Can compactness constrain the Gerrymander?

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? We operationalize compactness as a convexity constraint and apply a theorem of Kaneko, Kano, and Suzuki (2004) to the two party situation to show that for any population distribution a gerrymanderer can always create equal 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. Moreover 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. ∗I thank Mikio Kano for generous advice on this work.