Optimal gerrymandering : sometimes pack, but never crack

Standard intuitions for optimal gerrymandering involve concentrating one's extreme opponents in "unwinnable" districts ("packing") and spreading one's supporters evenly over "winnable" districts ("cracking"). These intuitions come from models with either no uncertainty about voter preferences or only two voter types. In contrast, we characterize the solution to a problem in which a gerrymanderer observes a noisy signal of voter preferences from a continuous distribution and creates N districts of equal size to maximize the expected number of districts she wins. Under mild regularity conditions, we show that cracking is never optimal -- one's most ardent supporters should be grouped together. Moreover, for sufficiently precise signals, the optimal solution involves creating a district that matches extreme "Republicans" with extreme "Democrats," and then continuing to match toward the center of the signal distribution.