Gerrymandering Explained: How Math Is Used For Political Gain To Win Elections

In 1812, Governor Elbridge Gerry of Massachusetts signed a bill creating a new voting district in Boston, the shape of which was so Byzantine, so strange and contorted in shape, that journalists likened it at the time to a slithery salamander. In honor of its creator, the new district was termed the “Gerry-mander” – and a political strategy was born.

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In the US today, however, gerrymandering isn’t just a niche way to tip the balance of a single local election. Entire states can be won or lost through the practice; it’s important enough to reach the Supreme Court over and over again, and it can send lawmakers into exile or house arrest in their efforts to prevent it.

And behind it all, math and computer science – endlessly churning out ever-more complex analysis and algorithms to beat or bolster new attempts at gerrymandering.

“What used to be a dark art is now a dark science,” Michael Li, a redistricting expert at the Brennan Center for Justice, told The Guardian in 2021. “Before, you weren’t sure about the data, but now you’re much more certain so you’re able to draw things in ways that can be more aggressive.”

How to win an election (even if you don’t have the votes)

It’s been more than two centuries since that first gerrymander – a term which, incidentally, you’ve been mispronouncing: Governor Gerry pronounced his name with a hard ‘g’, so it should really sound more like “gary-mander” than “jerry-mander” – and throughout that time, it’s been enthusiastically adopted by the ruling party and lambasted by the opposition no matter who those titles belonged to.

The result is a political map mottled with sometimes notoriously weird and squiggly districts. Think Maryland’s 3rd district pre-2019 – a splat on the land so famous it had its own nickname, “the praying mantis” – or Texas’s 35th, which links the Black and Hispanic majority parts of San Antonio with south and east Austin via about 80 miles of interstate highway.

But where do these weird shapes come from? With modern technology and techniques, there are plenty of methods to create them – but at base, it all comes down to "packing and cracking". 

“You pack your opponents into very few districts, so that they win those districts with an overwhelming majority,” explains Ellen Veomett, Associate Professor of Computer Science at the University of San Francisco and an expert in the math of gerrymandering, “and then the remaining opponents, you crack among the many, many, many, many other districts.”

So, suppose you have 100 voters in an area to be split into 10 districts, and you know 60 percent favor the green party and 40 percent favor purple.

Now, you could split them like this:

And get a pretty representative vote. But by drawing the boundaries slightly differently, you could also end up with this result:

And get a 10-0 result for the green party. If you’re really smart, though, you could do this:

It’s messy, sure – but it’s pretty great if you’re a purple supporter, because by packing half the green voters into their own districts and cracking the rest of them among districts in which purples just slightly outnumber them, you end up with purple winning 6-4. That’s right: you’ve literally reversed the original political bias of the population, just by careful redistricting.

Now, obviously this is a very simple example. “The extent to which packing and cracking can occur of course depends on where people live in the state,” Veomett points out, “and if they're clustered, like into giant clusters.” It’s practically impossible in Massachusetts for that reason, she explains: “Pretty much everywhere around the state, it's majority Democrat,” she tells IFLScience. “It’s very homogenized.” 

But when it can be applied, you might be surprised at just how powerful it can be.

“If the distribution of voters within the state is set up correctly, [and] your party holds more than 50 percent of the vote share, then you can win all the districts,” Veomett explains. “If your party has less than 50 percent – let’s say you have 40 percent of the vote share – you could, with 40 percent of the votes, get 80 percent of the seats.”

“So it's double the percentage,” she says. “That's the extreme to which it's possible.”

Spotting the con

Packing and cracking is effective, but it does have one drawback: it can be a little obvious. Search “most gerrymandered districts” online and Google abounds with weird squiggly geographical regions that nobody would assume had been drawn with pure intentions.

However, “With modern technology, you can gerrymander pretty effectively without making your shapes very weird,” Beth Malmskog, an associate professor of math and computer science at Colorado College, told Quanta Magazine back in 2023. Metrics such as the perimeter length – as a general rule, the stranger the shape of a district, the longer its perimeter will be – or the “compactness” of a district – a nebulous term, but you can basically think of it as how close the shape is to a circle – aren’t necessarily as useful as they once were in spotting a gerrymandered region. So what can we use instead?

For some, the answer here is “the efficiency gap”. First proposed in 2014 by polisci big hitters Nicholas Stephanopoulos and Eric McGhee, this figure tells us which party has more “wasted” votes – and, in doing so, kind of shows how far a given district setup sits from a “perfect” representation. 

Take the three green-purple voters maps we created earlier, for example. In the first, the purple party has 16 “wasted” votes – four in each of the four districts they won unanimously (remember, only six votes are needed to win, so the last four are unnecessary). The green party, meanwhile, has 24 “wasted”, for the same reason – making 24 – 16 = 8 “net” wasted, and an efficiency gap of 8/100 = 8 percent in favor of the greens.

In the second setup, the greens are in a much better position: they have zero wasted votes, with exactly enough in each district to win. The purples, meanwhile, have only wasted votes – all 40 of them were losers. The efficiency gap, therefore, is (40 – 0)/100 = 40 percent in favor of the greens – way further away from perfect representation.

And what about the final diagram, where we reversed the proportions of voters and representatives? In that one, the purples have four wasted votes, while the greens have a whopping 54. That’s an efficiency gap of a full 50 percent – the least representative of the lot.

On the face of it, then, this seems like a reasonable measure. But according to Veomett, it comes up short: “the efficiency gap is a very simple computation for a very complicated question,” she tells IFLScience. “It's just a calculation based on the outcome.” 

Other such shortcut metrics for spotting a gerrymandered district are similarly lacking. “Some of my research, which just came out, is about the mean-median difference and the partisan bias, which are two other metrics just like the efficiency gap,” Veomett says. “These other two metrics are actually older and have been used longer, and they’re also very flawed, and not particularly useful.”

“A single metric doesn't really tell you that much,” she adds. “Whether it be the efficiency gap, mean-median difference, partisan bias, declination – it’s too little information to really be particularly useful in telling you whether a map is gerrymandered.” 

Modern maps

If modern technology has made it easier to hide gerrymandering, then at least it has also provided new ways to uncover it. With computers, we can perform analyses of proposed districts on a scale no human could carry out, and we do: “Most of the researchers that I hang out with and talk to focus much more on the ensemble analysis method,” Veomett tells IFLScience. “[Though] the way the ensemble is created really matters.”

The idea is pretty simple – albeit not easy. Basically, you program a computer to make potential district maps of a region, taking into account all the relevant rules around size, shape, population, and so on; you let it create tens of thousands of possible results, and you compare those to the map you think may be gerrymandered. If what’s being offered is atypical compared to these thousands of legal maps your program found, then chances are it’s not legit.

Of course, discovering whether a district is unfair is only half the problem. When it comes to using math to draw better districts in the first place – well, it’s not as easy as you might think. Not least because the jury’s still out on what “fair” should actually look like.

“There's so many different ways to interpret what gerrymandering is, and what fair is,” Veomett tells IFLScience. “Some people would say the ideal is ‘proportional’. Some researchers would say, ‘well, what's fair is being sort of typical based on the political geography of the state.’” 

“I don't think, if you asked 10 people, even those that do research on this, you would get the same answer among two of them,” she says. “It’s a good question. It’s not naïve.”