Geometry Reveals the Tricks Behind Gerrymandering

Geometry Reveals the Tricks Behind Gerrymandering

Hardly anyone could have imagined that the ability to calculate the area of triangles or the volume of prisms in high school could be used to influence an election’s outcome. However, geometry can be a powerful tool for shaping the results of an election contest, at least in plural voting systems.

Designing a perfect election system for multiple parties is impossible, even with mathematical tools. However, if there are two dominant parties (as in the U.S.), then things should be quite clear. Right? Anyone who has been following the U.S. presidential election process in recent years knows that reality is . different. The actual layout of the voting districts is an important factor. A party that is actually losing may still win the majority of the representatives if it is well-designed. This issue was not absent in the U.S. midterm election.

Math plays an important role in determining election outcomes, particularly for the 435 seats in the House of Representatives. A party can win by choosing the boundaries of a congressional district. This is even if the vote count is not representative of the sentiment of the voters.

Here’s a highly simplified example: suppose a state consists of 50 voters, 20 of whom vote for a blue party and 30 for a red party. Some sections of Manhattan might have a grid-like pattern. Suppose there are 10 north-south avenues and five east-west streets. All the red voters reside on the two closest avenues. The other three avenues are home to the blue voters. Now it is time to divide the voters into five equal-sized electoral districts.

Deliberate choices about the boundaries of an electoral district can favor one party over another. Credit: Spektrum der Wissenschaft/Manon Bischoff (detail)

One could draw five horizontal boundaries: There would then be two electoral districts with only red voters, and three with only white voters. The votes in this district would result in three representatives from the blue party and two from the red party, a true reflection of voters’ opinions.

But, if the blue party wins the right to draw district boundaries, they might be inclined horizontally. All the districts would then look the same, with six blue voters and four red voters. The blue party wins each district and gets all five representatives. Something similar happened in New York state in 2012: 58 percent of people there voted for the Democrats, but the party got 21 of 27 seats (five more than would have been justified if the election districts had been drawn equitably).

The state legislatures and commissions that redraw the district lines might, however, make a different (somewhat more difficult) partitioning. They could do this by putting almost all blue voters in two districts. This would give the red party a majority of the remaining three districts with three red congressmen and two red congresswomen. However, more voters voted for the blue party. This is a common tactic used in U.S. congressional elections. For example, in Pennsylvania in 2012, Democrats received 51 percent of the vote, but only five of 18 seats.

The Gerry Mander’s body conforms to the shape of a Massachusetts electoral district drawn up by Governor Elbridge Gerry in the early 19th century. It reminded cartoonist Elkanah Tisdale in 1812 of a salamander. Credit: Bettmann/Getty Images

The deliberate redrawing of districts to gain a majority goes by the name of gerrymandering, a portmanteau of “gerry” and “salamander.” The former refers to Elbridge Gerry, the governor of Massachusetts in the early 19th century, who approved extremely odd-shaped voting districts that gave his party an advantage.

Even today, in most U.S. states, legislatures decide on the division of electoral districts about every 10 years (with the appearance of the new census). Redistricting is often used to the advantage of incumbent parties. This has been proven time and again. This can often be seen in strange-shaped electoral districts, similar to the one in Massachusetts at the beginning of the 19th century. Gerrymandering was first coined by a cartoonist who noticed that one of these districts looked like a salamander.

Restricting has been the subject of numerous legal challenges. In 1986, the U.S. Supreme Court even ruled that intentional gerrymandering is illegal. It has not yet touched an election district. It turns out that it is not easy to set fair districting rules. Even mathematicians are struggling to answer the question, and they are investing in huge computer power to solve it.

Boundaries for Maryland’s third congressional district. Credit: GIS shapefile data created by the United States Department of the Interior/Boundaries for Maryland’s 3rd United States Federal Congressional District (since 2013)/Wikimedia (detail)

How do you find gerrymandering Observing the Maryland district pictured above, you might suspect the designers had certain ulterior motives. It is especially striking because it is extremely jagged. One argument is that district borders should be “compact”, but it’s not clear what “compact” means.

One clue that gerrymandering might be present is the length the outer boundary. The more jagged a district is, the larger its perimeter. Redistricting literature often recommends drawing the smallest circle possible to include a district’s area and comparing it with the existing boundaries. The greater chance that a district has been redrawn to suit political ends, the further the boundaries of a district deviate from a circle. Gerrymandering may also be indicated by the average distance between residents in a precinct.

The division into electoral districts is not an easy task. Each state has its own rules for partitioning into electoral districts. A district should have roughly equal numbers of voters, be in the same area, not discriminate against any ethnic group, not cross county lines and follow natural boundaries such as rivers. These restrictions can lead to fractured districts, even without considering the voting habits of residents.

A compact voting district does not necessarily lead to equitable representation, as a 2013 study found. The study paid particular attention to the 2000 presidential election in Florida, in which about as many people voted for Democrats as Republicans, but the latter accounted for 68 percent of the votes in Florida’s congressional districts. Researchers used a nonpartisan algorithm to draw the most compact districts while still adhering to state rules.

Surprisingly, however, the computer also produced skewed outcomes, in which Republicans would largely have an advantage. Experts quickly discovered the reason: Most Democrats live in Florida cities. This means that they win in urban areas overwhelmingly while narrowly losing rural areas in each case. This “natural gerrymandering” means that more Republicans will be elected to the House of Representatives.

Florida is not an isolated case, as political scientist Jonathan A. Rodden noted. The problem is not the lack of “compactness” in a district. As Jonathan A. Rodden, a political scientist, noted , How can we measure this? In 2014, University of Chicago legal scholar Nicholas Stephanopoulos and Public Policy Institute of California political scientist Eric McGhee developed a metric for the problem, the efficiency gap. It is calculated by subtracting “wasted” votes from two parties and then multiplying by the total votes. In this example, a wasted vote for any party is one that results in a losing district to the other party or is greater than the margin required to win. The more impartial the result, the smaller the efficiency gap.

Calculating the efficiency gap, a measure of fairness in establishing electoral districts. Credit: Spektrum der Wissenschaft/Manon Bischoff (detail)

To visualize this, we can again use the initial example with the 50 voters (20 for red, 30 for blue) and calculate the efficiency gap for the different divisions. In the first case, when all boundaries were drawn vertically, the first and second districts (from the left) each have 10 red votes, wasting four each. The third, fourth and fifth districts, on the other hand, each have 10 blue votes, four of which are also wasted. Thus, the efficiency gap is as follows (the vertical bars indicate absolute value): |(2 x 4) – (3 x 4)|/50=2/25=0.08.

In the second division, each district is equal: blue always wins by six votes out of 10. All votes of red are used, but none of blue’s are wasted. The efficiency gap is 20/50=0.4, which is significantly higher than in the first division.

The third example is the most interesting: the two districts where blue wins 9 to 1, each have a surplus of three blue votes. In the three winning red districts, four blue votes each are wasted–so in total, (2 x 3) (3 x 4)=18 blue votes that are surplus ones. There are only two red votes left that were not used. This results in an efficiency gap of (18 – 2)/50=8/25=0.32.

The efficiency gap can be used to determine partisanship in particular voting districts. Sometimes, however, it is difficult to find better options due to natural circumstances such as when all the voters of a party reside in the same place. To investigate these possibilities, statistician Wendy Cho, along with computer scientist Yan Liu and geographer Shaowen Wang of the University of Illinois at Urbana-Champaign, designed an algorithm that divides maps into districts–based on the rules set by the state in question.

It is very difficult to find the best division of districts in order to ensure that each party has the equal chance of converting a vote into seats. This task falls under the category of “NP problems”, which computer scientists and mathematicians have known for decades cannot be solved with ordinary computers. It doesn’t necessarily mean that you won’t find a solution, but it could take a very long time. Cho and her coauthors decided to allow the computer to create a large number of splits, which are not always perfect.

For example, when they applied their program to the state of Maryland in 2011, they realized that almost all of the 250 million results gave an advantage to Democrats. The natural conditions and requirements for voting districts are such that Republicans are automatically at disadvantage. Cho and her colleagues compared Maryland’s actual apportionment with the computer’s output and were able to show that the official voting districts favored Democrats in more than 99. 79 percent of the 250 million computer-generated results.

Meanwhile, some U.S. states (mainly those where Democrats are in the majority) use independent commissions that redraw voting districts. These panels often resort to computer programs to find the fairest possible apportionment. In general, the apportionment of electoral districts this year appears to be the fairest in 40 years, as reported by the New York Times. When advantageous or detrimental districting decisions for both parties in all U.S. states are netted against one another, gerrymandering should result in only three extra seats for Republicans– down from 23 seats in 2012. In close elections, even three seats could make a difference. And news stories before the midterms depicted how gerrymandering is still very much a matter of public debate: Alabama’s state legislature redistricted to put many of the Black voters in the state into just one district, decreasing their electoral power, and resulting in a case that is now before the U.S. Supreme Court.


    Manon Bischoff is an editor at Spektrum der Wissenschaft. She primarily covers mathematics and computer science and writes the column the Fabulous World of Mathematics. Bischoff studied physics at Technical University of Darmstadt in G

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