In a democracy, voters get to pick their leaders. And every vote should have the same weight in selecting the winner. Indeed, under the U.S. Constitution, there is a general principal: One person, one vote. In some states, however, some votes may not carry the same weight.
In the United States, many elected officials at the state and federal levels represent people in particular districts, or areas. In most cases, states decide the boundaries for those districts. But, sometimes, the people who draw those boundary lines may deliberately give a big advantage to one political party. This is called gerrymandering, and it’s not fair.
Moon Duchin is a mathematician at Tufts University in Medford, Mass. “Gerrymandering is basically a math problem,” she explains. “It’s asking how you can divide something up so the division has certain properties.” In this case, one group wants the boundaries of the voting districts to include more people from one political party or from a certain interest group. In this way, district voters will likely give a win to a selected party or group.
Last year, some voters in Wisconsin legally challenged what they claimed were gerrymandered districts. Another case in Maryland challenged changes to one district’s boundary lines. The cases went all the way to the Supreme Court. The high court has just handed down its decisions.
On June 18, the court ruled against the Wisconsin challengers. All nine justices found that individual voters could not challenge the whole state’s voting district map. But seven justices left open the possibility for individuals to challenge the lines of their individual districts.
On the same day, the court also ruled against the Maryland challengers. The challengers had wanted the lower court to prevent use of the old voting district map until the case was finished. But that’s a matter of discretion for the trial judge.
Both decisions essentially sidestepped the basic question: When, if ever, do districts drawn to favor one party over another violate the U.S. Constitution?
Political parties in power want to stay in power. One way to do that is to encourage politicians in their party to support things their constituents want and need. Things like safety. Good schools. A clean environment. Good roads. Stable, well-paying jobs.
But sometimes political parties do more than that. They may redraw the boundaries of voting districts to make it much more likely that their supporters will win a majority. This is gerrymandering — the drawing of voting-district boundaries in such a way that one party gets a deliberate edge.
The practice goes back to at least 1812. That year, Massachusetts governor Elbridge Gerry’s team drew up a state voting map. One oddly shaped district looked like a fantasy monster. A Boston newspaper called it the “Gerry-mander.” And so the term was born.
Each state sets its districts for picking state and federal lawmakers. Districts should have roughly equal numbers of voters. In general, those voters should all live in the same neighborhoods or communities. Districts shouldn’t be drawn on the basis of factors such as race. (In fact, drawing districts in a way that dilutes the votes of racial minorities has been ruled unconstitutional. Doing it in a way that prevents minorities from electing their preferred candidate is also illegal.)
Wisconsin’s plan basically met those tests. Still, the plan gave a statewide edge to Republican candidates.
In 2012, for example, Republicans running for the state’s lawmaking branch got 48.6 percent of the popular vote statewide, or just under five in every 10 votes. Yet of 99 seats, Republicans won slightly more than six in every 10.
Like the original gerrymander, some voting districts may have odd — and tell-tale — shapes. But shape isn’t the only thing that matters. “You can do really unfair gerrymandering with shapes that look round and square and pretty,” Duchin notes. What’s more, oddly shaped districts might sometimes be fair.
“It’s not a beauty contest,” she says. Duchin and Mira Bernstein, also a mathematician at Tufts, wrote about the Wisconsin case and math in the October 2017 Notices of the American Mathematical Society.
Lawyers for both Maryland and Wisconsin did not want the courts to interfere. Maryland’s governor had even admitted that part of the reason for the challenged change was to give the Democrats an edge. Lawyers defending Wisconsin argued that voters could not challenge a statewide plan.
Story continues below image.
Cracking and packing
Math experts analyzed the Wisconsin state plan for the challengers. And those math experts found that two tricks led to gerrymandered vote results.
They refer to the first as “cracking.” It splits up a party’s supporters so they will be a minority in all or most districts. Their candidates might therefore lose in those places. The second trick is “packing.” It lumps a whole lot of one party’s backers into the same district. Now, their party’s candidate should win by a huge margin. However, many of the extra votes would be wasted; they wouldn’t affect the outcome of the election.
To see how cracking and packing can hurt, Duchin says, imagine a map with 42 grid squares. You want to divide it into six voting districts. Asterisks mark grid squares where voters prefer party X. Blank grid squares contain voters for party Z. If each party has 21 of the 42 squares, then you might expect each party to win in three districts.
But lines can be drawn to give one party’s supporters enough more than half the votes in most districts. Then party X might get five seats in the legislature, leaving Party Z with only one.
In an earlier case, the Supreme Court ruled that election results don’t have to match the proportion of voters statewide. Also, in real life, voters from different parties aren’t spread out evenly across a state. In some states, for example, people in large cities may be more likely to vote for Democrats. Republicans may be more common in rural areas. Social scientists ascribe such differences to what they call “political geography.”
Voters in both parties always have some “wasted” votes. Some votes will go beyond those needed for a candidate win. Other “wasted” votes would be votes for the losing side.
Adding to the problem: Not everyone votes. If one party is more likely to stay home on Election Day, their nonvoters may throw away the chance for their preferred candidate to win.
In 2012, 2014 and 2016, Wisconsin’s Democrats were significantly more likely to have “wasted” votes, the challengers said. In their view, that created an “efficiency gap.” Basically, they argued, the Republicans’ votes counted more than the Democrats’ did. In fact, Republicans did win a share of seats in the legislature that was far bigger than their statewide share of voters.
Story continues below image.
Now the Supreme Court has said that the individual voters in Wisconsin can’t challenge the statewide voting district map. Rather, a voter can only claim harm based on his or her district. “An individual voter in Wisconsin is placed in a single district,” Chief Justice John Roberts wrote. “He votes for a single representative.” If there’s a problem, then the solution is to change the boundaries of that district, not all the districts in the state.
Scouting for gerrymandering
The decision will make it tougher to challenge state voting district plans. But the court left open the chance for people to prove that their votes were watered down. And it doesn’t prevent voters from raising different arguments in future suits. But first, plaintiffs will need to show that their district’s lines were drawn unfairly.
In Duchin’s view, you need a range of math tools to do that.
Among them: computers. Algorithms are groups of rules to solve a problem in a series of steps. Using them, computers can come up with many thousands of possible ways to divide a state map into voting districts. Those computer techniques can also suggest many, many ways to split up the actual distribution of voters. (Remember, people are not equally distributed across a state.) All those other district maps form a huge sample set of plans. Then, Duchin says, you can compare plans and how fairly they would represent all voters.
Another important tool: statistics. In this field of mathematics, researchers study sets of data and information that can be gleaned from them. An outlier is a datum, a single piece of data, that is well outside the range of the rest of the set. It can be either higher or lower than those other data. But in any case, it’s not what you’d otherwise expect from a random sample.
Outliers happen. But they raise questions in voting rights cases.
One question: Compared with other possible plans, does the outlier plan give members of one political party a significant advantage or disadvantage? “If your plan performs the way lots and lots of other plans would, then it’s presumptively okay,” Duchin says. But if it waters down the value of votes from one party, compared with the other, “then your plan is really an abusive outlier,” she suggests. In other words, it would suggest someone went out of their way to hurt or help one party.
Wesley Pegden is a mathematician. He works at Carnegie Mellon University in Pittsburgh, Pa. His testimony about gerrymandering helped persuade judges to find that Pennsylvania’s congressional election map could not stand under state law. On March 19, the Supreme Court refused to block the new district map.
Like Duchin, Pegden likes the idea of seeing how voting district maps compare with other possible plans. But he suggests a different way to get a good sample of what other mapped districts would look like. His approach: Start with the plan that was drawn up. Then make random switches. At each step, pull a voting precinct (part of a district) in from a neighboring one, or give it up to a neighbor. This will create a whole lot of maps based on those small random changes.
Then see how those new maps compare to the one that politicians drew up. So, Pegden would ask: If one party was at a big disadvantage in the politicians’ plan, does its handicap get bigger, smaller or stay pretty much the same as random changes are made to produce new maps?
Here's a way to think about it, Pegden explains. Suppose you were asked to write numbers from one to 10 anywhere on a grid of squares. But you also want to make sure that any square someone might pick at random will be surrounded by squares with higher numbers. You can’t do it, Pegden says.
Likewise, when it comes to voting districts, “it is impossible,” Pegden says, for a series of random changes to map out new districts where each is fairer than the first — unless that first one had not been fair. That’s true regardless of an area’s political geography or other features, he notes.
Suppose almost all those maps made from little random-step changes would give election results with less bias for one party over the other. “Then that shows you that somebody worked hard to make that [first] map how it is,” Pegden says. And, presumably, the map produced by all of that hard work is unfair. His team wrote about the math and gerrymandering in the March 14, 2017 Proceedings of the National Academy of Sciences.
How could you draw districts that would be fair? Pegden and another group he works with suggest one approach. It’s like the “I cut, you choose” way to split a piece of cake. Under the plan, one party could “cut” the state’s area into districts. The other party could freeze the boundaries of one of those districts. Next, that other party could redraw all the other district lines. Now the first party would freeze one district’s lines. The process would go on until all the districts were frozen. The group plans to publish its work in a journal soon.
If an individual district was unfairly set, the Supreme Court said the remedy would be to change that district. But remember that districts need to have roughly equal numbers of voters. And the redrawn districts would have to avoid any legal problems that the first ones had. So it’s possible that challenges to enough single districts could lead to big changes in state maps.
Also, the courts are not the only place to work out complaints about voting district maps.
Voters can ask state lawmakers to change the way they set districts. Voters can also push to amend a state’s constitution.
The gerrymandering issue shows the importance of math “for everyday basic problems,” such as how to hold fair elections, Pegden adds. “It’s important for us all to feel like government in a democracy is basically fair,” he says.
algorithm A group of rules or procedures for solving a problem in a series of steps. Algorithms are used in mathematics and in computer programs for figuring out solutions.
application A particular use or function of something.
average (in science) A term for the arithmetic mean, which is the sum of a group of numbers that is then divided by the size of the group.
bias The tendency to hold a particular perspective or preference that favors some thing, some group or some choice. Scientists often “blind” subjects to the details of a test (don’t tell them what it is) so that their biases will not affect the results.
constituent An ingredient or building block of some material. (in politics) Someone represented by an elected political leader, such as a mayor, governor or senator.
data Facts and/or statistics collected together for analysis but not necessarily organized in a way that gives them meaning. For digital information (the type stored by computers), those data typically are numbers stored in a binary code, portrayed as strings of zeros and ones.
discrimination (in social science) An attitude of prejudice again people or things based on a bias about one or more of their attributes (such as race, sex, religion or age). It is not based on the actions of an individual but instead based on yet-unfounded expectations that are being applied broadly to a whole group.
environment The sum of all of the things that exist around some organism or the process and the condition those things create. Environment may refer to the weather and ecosystem in which some animal lives, or, perhaps, the temperature and humidity (or even the placement of components in some electronics system or product).
factor Something that plays a role in a particular condition or event; a contributor.
federal Of or related to a country’s national government (not to any state or local government within that nation). For instance, the National Science Foundation and National Institutes of Health are both agencies of the U.S. federal government.
field An area of study, as in: Her field of research was biology. Also a term to describe a real-world environment in which some research is conducted, such as at sea, in a forest, on a mountaintop or on a city street. It is the opposite of an artificial setting, such as a research laboratory.
geometry The mathematical study of shapes, especially points, lines, planes, curves and surfaces.
gerrymander The act of changing the boundaries of communities that will vote for the same set of candidates in political elections, usually with the goal of helping keep the same political parties in office.
grid (in mathematics or mapping) A network of lines that cross each other at regular intervals, forming boxes or rectangles, or an orderly field of dots that mark where each pair of lines intersect, or cross one another.
journal (in science) A publication in which scientists share their research findings with experts (and sometimes even the public). Some journals publish papers from all fields of science, technology, engineering and math, while others are specific to a single subject. The best journals are peer-reviewed: They send all submitted articles to outside experts to be read and critiqued. The goal, here, is to prevent the publication of mistakes, fraud or sloppy work.
outlier (in statistics) An observation that lies outside the range of the rest of the data.
politician A person who runs for or holds elected office in a town or larger governing body. For politicians, governing people (or organizations) and wielding power within government is a profession (job).
Proceedings of the National Academy of Sciences A prestigious journal publishing original scientific research, begun in 1914. The journal's content spans the biological, physical, and social sciences. Each of the more than 3,000 papers it publishes each year, now, are not only peer reviewed but also approved by a member of the U.S. National Academy of Sciences.
proportion The amount of a certain component of a mixture relative to other components. For example, if a bag contains 2 apples and 3 oranges, the proportion of apples to oranges in the bag is 2 to 3.
random Something that occurs haphazardly or without reason, based on no intention or purpose.
range The full extent or distribution of something. For instance, a plant or animal’s range is the area over which it naturally exists. (in math or for measurements) The extent to which variation in values is possible. Also, the distance within which something can be reached or perceived.
social (adj.) Relating to gatherings of people; a term for animals (or people) that prefer to exist in groups. (noun) A gathering of people, for instance those who belong to a club or other organization, for the purpose of enjoying each other’s company.
society An integrated group of people or animals that generally cooperate and support one another for the greater good of them all.
standard deviation (in statistics) The amount that each a set of data varies from the mean.
statistics The practice or science of collecting and analyzing numerical data in large quantities and interpreting their meaning. Much of this work involves reducing errors that might be attributable to random variation. A professional who works in this field is called a statistician.
Supreme Court The highest U.S. court to deal with legal questions about how to interpret the Constitution (the document that spells out how the national government works and what basic rights are), federal laws and treaties, and maritime law. Most cases at the Supreme Court are appeals of the decisions of lower courts. Constitutional issues deal with the rights of individuals, states or others. A few cases can also start there (such as disputes between two states). The Supreme Court has a Chief Justice and eight other justices (they are not called “judges” in this court). The president nominates individuals to serve on the court. Those appointments must then be approved (confirmed) by the U.S. Senate. Justices on this court serve for life, which means as long as they choose and are not found guilty of certain serious offenses.
tool An object that a person or other animal makes or obtains and then uses to carry out some purpose such as reaching food, defending itself or grooming.
transition The boundary where one thing (paragraphs, ecosystems, life stage, state of matter) changes or converts into another. Some transitions are sharp or abrupt. Others slowly or gradually morph from one condition or environment to another. (in gender studies) A term used to describe the process of permanently changing one’s outward gender in terms of name, behavior and expression to match one’s inner sense of his or her gender.
Journal: W. Pegden et al. A partisan districting protocol with provably nonpartisan outcomes. arXiv:1710.08781v1. Posted online October 24, 2017.
Journal: M. Bernstein and M. Duchin. A formula goes to court: Partisan gerrymandering and the efficiency gap. Notices of the American Mathematical Society. Vol. 64, October 2017, p. 1020. doi: 10.1090/noti1573.
Meeting: M. Duchin. Situating redistricting as a problem in law, math and politics. The Geometry of Redistricting Workshop. August 7, 2017. Somerville Theater and Tufts University, Somerville and Medford, Mass.
Journal: M. Chikina et al. Assessing significance in a Markov chain without mixing. Proceedings of the National Academy of Sciences. Vol. 114, March 14, 2017, p. 2860. doi: 10.1073/pnas.1617540114.