THIS time next year, US voters will be picking over the results of the 2018 mid-term elections. Will a beleaguered President Trump see the Republican majority harden in a further swing to the right? Or will the Democrats sweep to power in Congress and thwart his agenda?
It all comes down to where the votes are on the map – and how much they count for. Because in a “winner takes all” electoral system like that in the US and the UK, not all votes are created equal. You might be living in an area that’s a shoo-in for one party – meaning whether you support it or don’t, your vote is superfluous as far as the final result is concerned. Conversely, the electoral division you live in might be on a knife edge, giving disproportionate weight to the small number of votes that can tip the balance. That means the boundaries between seats matter, too.
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Gerrymandering – the drawing of electoral boundaries to benefit one party or group – is currently a hot political topic in the US, where the Supreme Court is considering a landmark case on the practice. In the UK, supposedly non-partisan boundary commissions are soon to propose what could be the biggest redrawing of its electoral map since the 1920s. So can boundaries ever be drawn fairly? And what exactly does fairness mean? To answer these questions, we need to put mathematics at the heart of politics.
Underpinning the democratic process is the notion that all votes should count equally: if it takes 1000 voters to elect a representative in one district, for example, it shouldn’t take 100 to elect another. It’s an admirable goal, and one of the reasons that regular censuses form an important part of democratic life.
But even within congressional or parliamentary districts of similar size, a single vote doesn’t always carry the same weight. The inordinate value of a few votes in finely balanced districts is exacerbated in winner-takes-all systems, where a candidate can represent nearly half the voters but still lose out by one or two percentage points.
This is the system used for parliamentary elections in the UK, as well as across the US (with the exception of a handful of state legislatures and the notoriously convoluted presidential elections). In both countries, a party can accumulate millions of votes and still secure no representation. “The most reasonable thing to do would be to abolish the system and start again,” says mathematician Peter Gritzmann at the Technical University of Munich, Germany. But that’s not going to happen any time soon, and even if it did, there’s no perfect voting system to take its place.
In the UK, at least, politicians generally don’t get to decide the electoral maps (see “Splintered Isle“). In the US, although a couple of states have handed over “redistricting” powers to an independent commission, those in power across most of the country have almost total freedom to redesign the maps. And redesign them they do. Republicans and Democrats are both guilty of gerrymandering, despite both sides agreeing that it’s wrong. The problem is that spotting the practice is tricky, and proving it even trickier.
The classic technique to hijack an electionis called packing and cracking: a gerrymanderer tries to create a small number of districts packed with their opponent’s voters, and draws other seats to spread the remaining vote so there isn’t quite enough for a majority. This results in their opponent winning a few seats with thumping majorities, while narrowly losing many more (see “Divide and rule“).
Packing and cracking has been used along party, racial and religious lines all over the world. In the US, racial gerrymandering in particular – the redistribution of African American or Hispanic voters so as to curb their influence – was so widespread that the government was forced to step in and outlaw it. The Voting Rights Act of 1965 protected the rights of minority citizens to have their voices heard, effectively institutionalising a form of benign gerrymandering to protect their ability to elect a representative of their choice.
“Voters don’t choose their elected officials. Elected officials choose voters”
But when political affiliation is at play in drawing electoral boundaries, the courts have been effectively toothless. The result is that voters don’t choose their elected officials; instead, elected officials choose their voters.
With advances in technology over the years, it is now possible to quickly assess thousands of possible electoral maps and their impact. Worried you are just not hitting it with the younger generation? Press a few buttons and moments later, you can have hundreds of maps that satisfy all of the legal requirements, while also packing or cracking the youth vote.
The case now being considered by the US Supreme Court has its roots in a redrawing of the electoral map of Wisconsin by Republican legislators in 2011. The benefit to their party was obvious, as a year later they won 60 per cent of the seats in the Wisconsin State Assembly with less than half the overall vote – a significantly improved result. When they repeated this feat at the 2014 elections, a group of Democrat voters sued. Nearly every similar lawsuit had failed, but surprisingly this group won. In November 2016, the state’s federal court in Madison said that the maps were so biased that they violated the constitutional rights of Democrat voters.
But what does fairness look like? For most people, a natural definition looks a lot like proportional representation, where the percentage of votes a party gets aligns with its share of seats. But that’s not the intent of a winner-takes-all system, where non-proportional results are part of the furniture. Somewhere like Massachusetts, for example, has a solid Democrat majority spread evenly across the state – so all nine members returned to the US House of Representatives are Democrats. “Even if you tried to gerrymander Massachusetts in favour of Republicans, it would be extremely hard,” says Mira Bernstein, a mathematician at Tufts University in Medford, near Boston.
The Supreme Court has previously ruled that proportional representation is not guaranteed by the constitution, ruling it out as a way of defining “fair” boundaries. “This does not prevent us from finding a good mathematical test for gerrymandering, but it does mean that such a test will be hard for most people to grasp,” says Bernstein.
One test will be obvious to anybody who has ever seen a gerrymandered district: it looks funny. This is hinted at by the word “gerrymander” itself, coined after the 1812 redistricting plan of Massachusetts governor Elbridge Gerry, whose redrawn maps included one notorious district shaped like a salamander. Ever since, mathematicians have tried to craft some measure that would reveal when a district was too weirdly shaped to be anything but the product of a political agenda at play.
Such “compactness metrics” sound good in theory. The trouble starts when you try to quantify what it is that makes one shape weirder than another. For some, the key property to test is isoperimetry, a measure of how efficiently a perimeter encloses the area within. By this yardstick, circles are maximally compact while long, snaking salamander shapes are not very compact at all. The logic starts to fall apart, however, when we confront the issue of how to measure a perimeter. Do you go around the boundaries of the district measuring distances down to the nearest kilometre? Centimetre? Nanometre? The smaller your ruler, the more imperfections you’ll be able to measure. Rather than a simple polygon consisting of a handful of straight lines, you could end up with a jagged shape of almost infinite perimeter.
A simpler version of the test measures convexity, or how closely the district’s area matches that created by placing a giant elastic band around it. Squares and rectangles? Very convex. Crescent moons and spiky polygons? Not so good.
Tests like convexity, or more sophisticated ones such as curvature, which take into account how a district’s residents are spread within it, are a step in the right direction. But ultimately, they fall at a crucial hurdle. Sometimes, districts just need to be a funny shape. Highways, rivers, mountain ranges and city boundaries all impose real constraints on map-makers, and can result in shapes no less weird than Gerry’s salamander for reasons that are perfectly noble. The 4th district in the state of Illinois, for example, is sometimes known as the “earmuffs” because of its distinctive shape. But to the two Hispanic communities it connects, the earmuffs offer them a hard-won constitutional right to decide on their own representative. Geometry alone would leave them vulnerable.
In 2014, Nicholas Stephanopoulos of the University of Chicago Law School and Eric McGhee of the Public Policy Institute of California developed an alternative test. Called the efficiency gap, it’s a simple way to hunt for signs of packing and cracking, and has accompanied the Wisconsin case all the way to the Supreme Court.
The efficiency gap is based around counting wasted votes for all political parties, a wasted vote being one that doesn’t contribute to electing a representative. So for example, all of a losing party’s votes are wasted, as are any of a winning party’s votes that it didn’t need for victory. Every system will have wasted votes, but if one party is wasting substantially fewer than another, it’s a sign gerrymandering could be afoot (see “How to spot a gerrymander“).
In Wisconsin, the efficiency gap was 13 per cent in favour of the Republicans, three times the average across the country. The lawyers in the original court case argued that anything over 8 per cent should be considered unconstitutional, and are hoping the Supreme Court approves their logic.
Fairest of them all?
The efficiency gap is a simple metric that may go on to shape the future of US democracy. But it’s not without its drawbacks. It turns a blind eye to gerrymandering committed by both parties, for example, so long as they offend equally, and struggles to produce meaningful results when one party has a genuinely dominant majority. For some, the efficiency gap also fails because it assumes that voters are entitled to a specific relationship between vote share and representation. “There is absolutely no reason to think that this, or any other, relationship is mandated by the constitution,” says Bernstein.
One way of distinguishing between natural and deliberate gerrymandering is to use the ability of computer simulations to generate thousands of different maps. A group at Duke University in Durham, North Carolina, recently used an algorithm to randomly draw 20,000 possible electoral maps for Wisconsin that satisfied all of the required criteria laid down in US law. In most of these, the Republicans won a majority, making it seem like the Democrats were just at a natural disadvantage.
But in the majority of the maps, Republicans secured a narrow advantage, replicating their 2014 margin of victory only in a very small number of maps. This means that the Wisconsin electoral map is a clear outlier and therefore is likely to have been gerrymandered. For mathematicians like Bernstein who worked on the algorithm, this statistical analysis is vitally important yet has been largely ignored in favour of the efficiency gap. “Fortunately,” says Bernstein, “the justices did notice, and they are the ones who matter.”
“If the court rules the Wisconsin map unconstitutional under a particular test,” says Joshua Douglas at the University of Kentucky, “then that will place an outer limit on the worst abuses in partisan gerrymandering.” This would have the biggest effects in swing states such as Wisconsin, North Carolina and Maryland. “The ruling would ultimately produce fairer maps, which also will likely give average Americans more confidence in the election process,” says Douglas.
Though the Supreme Court case has heard arguments from both sides, it will probably be months until it reaches a decision. Should it uphold the lower court’s decision, then the impact will be far-reaching. When states redraw their districts after the 2020 census, they may have to take account of the efficiency gap test, which many of them fail at present.
“The proposal allows for map-makers to break the efficiency gap requirement in exceptional cases, provided they offer sufficient justification,” says mathematician Dustin Mixon at Ohio State University. But they may just start taking the efficiency gap into their calculations to try to find another way to exploit the system, albeit in subtler ways. “This could in turn incentivise bizarrely shaped districts,” he says.
And while a rigorous mathematical test for gerrymandering will help level the playing field, and may even give Democrats an edge in the next few elections, it is unlikely to help the US overcome its partisan divides. Those with strong political views tend to vote with their feet, moving to live near those who hold the same opinions. Democrats cluster in cities, while Republicans dominate surrounding districts. At the end of the day, the problems pulling American society apart arise when people start gerrymandering themselves.
Splintered Isle
Reviews of the UK’s electoral map happen periodically, but what makes next year’s so significant is that the new map will do away with 50 constituencies, lowering the total to 600. Unlike the situation in the US (see main story), the map will be redrawn by independent boundary commissions rather than politicians. Their focus is meant to be exclusively on drawing up districts of equal size, and the guidelines prohibit them from considering the political implication of their choices.
Anyone can suggest improvements once the proposed map is published, but the commissions are under no obligation to accept them. “This is the nearest we have to gerrymandering in the UK,” says Ron Johnston at the University of Bristol.
When the proposed changes begin to emerge, there will no doubt be allegations of gerrymandering, with Labour, the largest opposition party, especially likely to complain. “Since World War II, the trend in the UK has been for people to move away from inner cities, where there are more Labour voters, and into suburbs, where there are more Conservative voters,” says Charles Pattie at the University of Sheffield. This means that Conservative constituencies tend to grow more populous, and so more votes are needed to elect a Conservative member of parliament. For Labour, the reverse tends to happen, meaning they get more bang for their buck.
Holding periodic boundary reviews readdresses this imbalance. “There is a strong case for saying that the only way in which parties can ‘guarantee’ a gerrymander in the UK is by not holding boundary reviews,” says Pattie.
But the fragile nature of the present minority Conservative government could lead to the latest review being mothballed. The Conservatives are reliant on the Northern Irish Democratic Unionist party to get their legislation through parliament, and the DUP are unlikely to back any vote that would reduce the number of representatives for Northern Ireland, as that could in turn weaken their influence on national politics.
Divide and rule
Imagine 36 voters, half supporting the pink party and the other half the green party. Keeping the districts the same size ensures each vote counts equally. Sometimes, simple divisions produce fair results
Geographical features such as rivers or coherent entities such as cities generally prevent such simple solutions. To protect the voting rights of minority groups, US law also allows the creation of districts where they form a majority
Divisions that take these realities into account can skew results in one party’s favour, as below, where the greens are a minority in most districts. If boundaries are deliberately shifted to favour one party then we have gerrymandering
How to spot a gerrymander
Tests to detect gerrymandering involve measuring the fairness of political boundaries. The efficiency gap, for instance, compares each party’s number of wasted votes. In this example, a party needs four votes to win a seat in each district, so all other votes in that area are wasted
Another way to check for gerrymandering is to use a geometric test such as isoperimetry or convexity. Essentially, these both work the same way: the odder the shape of an electoral unit, the greater the chance that it was intentionally drawn to favour one party
But no test is perfect. Geometric tests can’t distinguish between districts that are a funny shape because of restrictions imposed by roads, rivers or city boundaries and those designed for political gain. And the efficiency gap struggles to produce meaningful results when one party dominates.
This article appeared in print under the headline “Something is rotten in the states”
