In 2011, the Republican majority in the Wisconsin state Legislature devised a political map that would tip future elections in its favor. While the lawmakers didn’t use any new political powers in the mapmaking process, they did deploy new technologies: computerized statistical tools that allowed for districts to be tailored to Republican political advantage with great precision and to great consequence, in a way that would not have been possible a few decades ago.
The Supreme Court is now considering, in the Wisconsin gerrymandering case Gill v. Whitford, whether partisan gerrymandering can be ruled out of bounds. In a 2004 case, Justices Antonin Scalia and Anthony Kennedy disagreed on whether the courts could ever establish a “workable standard” that would help jurists make that determination. Academics from multiple disciplines, including mathematics, computer science, political science, and economics, have been working to build such a standard, upping the quantitative and technological counterweight to legislators’ new gerrymandering tools. It turns out, though, that it’s much easier to game the system than to diagnose or redesign it.
While most people have an intuitive understanding that, say, drawing oddly shaped district lines to split and dilute the voting power of a minority bloc is undemocratic, that intuition is difficult to define. District shapes and even voting patterns can’t tell the full story of why a district looks the way it does. “We want a lot of things out of our district lines,” says Justin Levitt, a law professor and civil rights attorney at Loyola Law School in Los Angeles. “We want lines that yield a roughly fair partisan balance. We want lines that are at least not anti-competitive. … We want districts that represent real communities. We want districts that reflect geographic features.”
These goals are often in tension. Consider the grid below, which represents a state in which members of two different political parties are concentrated in different geographic areas.
Let’s say four representatives will be assigned to this state. Consider these two possible districting schemes:
If your goal is to maximize geometric neatness, you would prefer the first district map. If your goal is to maximize proportional representation, you would prefer the second. It’s tricky enough to choose between those two approaches. But what these simple models don’t capture is uncertainty in the model of voter preference—that is, swing voters. Add a few “purple” blocks to the same map, and redistricting becomes a deeper question of trade-offs and competing political values.
Does it make more sense to design maximally competitive districts or districts that guarantee proportional representation for the blue party? Does the answer change if the blue voters are members of a racial minority group that’s historically been disenfranchised? How persuadable do the people in those purple squares need to be to justify a different answer?
The trade-offs in any districting scheme are unavoidable and fundamentally political. Thankfully, there’s a well-studied branch of mathematics that can help us choose how to make those trade-offs: the game theory of dividing assets.
In their monograph Fair Division: From Cake-Cutting to Dispute Resolution, political scientist Steven Brams and mathematician Alan Taylor trace the history of fair division problems to Hesiod’s Theogony, around the eighth century B.C. One popular formulation of the problem goes like this: Suppose Prometheus and Zeus want to split a piece of cake fairly. Zeus worries that Prometheus, if given the knife, will cut himself a bigger piece.
Prometheus has the same worry about Zeus. No additional deities are available to act as a neutral third party. However, if both parties agree beforehand that Prometheus will cut the cake and Zeus will then choose his preferred piece, or vice versa, it becomes impossible for the cake cutter to pick the better piece for himself.
A recent proposal from Wesley Pegden, Ariel Procaccia, and Dingli Yu at Carnegie Mellon University develops a variant on the cake-cutting problem, “I cut, you freeze,” in which two parties take turns dividing the political map. At each turn, the party that did not divide the map chooses one district to freeze into place, leading to a subsequent turn in which the remaining map is again divided.
When presented with the grid in the earlier example, the blue party could bring a competitive or risk-averse strategy into the game, with different outcomes to match. The risk-averse strategy would guarantee at least one blue-majority district.
The competitive strategy trades the safe blue district for two swing districts.
The advantage to the fair division approach is that it allows each party to define the values it brings to the districting process; values that strongly disadvantaged one of the parties, and therefore the democratic soundness of the system, would have less sway. Levitt, the law professor, says the system could “allow legislators control while putting guardrails up to prevent extreme outcomes.” In that sense, “I cut, you freeze” is a provocation, a reminder to policymakers and mathematicians alike that rigor can support, rather than replace, the hard questions about values—for which we still need politics.