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Math Trek

Mathematician answers Supreme Court plea

New, fair method for dividing states into congressional districts could reduce political squabbles.

2:47pm, February 24, 2009
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In 2003, Republicans in the Texas state legislature proposed a bill that would redistrict the state to increase the likelihood of Republican victories. The Democratic representatives, lacking the votes to defeat the measure, fled the state to deny a quorum. After two standoffs (one lasting 45 days), a Democrat broke down and returned to work, and Republicans pushed the measure through. In the next election, Texas Republicans gained six seats in the U.S. House of Representatives, for a total of 21 seats out of 32.

Democrats sued. The Republicans argued that the new districting was only redressing past wrongs, as Republicans had held fewer than half of the Texas congressional seats, even though they had 57 percent of the vote. In 2006, the case reached the Supreme Court.

“Because there are yet no agreed upon substantive principles of fairness in districting, we have no basis on which to define clear, manageable, and politically neutral standards,” Justice Anthony Kennedy had written two years earlier in a similar case in which the judges upheld the redistricting of Pennsylvania. “If workable standards do emerge … courts should be prepared to order relief.”

In the intervening two years, no such standards had presented themselves. The Texas redistricting was upheld.

The next time a redistricting case goes before the Supreme Court, a mathematician says he can provide a method that may satisfy the court. The solution, says Zeph Landau of the University of California, Berkeley lies in cutting cake.

Politicians figured out the power of redrawing district boundaries back in 1812, when Governor Elbridge Gerry lumped most of the Massachusetts Federalists into a single district, allowing his own part to take control of all the other districts in the state. Newspapers mocked the strange, salamander shaped districts, saying he had “gerrymandered” the state. Oddly shaped congressional districts are now common across the country.

By arranging the boundaries to lose big in a few districts and win the rest by small but safe margins, a party can as much as double its percentage of seats. So if, for example, 40 percent of people in the state voted Democratic, redistricting could in theory make 80 percent of the congressional seats Democratic. If, on the other hand, the Republicans drew the boundaries when they had 60 percent of the vote, they might be able to almost double their percentage and get every last seat, although these theoretical maximums often can’t be realized because of geographical constraints.

So what’s fair?

An entire field of mathematics is devoted to answering just this kind of question. For example, take the classic “I cut, you choose” method of dividing cake: If I cut a cake into two pieces I’d be equally happy with, and you pick which of the two you like better, then neither of us will prefer the other person’s piece to the one we have. The division will be fair in that sense even if our priorities are different. For example, I might really want the rose made of frosting, while you might care only about the size of your piece.

Landau and his collaborators, students Ilona Yershov and Oneil Reid of the City College of New York, realized that the mathematics of fair division could be used to solve the redistricting problem. They used a variation on another cake-cutting method: A third party wields the knife, moving left to right across the cake until one of us calls out, “Stop!” when it seems that both sides are equally good. Then the person who called out gets the left piece and the other gets the right one.

The researchers proposed that a variation of this method be used to divide the state into two regions such that neither political party preferred the other’s region. From there, each party would divide up its own region however it liked.

At first blush, this plan doesn’t seem to solve the problem at all. After all, if one party has only 40 percent of the vote, why should it get a full half of the control of the process of dividing the state into districts?

But the mathematicians showed that equally shared control will lead to about the right outcome even if the parties get very different proportions of the votes. If Democrats get only 40 percent of the vote, they can divide up their half of the state to get at most 80 percent of the seats in that region. If the Republicans get all the seats in their half, that means the Democrats would get about 40 percent of the total seats, which corresponds to their percentage of the total vote anyway.

“The idea is to set up the rules of the game so that cheating isn’t really possible,” Landau says.

Landau points out that any restrictions ordinarily applied to the entire state would continue to be applied to the two half-states. So, for example, districts would continue to be required to have approximately equal populations, and the Voting Rights Act would continue to require that for both half-states, the majority of the population in some districts be ethnic minorities.

This fair division method offers the alluring possibility that each party may feel it got the better deal. The reason goes back to the cake: If I care most about the rose made of frosting and you care most about the size of your piece, we each may think our piece superior to the other’s. Similarly, Landau points out, one political party might particularly want to be able to win the district with a stadium in it, while the other party cared more about a district with an important donor.

The team presented its findings in January at the Joint Mathematics Meetings in Washington, D.C., and the research will appear in an upcoming issue of Social Choice and Welfare.

Political scientist David Epstein of Columbia University praised the approach as innovative, but said it’s unlikely to be politically feasible. “The idea that any subset of people is going to have 100 percent dictatorial control of any portion of any state is totally incompatible with the democratic process,” he says. Still, he believes the idea could be useful in other settings, such as perhaps for sharing power within a corporation.

Landau points out that in the current scheme, the ruling party has nearly dictatorial control already, and his scheme assures that that control can’t be used unfairly. “The problem is that the underpinnings of its fairness aren’t quite transparent,” he says. “It requires a paper to explain it.”

What is clear in any case is that a solution is urgently needed. In the 2004 Pennsylvania case, Justice David Souter remarked, “The increasing efficiency of partisan redistricting has damaged the democratic process to a degree that our predecessors only began to imagine.”

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