Mathematical theorem detects gerrymandering in districts

Credit: Tami Tedesco/ Credit: Tami Tedesco/

A team of researchers from the University of Pittsburgh (Pitt) and Carnegie Mellon used a new mathematical theorem to show that Pennsylvania’s congressional districts are gerrymandered.

Gerrymandering was described by President Obama in his 2016 State of the Union address as when “politicians pick their voters, and not the other way around.” It is the biased process of re-drawing political districts to ensure that one party gets a majority of votes, and it is often used by the majority party in state legislatures to ensure re-election.

Two Carnegie Mellon researchers Alan Frieze, professor of mathematical sciences, and Wesley Pegden, assistant professor of mathematical sciences, worked with Maria Chikina of Pitt. They proved a new mathematical theorem that determines whether a given sample is nonrandom, and therefore biased.

The theorem depends on Markov chains, algorithms that can be used to develop models of randomness. A Markov chain begins with a fixed object and evolves it through a step-by-step process, making small, random changes to the object at each step. Markov chains are frequently used to model random processes in fields like thermodynamics, statistics, and genetics. They also form the basis of Google’s algorithm for ordering search results.

The researchers applied their Markov chain research to Pennsylvania’s congressional districts. If the congressional districts were unbiased, they would resemble the map produced by the algorithm. This process was not intended to create optimal districting, but to provide a reliable test for bias within the current districts. To ensure accurate results, the researchers gave their algorithm constraints that congressional redistricting should ideally abide by — each district must be contiguous, contain roughly the same number of people and maintain a reasonable ratio of perimeter to area.

In each step of the algorithm, one precinct was chosen at random to be changed over to another district, as long as the change would still adhere to the imposed constraints. Starting with the current congressional district map, the researchers applied the algorithm for over a trillion steps.
The map randomly generated by the algorithm was significantly different from the real map. “There is no way that this map could have been produced by an unbiased process,” Pegden said of the current districts.

These results confirm mathematically what political scientists have been arguing for years — districting has become a biased system allowing those in power to remain in power. Though gerrymandering is present in almost all states, Pennsylvania has been one of the worst offenders since its last redistricting in 2010; the state’s seventh district appeared on the Washington Post’s list of the country’s top ten most gerrymandered congressional districts. Gerrymandering has caused drastic bias in Pennsylvania’s state legislature.

The Democratic and Republican parties both received close to 50 percent of votes in the 2012 and 2014 elections, but the gerrymandered districts (decided by a Republican state legislature) gave 72 percent of Pennsylvania’s congressional seats to Republicans.

Efforts to reform congressional districting are usually faced with partisan groups working to maintain or increase the impact of gerrymandering. Analysis and investigation like the research above constitute an important component of the fight for honesty in state legislature. Former California governor Arnold Schwarzenegger started a campaign against gerrymandering

The research was published online in February in the Proceedings of the National Academy of Sciences.