Axioms and Divisor Methods for a Generalized Apportionment Problem with Relative Equality
Abstract
:1. Introduction
2. Literature Review
- Anonymity: For any permutation on , , i.e., the output of should not depend on the order of groups.
- Responsiveness: , for any .
- Scale-invariance: for any constant , where .
- Exactness: if , .
- Balancedness: , for any .
3. A Generalized Axiomatic System and a Framework of Generalized Divisor Methods
3.1. A Generalized Axiomatic System for the GAP
- Anonymity: For any permutation on , , i.e., the output of should not depend on the order of groups.
- Responsiveness: , for any .
- Scale-invariance: , , where .
- Balancedness: , for any .
- Relative-equality: if such that and , . Or equivalently, if there exists a common factor w such that are whole numbers for all i and summing up to S, then those whole numbers are the apportionment. This w is the same for all i.
3.2. A Framework for Generalized Divisor Methods
Algorithm 1 The proposed framework for generalized divisor methods |
Input:
A number of seats, sizes for group i, , a standard function , where , , and are constants, . Output:
A set of apportionment(s).
|
4. An Empirical Study with the European Parliament (EP)
4.1. On Existing Apportionment Methods for the EP
4.2. An Empirical Study Related to the EP
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Madison, J. Federalist No. 55: The Total Number of the House of Representatives. Available online: https://guides.loc.gov/federalist-papers/text-51-60#s-lg-box-wrapper-25493431 (accessed on 7 December 2022).
- Auerbach, C.A. The Reapportionment Cases: One Person, One Vote-One Vote, One Value. Supreme Court. Rev. 1964, 1964, 1–87. [Google Scholar] [CrossRef]
- Lijphart, A. Reforming the House: Three Moderately Radical Proposals. PS: Political Sci. Politics 1998, 31, 10–13. [Google Scholar] [CrossRef]
- Huntington, E.V. The Reapportionment Bill in Congress. Science 1928, 67, 509–510. [Google Scholar] [CrossRef] [PubMed]
- Willcox, W.F. The Apportionment of Representatives. Science 1928, 67, 581–582. [Google Scholar] [CrossRef]
- Chafee, Z. Congressional Reapportionment. Harv. Law Rev. 1929, 42, 1015–1047. [Google Scholar] [CrossRef]
- Davis, W. The 1941 Apportionment Bill. Science 1941, 93, 6. [Google Scholar] [CrossRef]
- Eagles, C.W. Democracy Delayed: Congressional Reapportionment and Urban-Rural Conflict in the 1920s; University of Georgia Press: Athens, Greece, 1990. [Google Scholar]
- Arnold, C. The mathematicians who want to save democracy. Nature 2017, 546, 200–202. [Google Scholar] [CrossRef] [Green Version]
- Taagepera, R. The size of national assemblies. Soc. Sci. Res. 1972, 1, 385–401. [Google Scholar] [CrossRef] [Green Version]
- Stigler, G.J. The Sizes of Legislatures. J. Leg. Stud. 1976, 5, 17–34. [Google Scholar] [CrossRef]
- Brams, S.J.; Straffin, P.D. The Apportionment Problem: Fair Representation. Meeting the Ideal of One Man, One Vote. Michel L. Balinski and H. Peyton Young. Yale University Press, New Haven, Conn., 1982. xii, 192 pp. $27.50. Science 1982, 217, 437–438. [Google Scholar] [CrossRef]
- Martínez-Aroza, J.; Ramírez-González, V. Several methods for degressively proportional allotments. A case study. Math. Comput. Model. 2008, 48, 1439–1445. [Google Scholar] [CrossRef]
- Grimmett, G.; Pukelsheim, F.; González, V.R.; Słomczyński, W.; Życzkowski, K. The Composition of the European Parliament. Available online: https://www.europarl.europa.eu/RegData/etudes/IDAN/2017/583117/IPOL_IDA (accessed on 3 April 2022).
- Eckman, S.J. Apportionment and Redistricting Process for the U.S. House of Representatives. Available online: https://sgp.fas.org/crs/misc/R45951.pdf (accessed on 3 April 2022).
- Balinski, M.L.; Young, H.P. Fair Representation: Meeting the Ideal of One Man, One Vote; Brookings Institution Press: Washington, DC, USA, 2001. [Google Scholar]
- Brams, S.J. Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures; Princeton University Press: Princeton, NJ, USA, 2008. [Google Scholar]
- Pukelsheim, F. Proportional Representation: Apportionment Methods and Their Applications; Springer: Cham, Switzerland, 2017. [Google Scholar] [CrossRef]
- Zhao, L.; Peng, T. An Allometric Scaling for the Number of Representative Nodes in Social Networks. In Proceedings of the NetSci-X 2020; Masuda, N., Goh, K.I., Jia, T., Yamanoi, J., Sayama, H., Eds.; Springer International Publishing: Cham, Switzerland, 2020; pp. 49–59. [Google Scholar] [CrossRef]
- Balinski, M.; Ramirez, V. Why webster? Math. Program. 2021, 1–8. [Google Scholar] [CrossRef]
- Palomares, A.; Pukelsheim, F.; Ramírez, V. Note on axiomatic properties of apportionment methods for proportional representation systems. Math. Program. 2022, 1–7. [Google Scholar] [CrossRef]
- Revel, M.; Lin, T.; Halpern, D. How Many Representatives Do We Need? The Optimal Size of an Epistemic Congress. Proc. AAAI Conf. Artif. Intell. 2022, 36, 9431–9438. [Google Scholar] [CrossRef]
- Patty, J.W.; Penn, E.M. Measuring Fairness, Inequality, and Big Data: Social Choice Since Arrow. Annu. Rev. Political Sci. 2019, 22, 435–460. [Google Scholar] [CrossRef] [Green Version]
- Zhao, L.; Tanimoto, A.; Lyu, W. The Most Convenient Number of Representatives. In Mathematics and Physics for Election, Voting, and Public Choice; Ohyama, T., Ed.; Kyoritsu Shuppan: Tokyo, Japan, 2022; pp. 99–122. [Google Scholar]
- Zhao, L.; Tanimoto, A.; Lyu, W. Standardizing Representation for Equality with a Population Seat Index. arXiv 2022, arXiv:2212.14790. [Google Scholar]
- Proportional Representation. Available online: https://history.house.gov/Institution/Origins-Development/Proportional-Representation/ (accessed on 27 March 2022).
- Taagepera, R.; Grofman, B. Mapping the Indices of Seats–Votes Disproportionality and Inter-Election Volatility. Party Politics 2003, 9, 659–677. [Google Scholar] [CrossRef]
- Madison, J. House Article the First, Congressional Apportionment Amendment. Available online: https://www.archives.gov/files/legislative/resources/bill-of-rights/CCBR_IIB.pdf (accessed on 17 April 2022).
- Penrose, L.S. The Elementary Statistics of Majority Voting. J. R. Stat. Soc. 1946, 109, 53–57. [Google Scholar] [CrossRef]
- Theil, H. The desired political entropy. Am. Political Sci. Rev. 1969, 63, 521–525. [Google Scholar] [CrossRef]
- Brooks, L.; Phillips, J.; Sinitsyn, M. The Cabals of a Few or the Confusion of a Multitude: The Institutional Trade-Off between Representation and Governance. Am. Econ. J. Econ. Policy 2011, 3, 1–24. [Google Scholar] [CrossRef] [Green Version]
- Koriyama, Y.; Laslier, J.F.; Macé, A.; Treibich, R. Optimal Apportionment. J. Political Econ. 2013, 121, 584–608. [Google Scholar] [CrossRef]
- Auriol, E.; Gary-Bobo, R.J. On the optimal number of representatives. Public Choice 2012, 153, 419–445. [Google Scholar] [CrossRef] [Green Version]
- Jacobs, K.; Otjes, S. Explaining the size of assemblies. A longitudinal analysis of the design and reform of assembly sizes in democracies around the world. Elect. Stud. 2015, 40, 280–292. [Google Scholar] [CrossRef]
- Godefroy, R.; Klein, N. Parliament shapes and sizes. Econ. Inq. 2018, 56, 2212–2233. [Google Scholar] [CrossRef]
- Statutes. Available online: https://www.ifors.org/statutes/ (accessed on 2 February 2023).
- Council, E. European Council Decision (EU) 2018/937: Establishing the Composition of the European Parliamen. Available online: https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32018D0937 (accessed on 19 April 2022).
- Campaign, U. A United Nations Parliamentary Assembly—Frequently Asked Question. Available online: https://www.democracywithoutborders.org/resources/FAQ_EN.pdf (accessed on 7 December 2022).
- Methods of Apportionment. Available online: https://www.census.gov/history/www/reference/apportionment/methods_of_apportionment.html (accessed on 2 February 2023).
- Grimmett, G.; Laslier, J.F.; Pukelsheim, F.; Gonzalez, V.R.; Rose, R.J.; Slomczynski, W.; Zachariasen, M.; Życzkowski, K. The Allocation between the EU Member States of the Seats in the European Parliament Cambridge Compromise. Available online: https://www.europarl.europa.eu/thinktank/en/document/IPOL-AFCO_NT(2011)432760 (accessed on 31 May 2023).
- Grimmett, G.R.; Oelbermann, K.F.; Pukelsheim, F. A power-weighted variant of the EU27 Cambridge Compromise. Math. Soc. Sci. 2012, 63, 136–140. [Google Scholar] [CrossRef]
- Ramírez, V.; Palomares, A.; Márquez, M.L. Degressively proportional methods for the allotment of the European Parliament Seats amongst the EU Member States. In Proceedings of the Mathematics and Democracy: Recent Advances in Voting Systems and Collective Choice; Springer: Berlin/Heidelberg, Germany, 2006; pp. 205–220. [Google Scholar]
- Cegiełka, K.; Łyko, J.; Rudek, R. Beyond the Cambridge Compromise algorithm towards degressively proportional allocations. Oper. Res. 2019, 19, 317–332. [Google Scholar] [CrossRef]
- Łyko, J.; Rudek, R. A fast exact algorithm for the allocation of seats for the EU Parliament. Expert Syst. Appl. 2013, 40, 5284–5291. [Google Scholar] [CrossRef]
- Łyko, E.; Łyko, J.; Maciuk, A.; Szczeciński, M. Asymmetrization of a Set of Degressively Proportional Allocations with Respect to Lexicographic Order. An Algorithmic Approach. Symmetry 2021, 13, 1269. [Google Scholar] [CrossRef]
- Cegiełka, K.; Dniestrzański, P.; yko, J.; Maciuk, A.; Szczeciński, M. A neutral core of degressively proportional allocations under lexicographic preferences of agents. Eurasian Econ. Rev. 2021, 11, 667–685. [Google Scholar] [CrossRef]
- Rudek, R.; Heppner, I. Efficient algorithms for discrete resource allocation problems under degressively proportional constraints. Expert Syst. Appl. 2020, 149, 113293. [Google Scholar] [CrossRef]
- 2020 Census: What Is Apportionment? Available online: https://www.census.gov/library/video/2021/what-is-apportionment.html (accessed on 20 March 2023).
- Eurostat. Detailed Database: Population and Housing Censuses. Available online: https://ec.europa.eu/eurostat/web/population-demography/population-housing-censuses/database (accessed on 3 December 2022).
- Eurostat. Population and Housing Census 2021—Population Grids. Available online: https://ec.europa.eu/eurostat/statistics-explained/index.php?title=Population_and_housing_census_2021_-_population_grids&stable=1#Distribution_of_European_population (accessed on 1 May 2022).
- Theil, H.; Schrage, L. The apportionment problem and the European Parliament. Eur. Econ. Rev. 1977, 9, 247–263. [Google Scholar] [CrossRef]
- Słomczyński, W.; Życzkowski, K. Jagiellonian Compromise—An alternative voting system for the Council of the European Union. In Institutional Design and Voting Power in the European Union; Cichocki, M.A., Życzkowski, K., Eds.; Routledge: Abingdon, UK, 2011. [Google Scholar]
- Słomczyński, W.; Życzkowski, K. Mathematical aspects of degressive proportionality. Math. Soc. Sci. 2012, 63, 94–101. [Google Scholar] [CrossRef] [Green Version]
- Grimmett, G.; Pukelsheim, F.; González, V.R.; Słomczyński, W.; Życzkowski, K. A 700-seat no-loss composition for the 2019 European Parliament. arXiv 2017, arXiv:1710.03820. [Google Scholar]
- Ramírez González, V. Composition of the European Parliament—The FPS-Method. Available online: https://www.math.uni-augsburg.de/htdocs/emeriti/pukelsheim/2017Brussels/RamirezGonzalez2017.pdf (accessed on 2 February 2022).
- Infographic: How Many Seats Does Each Country Get in in the European Parliament? Available online: https://www.europarl.europa.eu/news/en/headlines/eu-affairs/20180126STO94114/infographic-how-many-seats-does-each-country-get-in-in-the-european-parliament (accessed on 3 December 2022).
Adam’s | Jefferson’s | Webster’s | Hill’s | |
---|---|---|---|---|
k | ||||
i.e., rounding | up | down | off | at the geometric mean |
a | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|
b | |||||||
adj. R |
a | Rounding Rule | |||||
---|---|---|---|---|---|---|
6 | 0.9 | Adam’s | 7 | 96 | 1.88 | 11.68 |
4 | 0.8 | Webster’s | 6 | 95 | 1.77 | 10.12 |
a | Rounding Rule | S | |||||
---|---|---|---|---|---|---|---|
4 | 0.8 | Webster’s | 709 | 6 | 96 | 1.77271 | 10.0132 |
4 | 0.8 | Webster’s | 710 | 6 | 96 | 1.77271 | 10.0132 |
4 | 0.8 | Webster’s | 711 | 6 | 96 | 1.53206 | 10.0132 |
4 | 0.8 | Webster’s | 712 | 6 | 96 | 1.53206 | 10.0132 |
4 | 0.8 | Hill’s | 711 | 6 | 96 | 1.53206 | 10.0132 |
4 | 0.8 | Hill’s | 712 | 6 | 96 | 1.53206 | 10.0132 |
4 | 0.8 | Hill’s | 713 | 6 | 96 | 1.53206 | 10.0132 |
State | s | opt | opt | ||||
---|---|---|---|---|---|---|---|
Malta | 417,432 | 519,562 | 6 | 6 | 6 | 6 | 6 |
Luxembourg | 512,353 | 643,941 | 6 | 6 | 6 | 6 | 6 |
Cyprus | 840,407 | 921,033 | 6 | 6 | 7 | 6 | 7 |
Estonia | 1,294,455 | 1,319,629 | 7 | 7 | 8 | 7 | 8 |
Latvia | 2,070,371 | 1,893,223 | 8 | 8 | 9 | 8 | 9 |
Slovenia | 2,050,189 | 2,108,912 | 8 | 9 | 9 | 9 | 9 |
Lithuania | 3,043,429 | 2,810,761 | 11 | 10 | 10 | 10 | 10 |
Croatia | 4,284,889 | 3,871,833 | 12 | 12 | 12 | 12 | 12 |
Ireland | 4,574,888 | 5,105,761 | 13 | 14 | 14 | 14 | 14 |
Slovakia | 5,397,036 | 5,449,270 | 14 | 14 | 14 | 14 | 15 |
Finland | 5,375,276 | 5,533,179 | 14 | 14 | 15 | 14 | 15 |
Denmark | 5,560,628 | 5,840,045 | 14 | 15 | 15 | 15 | 15 |
Bulgaria | 7,364,570 | 6,519,789 | 17 | 16 | 16 | 16 | 16 |
Austria | 8,401,940 | 8,964,889 | 19 | 19 | 19 | 19 | 19 |
Hungary | 9,937,628 | 9,685,409 | 21 | 20 | 20 | 20 | 20 |
Portugal | 10,562,178 | 10,343,066 | 21 | 21 | 21 | 21 | 21 |
Sweden | 9,482,855 | 10,452,262 | 21 | 21 | 21 | 21 | 21 |
Greece | 10,816,286 | 10,481,735 | 21 | 21 | 21 | 21 | 21 |
Czechia | 10,436,560 | 10,524,167 | 21 | 21 | 21 | 21 | 22 |
Belgium | 11,000,638 | 11,554,767 | 21 | 23 | 23 | 23 | 23 |
The Netherlands | 16,655,799 | 17,475,443 | 29 | 30 | 30 | 30 | 30 |
Romania | 20,121,641 | 19,053,815 | 33 | 32 | 32 | 32 | 32 |
Poland | 38,044,565 | 37,019,327 | 52 | 51 | 51 | 52 | 51 |
Spain | 46,815,910 | 47,400,798 | 59 | 62 | 61 | 62 | 61 |
Italy | 59,433,744 | 59,030,133 | 76 | 73 | 72 | 74 | 72 |
France | 64,933,400 | 65,471,806 | 79 | 79 | 77 | 80 | 78 |
Germany | 80,219,695 | 83,239,650 | 96 | 95 | 95 | 96 | 96 |
Inequality index (6) | - | - | 1.77 | 1.77 | 1.40 | 1.77 | 1.39 |
State | p | 0.5-DPL | Power | Parabolic | Cambridge | Our Method |
---|---|---|---|---|---|---|
Malta | 434,403 | 6 | 6 | 6 | 6 | 6 |
Luxembourg | 576,249 | 6 | 7 | 7 | 6 | 6 |
Cyprus | 848,319 | 6 | 7 | 7 | 7 | 7 |
Estonia | 1,315,944 | 6 | 8 | 8 | 7 | 8 |
Latvia | 1,968,957 | 7 | 9 | 9 | 8 | 9 |
Slovenia | 2,064,188 | 7 | 9 | 9 | 8 | 9 |
Lithuania | 2,888,558 | 9 | 10 | 10 | 9 | 10 |
Croatia | 4,190,669 | 11 | 12 | 12 | 11 | 13 |
Ireland | 4,664,156 | 12 | 13 | 12 | 11 | 13 |
Slovakia | 5,407,910 | 13 | 14 | 13 | 12 | 14 |
Finland | 5,465,408 | 14 | 14 | 13 | 12 | 14 |
Denmark | 5,700,917 | 14 | 14 | 14 | 13 | 15 |
Bulgaria | 7,153,784 | 16 | 16 | 15 | 15 | 17 |
Austria | 8,711,500 | 18 | 18 | 17 | 17 | 19 |
Hungary | 9,830,485 | 20 | 19 | 19 | 18 | 20 |
Sweden | 9,998,000 | 20 | 20 | 19 | 18 | 21 |
Portugal | 10,341,330 | 21 | 20 | 20 | 19 | 21 |
Czechia | 10,445,783 | 21 | 20 | 20 | 19 | 21 |
Greece | 10,793,526 | 21 | 21 | 20 | 19 | 22 |
Belgium | 11,289,853 | 22 | 21 | 21 | 20 | 22 |
The Netherlands | 17,235,349 | 29 | 28 | 28 | 27 | 29 |
Romania | 19,759,968 | 32 | 31 | 31 | 31 | 32 |
Poland | 37,967,209 | 53 | 51 | 53 | 54 | 51 |
Spain | 46,438,422 | 62 | 60 | 62 | 65 | 60 |
Italy | 61,302,519 | 77 | 76 | 77 | 83 | 73 |
France | 66,661,621 | 82 | 81 | 83 | 90 | 78 |
Germany | 82,064,489 | 96 | 96 | 96 | 96 | 91 |
Inequality index (6) | - | 3.03 | 2.28 | 2.45 | 2.24 | 1.69 |
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Lyu, W.; Zhao, L. Axioms and Divisor Methods for a Generalized Apportionment Problem with Relative Equality. Mathematics 2023, 11, 3270. https://doi.org/10.3390/math11153270
Lyu W, Zhao L. Axioms and Divisor Methods for a Generalized Apportionment Problem with Relative Equality. Mathematics. 2023; 11(15):3270. https://doi.org/10.3390/math11153270
Chicago/Turabian StyleLyu, Wenruo, and Liang Zhao. 2023. "Axioms and Divisor Methods for a Generalized Apportionment Problem with Relative Equality" Mathematics 11, no. 15: 3270. https://doi.org/10.3390/math11153270
APA StyleLyu, W., & Zhao, L. (2023). Axioms and Divisor Methods for a Generalized Apportionment Problem with Relative Equality. Mathematics, 11(15), 3270. https://doi.org/10.3390/math11153270