The topic of apportionment is a central focus of study for legislatures around the world, whether the goal is to allocate seats to political parties, or to allocate seats to the member states of a federation. The first goal is sought by parliaments employing proportional representation for parties; the second goal is sought by the United States House of Representatives and the European Parliament. Many of the leading apportionment methods were created in the late 18th century in response to requirements listed in the United States Constitution. No apportionment method perfectly satisfies all desirable properties, particularly the properties of integrality, proportionality, and quota. The Largest Remainder method satisfies quota but suffers from other paradoxes; the divisor methods like the Greatest Divisors, Major Fractions (Arithmetic Mean), Equal Proportions (Geometric Mean), Harmonic Mean, and Smallest Divisor methods satisfy proportionality but may fail quota. Some apportionment methods like Greatest Divisor unfairly favor larger parties and states, and others like Smallest Divisor unfairly favor smaller parties and states. We introduce a new method for Congressional apportionment that creates the apportionment all at once, rather than determining seats one at a time. This method always satisfies quota. It partially resembles the familiar Huntington monotone divisor methods and indeed creates a quota-capped divisor method, but can be compared as well to largest remainder methods.
| Published in | Social Sciences (Volume 14, Issue 6) |
| DOI | 10.11648/j.ss.20251406.12 |
| Page(s) | 585-590 |
| 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), 2025. Published by Science Publishing Group |
Apportionment, Divisor, Largest Remainder, Proportional Representation, Quota
| [1] | Mendenhall, Peter Charles, and Switkay, Hal M., Consecutively Halved Positional Voting: A Special Case of Geometric Voting. Social Sciences, 12 (2023) pp. 47-59. |
| [2] | Constitution of the United States. |
| [3] | Balinski, Michel L., and Young, H. Peyton, Fair Representation: Meeting the Ideal of One Man, One Vote. Yale University Press, 1982. |
| [4] | Huntington, Edward. V., The Mathematical Theory of the Apportionment of Representatives. Proc. Natl. Acad. Sci. U.S.A., 7 (1921) pp. 123-127. |
| [5] |
Huntington, Edward V., A New Method of Apportionment of Representatives. Quart. Pub. Am. Stat. Assoc., 17 (1921) pp. 859-870.
https://www.tandfonline.com/doi/epdf/10.1080/15225445.1921.10503487 |
| [6] | Huntington, Edward. V., The Apportionment of Representatives in Congress. Trans. Am. Math. Soc., 30 (1928) pp. 85-110. |
| [7] | Balinski, Michel L., and Young, H. Peyton, A New Method for Congressional Apportionment. Proc. Natl. Acad. Sci. U.S.A., 71 (1974) pp. 4602-4606. |
| [8] | Mayberry, J. P., Quota Methods for Congressional Apportionment are Still Non-Unique. Proc. Natl. Acad. Sci. U.S.A., 75 (1978) pp. 3537-3539. |
| [9] | Birkhoff, Garrett, House Monotone Apportionment Schemes. Proc. Natl. Acad. Sci. U.S.A., 73 (1976) pp. 684-686. |
| [10] | Lang, Serge, Real and Functional Analysis. Springer, 1993. |
| [11] |
Agnew, R. A., Optimal Congressional Apportionment. The Am. Math. Monthly, 115 (2008) pp. 297-303.
https://www.tandfonline.com/doi/abs/10.1080/00029890.2008.11920530 |
| [12] | Balinski, Michel L., and Young, H. Peyton, The Webster Method of Apportionment. Proc. Natl. Acad. Sci. U.S.A., 77 (1980) pp. 1-4. |
| [13] | United States Census Bureau, 2018 Population Estimates. |
| [14] | Benford, F., The Law of Anomalous Numbers. Proc. Am. Phil. Soc., 78 (1938) pp. 551-572. |
| [15] |
Hill, Theodore P., The Significant-Digit Phenomenon. The Am. Math. Monthly, 102 (1995) pp. 322-327.
https://www.tandfonline.com/doi/abs/10.1080/00029890.1995.11990578 |
APA Style
Switkay, H. M. (2025). A Self-dual Pseudo-divisor Quota Method for Congressional Apportionment. Social Sciences, 14(6), 585-590. https://doi.org/10.11648/j.ss.20251406.12
ACS Style
Switkay, H. M. A Self-dual Pseudo-divisor Quota Method for Congressional Apportionment. Soc. Sci. 2025, 14(6), 585-590. doi: 10.11648/j.ss.20251406.12
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Download@article{10.11648/j.ss.20251406.12,
author = {Hal M. Switkay},
title = {A Self-dual Pseudo-divisor Quota Method for Congressional Apportionment},
journal = {Social Sciences},
volume = {14},
number = {6},
pages = {585-590},
doi = {10.11648/j.ss.20251406.12},
url = {https://doi.org/10.11648/j.ss.20251406.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ss.20251406.12},
abstract = {The topic of apportionment is a central focus of study for legislatures around the world, whether the goal is to allocate seats to political parties, or to allocate seats to the member states of a federation. The first goal is sought by parliaments employing proportional representation for parties; the second goal is sought by the United States House of Representatives and the European Parliament. Many of the leading apportionment methods were created in the late 18th century in response to requirements listed in the United States Constitution. No apportionment method perfectly satisfies all desirable properties, particularly the properties of integrality, proportionality, and quota. The Largest Remainder method satisfies quota but suffers from other paradoxes; the divisor methods like the Greatest Divisors, Major Fractions (Arithmetic Mean), Equal Proportions (Geometric Mean), Harmonic Mean, and Smallest Divisor methods satisfy proportionality but may fail quota. Some apportionment methods like Greatest Divisor unfairly favor larger parties and states, and others like Smallest Divisor unfairly favor smaller parties and states. We introduce a new method for Congressional apportionment that creates the apportionment all at once, rather than determining seats one at a time. This method always satisfies quota. It partially resembles the familiar Huntington monotone divisor methods and indeed creates a quota-capped divisor method, but can be compared as well to largest remainder methods.},
year = {2025}
}
TY - JOUR T1 - A Self-dual Pseudo-divisor Quota Method for Congressional Apportionment AU - Hal M. Switkay Y1 - 2025/12/11 PY - 2025 N1 - https://doi.org/10.11648/j.ss.20251406.12 DO - 10.11648/j.ss.20251406.12 T2 - Social Sciences JF - Social Sciences JO - Social Sciences SP - 585 EP - 590 PB - Science Publishing Group SN - 2326-988X UR - https://doi.org/10.11648/j.ss.20251406.12 AB - The topic of apportionment is a central focus of study for legislatures around the world, whether the goal is to allocate seats to political parties, or to allocate seats to the member states of a federation. The first goal is sought by parliaments employing proportional representation for parties; the second goal is sought by the United States House of Representatives and the European Parliament. Many of the leading apportionment methods were created in the late 18th century in response to requirements listed in the United States Constitution. No apportionment method perfectly satisfies all desirable properties, particularly the properties of integrality, proportionality, and quota. The Largest Remainder method satisfies quota but suffers from other paradoxes; the divisor methods like the Greatest Divisors, Major Fractions (Arithmetic Mean), Equal Proportions (Geometric Mean), Harmonic Mean, and Smallest Divisor methods satisfy proportionality but may fail quota. Some apportionment methods like Greatest Divisor unfairly favor larger parties and states, and others like Smallest Divisor unfairly favor smaller parties and states. We introduce a new method for Congressional apportionment that creates the apportionment all at once, rather than determining seats one at a time. This method always satisfies quota. It partially resembles the familiar Huntington monotone divisor methods and indeed creates a quota-capped divisor method, but can be compared as well to largest remainder methods. VL - 14 IS - 6 ER -
Department of Arts and Sciences, Goldey-Beacom College, Wilmington, USA
