Gerrymandering and its discontents

Flatland has recently decided to elect a representative congress. Each district will get to send one local representative to this congress, in order to better act on the needs of the citizens. This is all well and good, but the citizens are having some trouble deciding where the lines between districts should be drawn.

Eventually they decide to let current president, Triangle, decide how to divide up the districts. He does so, and divides up the land as follows (see Figure 1). It all looks nice and fair, with evenly sized districts not dominated by any one political party, but the citizens begin to get suspicious when the election results come in, and the congress is made up of mostly triangle representatives.

Now, Triangle has the majority of supporters in these districts, but a proportional representation would also have at least one square representative. Here, Triangle used a technique called “cracking.” He spread out Square’s supporters, making sure no one district had enough votes to win, and thus ensuring all of Square’s votes were wasted.

Square makes a big deal out of the fact that he didn’t get any representation in those districts, despite having a significant number of supporters. The citizens agree and let Square lay out the district lines for the next election, which he does like so (see Figure 2).

Now, this time the election results come in, and square gets two seats! The citizens aren’t happy at all with this result either, since Square is a minority party. Square’s supporters shouldn’t be taking the majority of seats in those districts!

Square could win a majority, though, by using a technique called “packing.” Square packed most of Triangle’s supporters into a single district, spreading the remaining voters out so thin that they couldn’t win in any other districts.

The same group of voters can thus give either political party a majority of representation, depending on where the district lines are drawn. This technique of manipulating voting lines to determine outcomes of elections is known as gerrymandering. It’s particularly prevalent in first-past-the-post voting systems (which the US employs), as there is only one representative from each district.

The citizens of Flatland quickly realize that whomever they put in charge, there will be bias in dividing up the districts, but they’re sure that a solution must exist, so all the citizens go home and sleep on the problem. The next day, Hexagon excitedly announces that she has a solution!

Hexagon’s solution is called the shortest line algorithm. It works by dividing up any plot of land mathematically, in order to ensure that no single shape has power of the process, and it’s easy to see why divisions are where they are. The algorithm works by drawing the shortest possible line across the district that divides it in half.

Now the two halves are divided in half. This process continues until the entire land has been divided into the proper number of districts.

Flatland attempts to divide the country this way for the next election, and it proves to be incredibly successful, with a representative congress! However, in the next election, Square ends up with a huge majority in congress even though the distribution of voters has not changed much.

All of Flatland is puzzled. It seems like gerrymandering, but how could there be bias when the districts were mathematically selected? Citizens investigate, and find the voting distribution looked like this (see Figure 3).

All of Flatland then discovers that the shortest line algorithm is not always fair. Sometimes, it can randomly produce a biased result. Now, randomly being biased is certainly better than having an innate bias, but it’s not good enough for the citizens of Flatland. They want a system that can always result in a representative congress.

To achieve this goal, they go about a very unorthodox strategy: In order to eliminate the effects of gerrymandering entirely, the government decides to hire someone to gerrymander for them! Sphere, who lives in a faraway kingdom, has no interest in the politics of Flatland, so he’s the perfect candidate for Flatland to hire.

Sphere’s job is to draw the district lines in a way that allows Flatland to come up with a representative congress every election. So he gets to work, packing voters of one type together, cracking some of another type out, and eventually presents his plan to Flatland (see Figure 4).

It works perfectly, and Flatland achieves a representative congress with no random bias! However, the citizens of Flatland still are not happy. The idea of purposely changing districts to manipulate the outcomes of elections, even if it’s being done with fairness in mind, doesn’t seem very fair.

In order to come up with a better solution, however, all of Flatland will have to completely rethink how they go about voting.

Stay tuned next week, as Flatland explores multiple voting, approval voting and single transferable vote systems! All graphics by Sam Beckmann.

Post Author: westanderson

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