How far can gerrymandering go?

Question

Gerrymandering can keep a relatively unpopular party in power for longer than it otherwise would be. Assuming the unpopular party becomes even more unpopular, just how unpopular would they need to be before the stratagem of gerrymandering itself is insufficient to support them?

For example, in 2016 gerrymandering enabled Republican candidates who received 50% of the popular vote to win 9 out of 12 available Congressional seats in North Carolina. That is, half the votes, but 3/4 of the seats and power. And suppose that as long as they hold 3/4 of the power, they'll continue to gerrymander.

Now suppose in the future, the Republicans become more unpopular, and receive only 25% or 10% or even 1% of the North Carolina vote, can gerrymandering alone sustain their hold on the majority of seats, or is there some necessary numeric limit that would eventually be reached?

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Note: for the purposes of this question, please ignore other concomitant stratagems by which unpopular parties might squeak by, and also suppose there are only two parties contending. For US voting systems Reynolds v. Sims, 377 U.S. 533 (1964) et al would apply, so that the gerrymandered districts must have approximately equal populations.

Answer 1

This is a rather simple mathematical exercise. If you allow me total freedom to draw districts within the current requirements, I can place anyone in any district I want provided they have equal population in the end. Connectedness in two dimensions is not enough of a barrier to stop this. In that case, my best bet is to fill as many districts as possible with 50%+1 of my supporters, and fill in the rest with my opponents. Since at least half the population of each district I win must consist of my supporters, we can easily see there is an upper limit on how many districts I can win:

A party with x% of the vote cannot win more than 2x% of the districts no matter how they are drawn.

In practice there's no way I can reach this limit--no court in the land would allow me to draw the kind of tortured districts needed for arbitrary assignment. 50%+1 is also nowhere close to safe--I'd need a small buffer in my districts so that changing demographics and opinions don't turn it into a gerrymander against me. I'd guess that overall this limits one's practical chances to 1.5x%, but this just an estimate, not a mathematical truth.

Answer 2

The problem here is that "one person one vote" can be based on residents rather than citizens or voters. So if you could assign people purely arbitrarily, you could fill up your districts with non-voters such that only a small number of voters are required. In theory, one voter and eight hundred thousand non-voters per district would work. Pack all the other side's actual voters into one district (no non-voters and none of your voters).

However, you then run into a few problems.

1. It's not the number of federal House seats that determines your ability to gerrymander. It's the number of state legislative seats. So you need more than just twelve federal seats. You need a larger number of state seats.
2. You can't gerrymander statewide races. So to make this work, you have to disempower the governor's ability to veto a districting plan.
3. Most states don't have enough non-voters to make this work.
4. You can't actually assign people arbitrarily per district.

Realistically, this would never work. But it is theoretically possible with just a few hundred voters given sufficient non-voters.

In actuality, Republicans in North Carolina received slightly more votes than Democrats. But instead of winning seven to six, they won ten to three in North Carolina. But this didn't overcome the Democratic-favoring gerrymander in California, where Republicans won about 33% of the vote but only 13% of the seats. That's a surplus of eleven for the Democrats. Even netting out the three for the Republicans from North Carolina, that's still an advantage of eight seats for the Democrats from those two states.

Apparently in the last House election, the gerrymanders almost exactly cancelled out. The Democrats won about as many House seats (54%) as their share of the vote (53.4%). Which is surprising because Democrats face a natural handicap in that their voters are more concentrated than Republican voters.

Source: Wikipedia from Johnson, Cheryl L. (February 28, 2019). "Statistics of the Congressional Election of November 6, 2018". Clerk of the U.S. House of Representatives.

Answer 3

One of the problems is that people look at Gerrymandering from the lens of Republicans vs Democrats. It's not - it's a terrible example of the two parties working together.

Imagine you belong to Team Yellow, and you've been elected - but it was kinda close. There's a heavy Pro-Purple area along the northern edge of your district that nearly cost you your election. Meanwhile, Team Purple elected in a guy who has a section of Pro-Yellow people that was giving him grief. So when it comes to redistricting, what do you do? You 'trade' your pro-purple area for his pro-yellow area - entrenching both of you into your spots. It's not Yellow-Vs-Purple at all - it's about both reps working to make their reelection easier.

This helps highlight why the premise of the question is wrong. While the Republican party is probably happy with the current state of things, the Democrat representatives are probably much happier than their Republican counterparts. Because they don't have to worry about reelection. The Republican reps are likely thinking about doing the opposite as what the question would imply. If they could do a redistrict right now, they'd probably "sacrifice" a district in order to more solidly claim the other 8 - several of which they came close to losing (better to cleanly have 8 than gamble for 9.) Which the Democrats reps would probably frown at - it'd make their reelection efforts harder. The "party" doesn't mean much if you're thrown out of office, after all - no matter which party you're part of.

Answer 4

In theory, if the stars align perfectly, one can gerrymander a 50-50 vote to win 11 out of 12 districts. This can be done by drawing the district boundaries so that one district has 61% of voters supporting your opponents, and the other 11 have only 49% supporting your opponents.

In practice, it will come down to how tightly packed, geographically, your opponents are. If they are evenly spread, then it becomes much more difficult than if they are densely packed in a few geographic areas. Thanks to a combination of voter registrations (where the state knows voter affiliations in each house), and cheaper computing power allowing for more complex modelling, a willing party can find a way to optimise the boundaries for their benefit.

Answer 5

Actually, the key to successful gerrymandering is not spreading out the votes of the controlling party but concentrating the votes of the minority.

Say that voters have equal inclination to vote for each party, (that is the electorate is split 50-50 but with guaranteed loyalty to their party), but that one party completely controls redistricting. In that case with perfect knowledge of the electorate it would be possible for the majority to win all but one seat. The majority makes it so the minority wins that one seat with a 100% margin while all others go to the 'majority' party with much narrower margins.

Of course reality does not feature such perfect knowledge but it should give a basis for thinking about whatever terms you choose for modeling reality.

Answer 6

In gerrymandering we have 2 divisions to manipulate; blue vs red voters, and voters vs non-voters.

10% of the vote, 83% of the seats

Consider 12 seats, a 29% turnout and a 90% blue (10% red) vote. If we pack the (26.1%) blue voters into 3 seats, reds win 9/12 seats. However, what if we pack 2 blue seats? That would absorb 16.7% out of the 26.1% of blue voters, leaving 9.43% blue voters vs 10% red voters. Distributing the remainder evenly would yield a win in all the remaining seats, for a total of 10/12.

This strategy clearly depends on the geographical distribution of voters vs non-voters, and red vs blue. In a world where they were all evenly distributed it would be impossible to gerrymander an unfair victory. But the more uneven the distribution, the greater the opportunity. So enacting policies which simply increase the clustering of similar groups (say, localised taxes which can align with whichever is the ruling local group, or tax breaks for family groups) increases the degree to which this approach can work.

Another limitation would be the safety margin you wish to retain - in the above example, 10 out of 12 seats are won on a margin of 1.5% (51.5/48.5). If the weather makes a difference of 2% this would be a risky strategy.

It is clear, however, that gerrymandering can have surprisingly large effects as long as there is an uneven distribution and low turnout. Raising turnout 1% (without changing the ratio of red/blue) would yield an additional seat for blue.

Can 1% win a majority? Yes

Let's change our scenario to 11 seats to make the majority apparent, we'll make it 1% of votes red and 99% blue, leaving only the turnout as a variable, say T. The requirement is to win 6/11 seats which means the maximum blue vote is 55% packed into the 5 seats lost, plus T * 1% in the remaining seats (which we will match with the T * 1% blue). So (0.55 + 0.01T)/(0.01T) = 99 -> 0.55 + 0.01T = 0.99T, T = 0.561. So turnout of 56.1% or less can permit a red vote of 1% to win a majority over 99% (in theory).

Answer 7

While it's not in the USA, there was a period where some parliamentary seats in the UK were voted on by 7 voters. And about 1% of the population voted on 37% of the seats (152/406).

https://en.wikipedia.org/wiki/Rotten_and_pocket_boroughs

Summarizing the information in the link, towns in the UK had a set number of seats assigned to them. If the number of people in the town shrank (closed industries, the town literally falling into the sea, and so on), then the few remaining people in the town effectively controlled the seat. In some cases, one person controlled the seats due to being the landlord of over half the houses in the town, and the ballot not being secret...

The gerrymandering here is not correcting the district boundaries when the population changes. In theory, you could gerrymander by altering existing boundaries so that almost all districts contain one voter, with the rest of the country being put together in another district. This means that in the extreme case you could have two districts with one voter each, and the rest of the population in the other district. Those two voters would control which party/faction got into power.

Answer 8

100% turnout, Identical sized districts, Infinite population = 25% minimum.

E.g. 1200 voters in 12 districts:

• 49 blue votes, 51 red votes in 7 districts
• 100 blue votes in 5 districts
• Total 843 blue voters, 357 red votes. (29.75% to stay in power)

1.2 million voters in 1200 districts:

• 499 blue votes, 501 red votes, in 601 districts
• 1000 blue votes in 599 districts
• total 898899 blue votes. 301101 red votes (25.09% to stay in power)

As you increase the population size, you can get an asymptote to 25%

100% turnout, district sizes allowed to vary by up to a factor of 2. = 1/6 minimum

If you divide a state into 1 division per N people, and then round to the nearest integer, the division sizes will vary by a factor of up to 2. (Montana has 994k people per district, Rhode Island 1st has 526k.)

Eg 1200 districts:

• 249 blue voters, 251 red voters, in 601 districts of 500 people.
• 1000 blue votes in 599 districts of 1000 people.
• 150851 red voters, 748649 blue. Red wins with 16.77%

Increasing the number of votes or districts to infinity and the ratio will go to 1/6

<100% Turnout - no lower bound (a single vote can win)

Uniform turnout won't affect the answer (if every voter has a 50% chance of not voting - then the vote counts half evenly, same result). So you need to weight the turnout.

Turnout at extremes can vary from 0% to 100%, which can effect the vote entirely - 1 vote can win an election if the other sides turnout is 0%

A 1% turnout for blue, and 100% turnout for red:

• 249 blue votes from 24900 supporters, 251 red voters from 251 supporters, in 601 districts of 25151 people.
• 503 blue votes from 50302 supporters, in 599 districts
• Total 45095798 blue supports, 150851 red supporters. Red wins with 0.33% of the vote.

Increasing the number of votes or districts to infinity and the ratio will go to 1/300 (with a 100:1 turnout discrepancy)

Uneven district sizes - no lower bound (2 votes can win).

With 100% turnout, and 3 districts:

• 1 red vote and 0 blue votes in 2 districts
• infinite blue votes in 1 district
• Red wins 2 to 1 with ~0% of the vote.