Can Compactness Constrain the Gerrymander?

Abstract

Gerrymandering – the manipulation of electoral boundaries to maximize constituency wins – is often seen as a pathology of democratic systems. A commonly cited cure is to require that electoral constituencies have a ‘compact’ shape. But how much of a constraint does compactness in fact place on would-be gerrymanderers? The author operationalizes compactness as a convexity constraint and applies a theorem of Kaneko, Kano and Suzuki to the two-party situation to show that for any population distribution a gerrymanderer can always create equal (population)-sized convex constituencies that translate a margin of k voters into a margin of at least k constituency wins. Thus, even with a small margin a majority party can win all constituencies. In addition, it is shown that there always exists some population distribution such that all divisions into equal-sized convex constituencies translate a margin of k voters into a margin of exactly k constituencies. Thus, a convexity constraint can sometimes prevent a gerrymanderer from generating any wins for a minority party. These results clarify that the heart of the problem with outcomes that deviate from proportionality in single member constituency systems is not the manner in which constituencies are drawn but the mode of preference aggregation.

Keywords:

• gerrymandering
• single member constituencies
• first past the post
• redistricting
• majority rule
• proportional representation
• ham sandwich theorem

Acknowledgements

The author is deeply grateful to Eddie Hyland for opening a universe of questions on aggregation and representation and forcing the author to put some discipline on his thinking about them. The author also thanks Mikio Kano for very generous advice on this work.

Notes

Note that other recent work on gerrymandering allows for more continuous party affiliation but removes geographical constraints (Friedman & Holden, 2008).

Note that g, h and a are all guaranteed to be integers here because k odd (even) implies n odd (even), which in turn implies m odd (even).