# Do more legislative seats make Gerrymandering harder?

In a system of single member districts, elected for an entire legislative body all at once, is it easier to gerrymander the map to favor one party over another in a two party system, with more districts, or with fewer districts, or does it not matter?

For example, in Virginia, there are 100 seats in the state house, and 40 seats in the state senate, with all seats in both houses filled by the 2023 general election. Is it easier to gerrymander the state house, or the state senate, or is there no difference, if one party is in full control of the process?

Definition

• A legislative body is "easier to gerrymander" if it is possible to cause more seats in the legislative map for the body to favor the party drawing the map, taking the geographical distribution of political preferences and turnout likelihood in the territory served by the legislative body constant.

Assumptions

• Assume that the districts must be contiguous and have effectively equal populations, but that there are no other regulations for map drawing which is done by a single political party that controls all levers of power.

• Assume that a legislative body can have no less than five and no more than 650 members.

• Assume that the population of the overall territory served by the legislative body is at least 6.5 million.

• Assume that politically preferences are inhomogeneous to an extent within the range of inhomogeneity found in larger U.S. states.

• Assume that expected turnout is not necessarily uniform from one district to another with variation in turnout rates within the range found in larger U.S. states and is fairly strongly correlated with political preferences.

• Assume that there are only two political parties, that each district has one candidate from each party, and that there are no independent or write-in candidates. These assumptions, among other things, eliminate any differences between systems that require a plurality to win, systems that require a majority to win, and instant runoff voting, for example.

Comment

This answer should have a mathematically determinable answer, or set of answers that involves varying factors like turnout and inhomogeneity of political opinion with geography.

But, I've never seen the answer to this question clearly stated, and I don't have a good intuition one way or the other.

In a completely homogeneous distribution of political preferences, all maps have the same outcome (100% of the seats will favor the party with a larger share of the total).

In general, single member districts tend to produce legislative majorities that are greater than the percentage of voters that favor the party that gets the most legislative seats. Single member districts tend to amplify differences in popular vote support relative to proportional representation. The most common example of this in the real world is the electoral college.

I'm aware that this question glosses over (perhaps inappropriately) the choices that a map maker in full control has between making their districts safer in the event of a shift in political opinion after the map is drawn, and winning more districts that may be more competitive.

• This question is essentially a variation of the Knapsack Problem, and you'd probably do better to ask over at a Maths or Combinatorics stack. I'll look at it for a bit, but my math skills are rusty, so I can't promise anything. Commented Jun 21, 2023 at 21:20
• It is not a simple unconstrained mathematical problem because the answer depends on the distribution of voters. If voters are entirely evenly distributed, then gerrymandering is impossible as there is no way to selectively divide the population by their politics. Only when they clump geographically can gerrymandering work and the specific pattern of clumping determines how far it can go in creating bias. Commented Jun 22, 2023 at 10:13
• @NoDataDumpNoContribution Not very practical. Imagine all the people living in houses with even numbers vote republican and those in odd-numbered houses vote democrat: the districts would have to zig zag between neighbouring houses. Commented Jun 22, 2023 at 12:28
• @matt_black I think that you have hit the nail on the head. The (very thoughtful and well considered and analyzed) answers so far don't really grapple with the fact that the primary way that the number of seats could impact the extent to which gerrymandering is available is the way that this interacts with the kind of geographic clustering that is present in the underlying geographic distribution of voters. Heuristically, it seems like a key factor is typical homogeneous cluster size v. legislative district size. If districts are smaller than clusters it's hard to split and crack very much. Commented Jun 22, 2023 at 17:08
• It would be nice to see an answer which considers the requirement that districts must be contiguous regions. Additionally, an important part of how "easy" it is to gerrymander, is how likely it is that a gerrymandered map will be upheld by a court. It could be the case that gerrymandering can achieve the same level of bias for small and large numbers of districts, but perhaps the district boundaries required to achieve that level of bias would need to be much squigglier when there are more or fewer districts, and maps with such complex geometry are more likely to be thrown out when challenged. Commented Jun 22, 2023 at 18:22

A map is a function assigning voters to districts such that each district has equally many voters. A seat goes to the majority candidate in the district.

The map with the highest number of seats for the Party combines a slim majority for the Party candidate in as many districts as possible, with 100% majority for the Opposition candidate in as few districts as possible. Therefore, we have:

``````Party share of seats * 50% < Party share of the vote
``````

and since seats are an integer:

``````Party share of seats = floor(2 * Party share of the vote * districts) / districts
``````

For instance, a party with 40% of the vote can get 80% of the districts, rounded down.

As you can see, the only effect of having more districts is that rounding down is less relevant. More districts therefore confers a slight advantage to the gerrymandering side (or enables gerrymandering in the first place, if moving from one to more than one district).

(we are assuming that there remain many voters in each district. If you make districts so small that they contain few voters, rounding issues with the share of the vote can make gerrymandering less effective, and wholly eliminate gerrymandering if each voter gets their own district. However, since the purpose of elections is choosing few representatives for many voters, that remedy is not available in practice)

That is, more districts is not an effective remedy against gerrymandering.

• I would say on the scales where gerrymandering typically is an issue, more districts make it slightly worse. So not only is it not a remedy, the tendency is to make the issue worse. One could argue that less districts is an effective remedy to gerrymandering and would then have to figure out whether the effect is big enough to be noticeable. Commented Jun 22, 2023 at 6:42
• The formula looks wrong. Should it be `seats = floor(2 * share * districts)` instead of what you have now? Commented Jun 22, 2023 at 7:43
• @meriton, `Party share of the vote / districts` fails to balance the math units as that is `votes/area / area`. `votes/area * area` makes more sense to returns a vote value. Commented Jun 22, 2023 at 19:16
• Oh, of course. Guess that's what I get for doing math before waking up entirely :-) Fixed. Commented Jun 22, 2023 at 20:44
• Note that smaller districts are more difficult to model. If I want to know the voting intentions of a region, the number of samples (size of the poll) I need to do isn't really a function of the district size (except in the limit of small districts). If there is 1 district, I can do a 1000 person poll and know the entire place +/- 3%ish. If there are 100 districts, to get +/-3% I need to poll 100,000 people (1000 per district).
– Yakk
Commented Jun 23, 2023 at 20:04

Suppose you are drawing d districts in a population of a voters for party A and b voters for party B.

Suppose that A is the majority party and we are trying to gerrymander for the benefit of party B.

The optimal strategy is to draw districts that have a 100% composition of party A voters, until the population remaining outside of the districts we have drawn is majority party B, and then draw the remaining districts "fairly" according to the composition of voters.

For example, let's say we have 61 party A voters and 39 party B voters, and we are drawing 4 districts. The optimal strategy is to make one district with 25 party A voters, and three districts each with 12 party A voters and 13 party B voters.

Now let's examine what happens as we vary the number of districts.

The graph below examines a situation with 60% party A voters and 40% party B voters. The X axis represents the number of districts. The gray line represents the "fair" percentage of districts that go to party B, and the yellow line represents the optimal gerrymandered percentage of districts that go to party B.

As you can see, increasing the number of districts doesn't really make gerrymandering harder or easier, it just decreases the average amount of rounding error away from the "ideal" 80% of districts.

Edit: As requested in the comments, here's my function for finding out the number of districts that can go to party B:

``````function optimize(a,b,d) {
const districtSize = (a + b) / d;
while(a > b) {
a -= districtSize;
}
}
``````
• @JoeW The gray line is just `floor(d * a / b) / d`. The yellow line was calculated using a program that follows the algorithm I described in the third paragraph. (It's probably not too hard to make a formula but I was too lazy to work out the math.) Commented Jun 22, 2023 at 15:29
• It is still good to put in your answer so that people know how the graph is calculated. Commented Jun 22, 2023 at 15:45
• @JoeW OK, see edit Commented Jun 22, 2023 at 15:47
• @PeterOlson - That would have been based on the 2010 redistricting, carried out in 2011 when Republicans held the govnernor's mansion and state senate majority in Wisconsin. So no, that was the after-affects of a redistricting in favor of what was, at the time it was carried out, the majority party. Commented Jun 22, 2023 at 18:41
• @T.E.D. Did I ever say Party A is trying to gerrymander for the benefit of party B? Presumably it's Party B that is trying to gerrymander for the benefit of itself. Commented Jun 22, 2023 at 19:24

On it's whole, the number of districts in a state doesn't matter as the two methods for Gerrymandering (cracking and packing) will still allow for this to happen. In Cracking Gerrymandering techniques, the idea is to draw your map lines so a large amount of the communities are split into two districts such that the communities supporting the opposition party are outnumbered by the size of the population that supports the party in their new districts, even though those communities have different needs and priorities than the ones that are separated at the border.

Packing is the exact opposite and instead, draws the lines so that most of the opposition party is packed into one district and the dominate party is given district bounds that will make it so the dominant party overwehlm's the remaining centers.

More districts will not change this, as the game isn't to draw the lines to totally eliminate the opposition party from winning... but to maximize the number of likely districts to vote for the dominant party and minimize the districts reliablely opposed.

For example, Maryland is one of the worst gerrymandered states in the union (and the one I'm most familiar with). By the numbers, 2/3rds of registered voters are Democrats and 1/3rd are Republicans and has 8 representative seats in the house. If those seats were evenly divided, this would mean Dems would get 5-6 seats and Republicans would get 2-3 seats (depending on which party is doing well in a given year). However, in practice it's almost always a 7-1 split between Dems and Republicans due to how the district lines are drawn. The lines are drawn not to eliminate Republican votes, but to make sure that the they are not represented as evenly.

As such, if we doubled the districts in Maryland, it wouldn't address the problem, as the problem is about drawling the lines to disfavor the minority and empower the majority. It can still be done.

• If the voters in Maryland are positioned randomly (no clumping) and the districts are also random (zero gerrymandering), a 2/3 Democratic population would expect 100% of districts being Democratic.
– Yakk
Commented Jun 23, 2023 at 20:06
• The new map looks un-gerrymandered, and is 6/1/1 (1 toss-up) instead of 7/0/1. My point is that district based voting does not, and should not, be expected to model PR.
– Yakk
Commented Jun 23, 2023 at 20:25
• @Yakk: They are not. Republicans are much stronger in the rural areas Bordering PA and the Eastern Shore than they are in the Central Maryland in the Baltimore-Washington Corridor which has the bulk of the Democrat supporters (and is the most densely populated part of the state). Commented Jun 26, 2023 at 12:06
• I don't dispute your conclusion, I dispute your logic. Your "If those seats were evenly divided" argument provides almost no support for the region being gerrymandered or not. Yet, you structured your post as if it was supporting the gerrymandering argument. I mean, I can look at the old map and see the obvious gerrymander; but I can't tell that from the state-wide support and the election votes. I look at the new map, and anyone who managed to gerrymander with that map is an artist.
– Yakk
Commented Jun 26, 2023 at 14:30
• @Yakk: Fair point. I'm quite familiar with the politics of the state, and can tell which districts will probably break for which party based on knowledge of the region. MD also is a good state to study because it's small in area, middle road in population and has a good mix of Urban/Rural divide. it's easy to see the "all politics is local" in a drill down of the state. Commented Jun 26, 2023 at 16:59

The goal of gerrymandering is to distribute your votes to win the most districts while rendering the votes of the opposing party meaningless.

The easiest way to do that is to win a narrow race. In that case you need 50+x% of the votes to win which means the 50-x% of the opposing party were cast without having any influence.

Now x could be as small as 1 vote or could be as large as 5% or more depending on the uncertainty of the result.

So the idealistic calculation would be:

``````votes required to win a seat = ((population/seats)/2)+x

``````

So if you are the party that is leading with regards to the total number of votes the relevant calculation comes down to

``````excess votes - (x * number of seats)
``````

if that is above 0 you'll be able to win all the seats. And while that scales with the number of seats, there's another mechanism to invalidate the votes of the opponent and that is over-winning. Like if you know that you'll mathematically lose at least one district then you can also lose it 100:0 so that lots of votes are wasted there and increasing the difference in other districts.

Same for when you have fewer total votes than the opponent you can make them waste that excess in a few districts and win the rest by a small margin.

Now the number of districts is a double edged sword. If the margin to win X is small with respect to the number of excess votes, it won't have any significance. And the one leading in votes AND doing the gerrymandering will win all the seats. If the margin is small it might increase the chance of winning a low number of seats but under ideal conditions that is about 1-2 seats which in their significance need to be divided by the total number of seats. As 99 seats going to party A and 1 seat going to party B with the same voting right are still a de facto 100% for party A.

If you have less than 50% and want to secure a win by gerrymandering you're making even more excessive use of that losing by high margins. So a number of seats will be lost anyway but you're able to score a disproportionate number of seats in the rest of the races.

Again the outcome depends on how many seats you have to lose by a margin in order to secure an excess vote count to win the rest. If x is 1 vote you might turn in 99% v 1% into a 1% v 99% with more seats meaning a larger lead. While a bigger X might mean the distribution might stay closer to the original.

That would be the calculation of the ideal case, the real cases are likely more difficult and require more information on the particular situation.

Gerrymandering is an information problem.

The requirement that districts be contiguous is only a modest barrier. If you know exactly what how ever person is going to vote, you could Gerrymander a 65% opposition state into majority for the 35%.

Suppose you have only 10 districts. You make 4 districts each with near 100% support for the 65% (say 90%), using up 36% vs 4% of the voters.

29% opposition and 31% friendly remain -- which you can distribute carefully over 6 districts.

The hard part is knowing exactly how every person will vote with perfect confidence. And having that level of knowledge is harder when there are more districts.

A random sample of 1000 people in a district is enough to know A/B questions about the members of that district with about a 3% error, 19 times out of 20. This doesn't scale with the population of the district much, unless the district is so small that 1000 is almost as big.

With exceedingly huge districts you run into an inability to rearrange districts, but as you can see even with only 10 it isn't impossible to arrange a Gerrymander'd coup.

Given enough information and a free hand to draw a map, you need more than a 2:1 majority to prevent Gerrymandering from flipping the map. As the restrictions on the map get higher and the information more expensive, this ratio falls.

In the limit, you could imagine allocating individual lots to one or the other district. But you don't usually have to do that.

Do more legislative seats make Gerrymandering harder?

That depends on how you increase the seats. As other answers say, simply adding more districts may/will not reduce the ease of gerrymandering, but there are other ways to increase the amount of seats in a branch of government.

In a Netflix series, "Explained", there's an episode about voting, called "Whose Vote Counts, Explained". This episode goes through a bunch of things with how votes are counted, who can vote, who is typically suppressed, etc., but the episode also talks about gerrymandering.

I'm not going to be able to explain it as well as they did, but I'll give you the basics.

If we increase the districts as well as increase the representation from those districts plus we use ranked choice voting, then we have less of a chance gerrymandering works.

Every district in the US has only one representative for House of Representatives and each state has two Senators. But if we increase the House districts from 1 rep to 3 or 4, then we can have more of a chance for a mix of representation for that district. And if we increase the Senators from 2 to 8, also get more of a mix. Adding in the ranked choice voting system, we also have less of a chance for a two-party system automatically dismissing anyone that doesn't fit their "party values". This is because we now have other 3rd, 4th, etc. parties that will be voted for, instead of them losing votes with people assuming they are "wasting" votes on anyone other than the main two parties.

So, now with the changes implemented, instead of the whole district being represented by 1 Republican or 1 Democrat, now we might have 2 Republicans and two Democrats, or 1 Republican and 3 Democrats, or 1 each of the Green, Independent, Republican, and Democrat parties, or one of many other combinations. Maybe we still end up with 4 of a single party representing the district, but that's still likely to be more representative of the district than the single representative we have now.

According to Google, the US has a population of roughly 331.9 million people. There are 100 seats in the US Senate and 435 seats in the House of Representatives. This means each Representative represents around 763,000 people. If we increased the amount of House districts by 4 and increased the number of Representatives in each district by 4, we can make it so each Representative represents (roughly) 48,000 people. Since we would be going away from a strict two-party system, not only would it be harder for a single party to get a majority of 6960 Representatives, but it would also be more representative of the people who actually voted. And, since there would be less incentive to vote for a non-representative two-party system, we'd likely see greater voter turnout.

The same goes for increasing the amount of Senators in each state from 2 to 8, although the amount of people represented by each Senator depends on how many people are in their state.

And, as an aside, if we add in Puerto Rico, Washington D.C, and any other US protectorates that want to become states, we make it even harder for a single party to gerrymander themselves into a majority role. This also changes the role of the electoral college, but that's a different kind of gerrymandering.

Getting more people to vote and for more than just a two-party system dramatically changes the idea of gerrymandering. If you don't have the ability to make a 50% chance guess at who people are going to vote for, but rather a less than 25% chance, then gerrymandering makes considerably less sense and is that much less effective, as well as being that much harder to even attempt.

So, yes, adding more legislative seats can make gerrymandering harder, but it has to be done a certain way, along with other changes, to have any sort of real effect.