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Social Choice and Welfare

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01-01-2015

Axiomatic districting

Authors: Clemens Puppe, Attila Tasnádi

Published in: Social Choice and Welfare | Issue 1/2015

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Abstract

We study the districting problem from an axiomatic point of view in a framework with two parties, deterministic voter preferences and geographical constraints. The axioms are normatively motivated and reflect a notion of fairness to voters. Our main result is an “impossibility” theorem demonstrating that all anonymous districting rules are necessarily complex in the sense that they either use information beyond the mere number of districts won by the parties, or they violate an appealing consistency requirement according to which an acceptable districting rule should induce an acceptable districting of appropriate subregions.

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See, e.g., Tasnádi (2011) for an overview.

Since a districting forms a partition of the given region, it is evidently not possible to move from one districting to another districting by changing only one district.

Observe that overall determinacy, i.e. that \(\delta _A (F_\varPi )\) and \(\delta _B (F_\varPi )\) be singletons for every problem \(\varPi \), is a strictly stronger requirement than two-district determinacy; for instance, the least biased solution satisfies two-district determinacy but can easily be shown to violate overall determinacy.

To verify this, observe that if there exist admissible districtings \(D,D'\in \mathcal{D}_\varPi \) with \(\delta _A(D)=2\) and \(\delta _A(D')=1\), then one must have \(0.5<\mu _A(X)/\mu (X)<0.75\). Thus, \(D'\) must be chosen both by \(M\! E\) and \(L\! B\).

Clearly, this requirement has to be restricted to subregions that are unions of districts, since a given districting does in general not induce an admissible sub-districting on other subregions.

We would like to thank Dezső Bednay for suggestions that improved our original proof.

For a definition of overall determinacy see Footnote 3.

We call two subsets of the plane neighboring if they share a common boundary of positive length.

If \(\mu _A(e)\ne \mu (e)/2\), then \(\mu _A(e')\ne \mu (e')/2\) can be guaranteed by exchanging sets of sufficiently small measure \(\mu \) between \(d\) and \(e\). In addition, if \(\mu _A(e)=\mu (e)/2\) and \(\mu _A(e')=\mu (e')/2\), then we can repeat the exchange of territories between \(e'\) and \(d'\) to ensure that both sets satisfy (1).

Both pictures only show the two districts involved in a territorial exchange and not the entire districtings.

It might happen that \(d'\) or \(e'\) violate (1) since we only took care of the shapes and sizes of the two districts. However, Lemma 2 ensures that through an appropriate territorial exchange between \(d'\) and \(e'\) we can also ensure (1). In what follows we will carry out all territorial exchanges between districts so as to satisfy (1) without explicitly mentioning Lemma 2 each time.

District \(e'\) in Fig. 7 is not drawn in the most efficient way in the sense that it is possible to draw \(e'\) such that it allows for a larger reduction of the separated areas. However, the purpose of Fig. 7 is only to illustrate the possibility of the reduction of separated areas.

In fact the number of required iterations is at most \(\lceil t\mu (Y)/\mu (X) \rceil +1\), where \(Y\) stands for the area “intertwined” by \(d\cup e\).

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Title: Axiomatic districting
Authors: Clemens Puppe
Attila Tasnádi
Publication date: 01-01-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 1/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0824-9

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