles Mendenhall

Research Centre, Geometric-voting.org.uk, Nottingham, UK
Hal M. Switkay

Department of Arts and Sciences, Goldey-Beacom College, Wilmington, USA

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APA Style

Peter Charles Mendenhall, Hal M. Switkay. (2023). Consecutively Halved Positional Voting: A Special Case of Geometric Voting. Social Sciences, 12(2), 47-59. https://doi.org/10.11648/j.ss.20231202.11

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ACS Style

Peter Charles Mendenhall; Hal M. Switkay. Consecutively Halved Positional Voting: A Special Case of Geometric Voting. Soc. Sci. 2023, 12(2), 47-59. doi: 10.11648/j.ss.20231202.11

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AMA Style

Peter Charles Mendenhall, Hal M. Switkay. Consecutively Halved Positional Voting: A Special Case of Geometric Voting. Soc Sci. 2023;12(2):47-59. doi: 10.11648/j.ss.20231202.11

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@article{10.11648/j.ss.20231202.11,
  author = {Peter Charles Mendenhall and Hal M. Switkay},
  title = {Consecutively Halved Positional Voting: A Special Case of Geometric Voting},
  journal = {Social Sciences},
  volume = {12},
  number = {2},
  pages = {47-59},
  doi = {10.11648/j.ss.20231202.11},
  url = {https://doi.org/10.11648/j.ss.20231202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ss.20231202.11},
  abstract = {The Borda count is a positional voting system that favors ‘consensual’ candidates with broad support while plurality is instead biased towards ‘polarizing’ ones with strong support. Our article focusses first on developing indices for quantifying system bias and then on vector analysis and design, while seeking to find an intermediate vector evenly balanced between consensus and polarization. The bias indices are based on the preference weightings of a normalized vector that represents a class of affine equivalent ones. The use of weightings that form a geometric progression evolves from this development. Such a ‘geometric voting’ vector can represent any positional voting vector with three preferences. With its common ratio as the sole variable, this vector can also span the whole spectrum of system bias continuously regardless of the number of preferences it employs; as demonstrated by our case study of the 1860 US presidential election with four candidates. Using this variable vector as an analytical tool, it establishes the ‘consecutively halved positional voting’ vector as the optimum one for balance. In our case study of the 2019 Nauru general election, this balanced vector is compared to its Dowdall rival that comprises a harmonic progression of weightings and several advantages are identified.},
 year = {2023}
}

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