Optimality and fairness of partisan gerrymandering

We consider the problem of optimal partisan gerrymandering: a legislator in charge of redrawing the boundaries of equal-sized congressional districts wants to ensure the best electoral outcome for his own party. The so-called gerrymanderer faces two issues: the number of districts is finite and there is uncertainty at the level of each district. Solutions to this problem consists in cracking favorable voters in as many districts as possible to get tight majorities, and in packing unfavorable voters in the remaining districts. The optimal payoff of the gerrymanderer tends to increase as the uncertainty decreases and the number of districts is large. With an infinite number of districts, this problem boils down to concavifying a function, similarly to the optimal Bayesian persuasion problem. We introduce a measure of fairness and show that optimal gerrymandering is accordingly closer to uniform districting (full cracking), which is most unfair, than to community districting (full packing), which is very fair.