Abstract
An apportionment method is proposed that generalises Hamilton’s method for matrices, optimising proportionality in both directions, both for rows and columns. The resulting matrix respects fixed totals for rows and columns even when such totals do not satisfy standard criteria (monotonicity, maximum or minimum Hare), for example following the allocation of majority prizes to parties or coalitions.
Optionally, if required, the result can also respect the minimum Hare quotae for rows and columns. The algorithm may easily be expressed on the basis of rules.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
From Gambarelli and Palestini [3].
- 2.
For more information about calculation methods used for these elections, please refer to the law in force at the time of the elections [11].
- 3.
Hare 5.13 obtained by dividing the 388,046 district votes by the 7,332,134 national votes and multiplying by 97 total seats to be assigned to the party at a national level.
- 4.
Hare 8.18 obtained by dividing the 658,475 district votes by the 8,691,406 national votes and multiplying by 108 total seats to be assigned to the party at a national level.
- 5.
Hare 6.62 obtained by dividing the 532,699 district votes by the 8,691,406 national votes and multiplying by 108 total seats to be assigned to the party at a national level.
References
Demange, G.: On allocating seats to parties and districts: apportionments. In: Fragnelli, V., Gambarelli, G. (eds.) Open Problems in Applications of Cooperative Games - a Special Issue of International Game Theory Review, vol. 15, no. 3 (2013)
Gambarelli, G.: Minimax apportionments. Group Decis. Negot. 8(6), 441–461 (1999)
Gambarelli, G., Palestini, A.: Minimax multi-district apportionments. Homo Oecon. 24(3/4), 335–356 (2007)
Gambarelli, G., Stach, I.: Power indices in politics; some results and open problems. Essays in honor of Hannu Nurmi. Homo Oecon. 26(3/4), 417–441 (2009). ISBN 978–3-89265-072-0, ISSN 0943-0180
Hamilton, A.: Proposal of apportionment approved by U.S.A. Congress (1791)
Maier, S., Pukelsheim, F.: BAZI: a free computer program for proportional representation apportionment. Preprint Nr. 042-2007. Institut fur Mathematik, Universitat Augsburg (2007). www.opus-bayern.de/uni-augsburg/volltexte/2007/711/
Pukelsheim, F.: Current issues of apportionment methods. In: Simeone, B., Pukelsheim, F. (eds.) Mathematics and Democracy: Recent Advances in Voting Systems and Collective Choice. Studies in Choice and Welfare, pp. 167–176. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-35605-3_12
Pukelsheim, F., Ricca, F., Scozzari, A., Serafini, P., Simeone, B.: Network flow methods for electoral systems. Networks 59, 73–88 (2011). Special Issue on the INOC 2009 Conference, April 26–29, 2009, Pisa, Italy
Website of the Italian Ministry of Interior “Historical Archives of the Elections”, edited by the Office IV - Election Informatics Services. (http://elezionistorico.interno.it). Accessed 19 Dec 2015
Minutes of operations of the Central National Electoral Office of March 20, 2018 (http://www.cortedicassazione.it/corte-di-cassazione/it/elezioni.page). Accessed 26 Apr 2018
Italian Law December 21, 2005, n. 270 Changes to the rules for the election of the Chamber of Deputies and the Senate of the Republic (2005)
Acknowledgments
This paper is under the patronage of MIUR. The authors wish to thank Luciano Violante for his valuable comments on a previous version of this paper, and Angelo Uristani for useful discussions on a local level seat assignation method.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendices
Appendix A: Legislation
A draft legislative rule, referring to the case of the minimum Hare quotae, could be the following:
-
Table of the row Hare quotae is determined, each element of which is the product of the votes obtained by that party in that district for the total seats in that district, divided by the total votes in that district;
-
Table of column Hare quotae is determined, each element of which is the product of the votes obtained by that party in that district for the total seats of that party, divided by the total votes in that party;
-
An assigned seats matrix is prepared, initially assigning the minimum value truncated, between the row Hare and the column Hare for each position;
-
The reference matrix is then prepared, assigning the maximum value between the row Hare and the column Hare for each position;
-
At each step, the method assigns a seat to the position in which there is the greatest difference between the “reference matrix” element and the “assigned seats matrix” element. The assigned seat is updated in the “assigned seats matrix”;
-
The method checks the residual availability of seats that may be assigned to each party only for those seats for which a given party is represented and when residual availability is equal to the remaining number of seats to be assigned to the given party at a national level;
-
The system continues in the assignment loop until all the seats have been allocated.
Appendix B: Automatic Calculation Software
The latest version of the bi-proportional apportionment software described in this paper is available at the following web address:
http://dinamico2.unibg.it/dmsia/staff/gampubl.html#software.
Rights and permissions
Copyright information
© 2019 Springer-Verlag GmbH Germany, part of Springer Nature
About this chapter
Cite this chapter
Bezzi, M., Gambarelli, G., Zibetti, G.A. (2019). Bi-proportional Apportionments. In: Nguyen, N., Kowalczyk, R., Mercik, J., Motylska-Kuźma, A. (eds) Transactions on Computational Collective Intelligence XXXIV. Lecture Notes in Computer Science(), vol 11890. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60555-4_10
Download citation
DOI: https://doi.org/10.1007/978-3-662-60555-4_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-60554-7
Online ISBN: 978-3-662-60555-4
eBook Packages: Computer ScienceComputer Science (R0)