system
Given the following post, respond as if you're commenting on Reddit:
human
What is that largest percent majority that could still loose an election due to gerrymandering?

I saw a meme that visually depicted how gerrymandering can cause a 60% majority to loose an election to a 40% minority, which made me wonder to what extent gerrymandering could theoretically change the outcome of an election.
gpt
First, we have to specify a few assumptions:

- A population is separated into districts and each district elects 1 representative.

- Each district has the same number of inhabitants.

- There are only 2 candidates per district, one from each party.

- The candidate that obtains more than 50% of the votes wins.

Some of these assumptions align with the US electoral system, others are idealized situations that we use for the purpose of this thought experiment. Lets say that we're looking at parties A and B and that we are trying to optimize the outcome of party A.

The question is: What is the lowest percentage of support among the population as a whole that could still win a majority of the districts, given the right districting scheme?

In order to achieve the maximum effect of gerrymandering, the goal is to create districts in such a way that supporters of party B are packed into as few districts as possible, with B-majority districts ideally having 100% party B support. Party A supporters are spread across districts in such a way that districts have a 50% + 1 voter A-majority.

Using this system, you can fairly easily see the smallest overall rate of support for A that still results in a 50/50 outcome: B-majority districts have 0 A-supporters, whereas A-majority districts have 50% + 1 A-supporters and 50% - 1 B-supporters. If the number of people per district is large enough, the +1 and -1  becomes negligible and given an equal amount of A-majority and B-majority districts that means that 75% (minus a negligible amount) of the total population supports B and 25% (plus a negligible amount) supports A while still resulting in a 50/50 outcome.

In order to achieve an actual victory in number of districts for party A, they'll need to win 1 more district than the 50/50 outcome and so the number of voters they need will slightly exceed the bottom value of 25%, with the amount depending on how many districts there are (the more districts, the smaller the deviation).

So in the limit case with many districts that have a large population, a 75% majority in popular vote could still be insufficient to win a majority of the districts given the right districting scheme.