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In mathematics and social choice, apportionment problems are a class of fair division problems where the goal is to divide (apportion) a whole number of identical goods fairly between multiple groups with different entitlements. The original example of an apportionment problem involves distributing seats in a legislature between different federal states or political parties.[1] However, apportionment methods can be applied to other situations as well, including bankruptcy problems,[2] inheritance law (e.g. dividing animals),[3][4] manpower planning (e.g. demographic quotas),[5] and rounding percentages.[6]
Mathematically, an apportionment method is just a method of rounding real numbers to integers. Despite the simplicity of this problem, every method of rounding suffers one or more paradoxes, as proven by the Balinski-Young theorem. The mathematical theory of apportionment identifies 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.[citation needed] It was later developed to a great detail by the mathematician Michel Balinski and the economist Peyton Young.[7]
- ^ COTTERET J. M; C, EMERI (1973). LES SYSTEMES ELECTORAUX.
- ^ Csoka, Péter; Herings, P. Jean-Jacques (2016-01-01). "Decentralized Clearing in Financial Networks (RM/16/005-revised-)". Research Memorandum.
- ^ 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.
- ^ Chakraborty, Mithun; Schmidt-Kraepelin, Ulrike; Suksompong, Warut (2021-12-01). "Picking sequences and monotonicity in weighted fair division". Artificial Intelligence. 301: 103578. arXiv:2104.14347. doi:10.1016/j.artint.2021.103578. ISSN 0004-3702. S2CID 233443832.
- ^ 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.
- ^ 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, 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.