Mathematics of apportionment describes mathematical principles and algorithms for fair allocation of identical items among parties with different entitlements. Such principles are used to apportion seats in parliaments among federal states or political parties. See apportionment (politics) for the more concrete principles and issues related to apportionment, and apportionment by country for practical methods used around the world.
Mathematically, an apportionment method is just a method of rounding fractions to integers. As simple as it may sound, each and every method for rounding suffers from one or more paradoxes. The mathematical theory of apportionment aims to decide what paradoxes can be avoided, or in other words, what properties can be expected from an apportionment method.
The mathematical theory of apportionment was studied as early as 1907 by the mathematician Agner Krarup Erlang. It was later developed to a great detail by the mathematician Michel Balinsky and the economist Peyton Young.[1][2][3] Besides its application to political parties,[4] it is also applicable to fair item allocation when agents have different entitlements.[5][6] It is also relevant in manpower planning - where jobs should be allocated in proportion to characteristics of the labor pool, to statistics - where the reported rounded numbers of percentages should sum up to 100%,[7][8] and to bankruptcy problems.[9]
^Balinski, Michel L.; Young, H. Peyton (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.
^Balinski, Michel L.; Young, H. Peyton (2001). Fair Representation: Meeting the Ideal of One Man, One Vote (2nd ed.). Washington, DC: Brookings Institution Press. ISBN 0-8157-0111-X.
^Balinski, M.L.; Young, H.P. (1994-01-01). "Chapter 15 Apportionment". Handbooks in Operations Research and Management Science. 6: 529–560. doi:10.1016/S0927-0507(05)80096-9. ISBN 9780444892041. ISSN 0927-0507.
^COTTERET J. M; C, EMERI (1973). LES SYSTEMES ELECTORAUX.
^Chakraborty, Mithun; Segal-Halevi, Erel; Suksompong, Warut (2022-06-28). "Weighted Fairness Notions for Indivisible Items Revisited". Proceedings of the AAAI Conference on Artificial Intelligence. 36 (5): 4949–4956. arXiv:2112.04166. doi:10.1609/aaai.v36i5.20425. ISSN 2374-3468.
^Diaconis, Persi; Freedman, David (1979-06-01). "On Rounding Percentages". Journal of the American Statistical Association. 74 (366a): 359–364. doi:10.1080/01621459.1979.10482518. ISSN 0162-1459.
^Balinski, M. L.; Demange, G. (1989-11-01). "An Axiomatic Approach to Proportionality Between Matrices" (PDF). Mathematics of Operations Research. 14 (4): 700–719. doi:10.1287/moor.14.4.700. ISSN 0364-765X.
^Csoka, Péter; Herings, P. Jean-Jacques (2016-01-01). "Decentralized Clearing in Financial Networks (RM/16/005-revised-)". Research Memorandum.
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