The Mathematics And Strategy Behind Gerrymandering​

If you are unfamiliar with the term, Merriam Webster defines gerrymandering as “the practice of dividing or arranging a territorial unit into election districts in a way that gives one political party an unfair advantage in elections.” But rather than trying to make sense of that long string of words, let’s start with a simpler, visual representation.

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The Basic Idea: the Gerrymandering Activity

In these grids, let each column represent a district. If 15 red voters and 10 blue voters are split into their own districts, each party wins the amount of districts proportional to the population they represent (left) – a fair, majority wins system. However, I can easily rearrange this board so that red wins every single district (right). But since red was the majority in the first place, that rearranging doesn’t make any difference in the outcome of the election.

But what if I want blue, the minority, to get a majority of the 5 districts – is that possible? Well, if I clump most of the red voters into two districts, then spread the rest out over the other three districts, red no longer wins the majority, even though they represent a majority of the population of the whole state.

You can see from that grid that there’s room for one blue to switch over to red, so that there are only 9 blue voters and 16 red voters. Even when blue makes up just 36% of this population, it can still win a majority of districts!

Okay, let’s make this a little more challenging. In the real world, you can’t just move everyone around to create the ideal district map. People are bound to specific geographical locations. So, take this map below, where the locations of the voters are fixed. Without the bounds of columns, is it possible to draw five equally sized districts so that blue again wins a majority? 

The answer is yes! It’s actually incredibly easy, and there are many different ways to solve this problem. I could keep the columns (or rows, for that matter) as the five districts, or just make up my own, more complex map.

But that grid contains 14 red voters and 11 blue voters, so it wasn’t as difficult to draw the districts. What about 15 red and 10 blue?

Ok, but what about 16 and 9 again?

While these problems got a little more difficult, they were still possible – with enough distortion of the districts. You’ll notice I could no longer draw straight lines for the grids with less blue voters, and the districts look much more messy.

Had I reduced the number of blue voters to 8, it would have been impossible to draw the districts such that blue won. How do I know that? Well, blue must have at least 3 voters within a district to win it, and needs to win 3 districts to win overall, so that gives us (3 voters) x (3 districts) = at least 9 blue voters total.

Strategies and Application

You might have noticed a few patterns from the activity, especially when we had 9 blue voters. Often, one or two of the districts contained all or mostly red voters. That allowed for blue to take a majority, as the low number of remaining red voters were spread thin among the rest of the districts.  So, in order for the minority party to win, they must split the other party as much as possible so their party still has a majority in every district, then clump the rest together so more of their votes are “wasted” votes (or extra votes spent on a candidate who already has a majority in that district).

So far we’ve only seen these strategies used in a child’s game, but once you start looking at the maps of congressional districts in the U.S., you’ll realize how real-world political parties use these strategies today to their advantage. That’s when we start to see fishy district lines like these:

The map below shows a rough outline of voting preferences for Colorado based on polling from the 2012 election. Let’s say for the sake of simplicity that each box is a single voter. Then, in this map, there are 140 total voters – 52 blue, 88 red – and seven districts comprised of 20 people each. 

Again, say I want the blue party to win the election. Blue then needs to win the majority of 4 districts to win the election. To win a majority within a district, there must be 11 blue voters. Therefore, I must have a minimum of (11 voters) x (4 districts) = 44 blue voters on this map. Since I have 52, the odds are looking good for blue. Here’s one solution I found:

But wait, could I have done better? Could blue have won 5 districts instead of just 4? As of now, I have one district with 17 blue voters and only 3 red voters. Can I use those excess blue voters to win another district?

Well, let’s look at the math. If I want to win 5 out of the 7 districts, I need to a majority (at least 11 blue voters) in each district. That gives us (11 voters) x (5 districts) = 55 blue voters. Unfortunately, this particular map only contains 52, so winning 5 districts would be impossible.

Sure, I could have more evenly split the reds among the blues to better guarantee success (just because there are 11 blue voters doesn’t mean all 11 will vote blue…), but overall, the best 52 blue voters can do is win 4 districts.

To see a more accurate map of Colorado’s (and other states’) districts and play around with how they could be drawn, visit FiveThirtyEight’s “Atlas of Redistricting.” Explore maps that are gerrymandered in favor of either republicans or democrats, a map that promotes highly competitive elections, and more!

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