ABSTRACT
Deciding how to allocate the seats of a house of representatives is one of the most fundamental problems in the political organization of societies, and has been widely studied over already two centuries. The idea of proportionality is at the core of most approaches to tackle this problem, and this notion is captured by the divisor methods, such as the Jefferson/D'Hondt method. In a seminal work, Balinski and Demange extended the single-dimensional idea of divisor methods to the setting in which the seat allocation is simultaneously determined by two dimensions, and proposed the so-called biproportional apportionment method. The method, currently used in several electoral systems, is however limited to two dimensions and the question of extending it is considered to be an important problem both theoretically and in practice. In this work we initiate the study of multidimensional proportional apportionment. We first formalize a notion of multidimensional proportionality that naturally extends that of Balinski and Demange. By means of analyzing an appropriate integer linear program we are able to prove that, in contrast to the two-dimensional case, the existence of multidimensional proportional apportionments is not guaranteed and deciding its existence is NP-complete. Interestingly, our main result asserts that it is possible to find approximate multidimensional proportional apportionments that deviate from the marginals by a small amount. The proof arises through the lens of discrepancy theory, mainly inspired by the celebrated Beck-Fiala Theorem. We finally evaluate our approach by using the data from the recent 2021 Chilean Constitutional Convention election.
- Michel Balinski and Gabrielle Demange. 1989 a. Algorithms for proportional matrices in reals and integers. Mathematical Programming, Vol. 45, 1--3 (1989), 193--210.Google ScholarCross Ref
- Michel Balinski and Gabrielle Demange. 1989 b. An axiomatic approach to proportionality between matrices. Mathematics of Operations Research, Vol. 14, 4 (1989), 700--719.Google ScholarDigital Library
- József Beck and Tibor Fiala. 1981. Integer-making theorems. Discrete Applied Mathematics , Vol. 3, 1 (1981), 1--8.Google ScholarCross Ref
- Gabrielle Demange. 2013. On allocating seats to parties and districts: apportionments. International Game Theory Review, Vol. 15, 03 (2013), 1340014.Google ScholarCross Ref
- Günter Rote and Martin Zachariasen. 2007. Matrix scaling by network flow. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA).Google Scholar
Index Terms
- Multidimensional Apportionment through Discrepancy Theory
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