Math in Society, 2nd Edition: Tools for decision making

After a reapportionment, it is up to each state government to divide their population into equal districts which each elect their representative to the U.S. House. Each representative should represent approximately the same number of people, but their areas may not be the same. Due to different population densities in urban and rural areas, you can see in the map of Oregon that the districts are not geographically equal.

Currently in most states, the state legislature draws the district boundaries. That means the party in power is in charge of drawing the new districts and they could redraw the lines to help their own party. This is called gerrymandering. Gerrymandering, pronounced, “Jerrymandering,” is when districts are drawn to the political advantage of those drawing the boundaries.

Gerrymandering has occurred by both parties. Gerrymandering by race is illegal due to the Voting Rights Act of 1965, so people of color cannot be spread out to dilute their vote. Groups with similar interests may want to be put together to form voting blocks, but that is controversial. Gerrymandering by political party has been the subject of many lawsuits and Supreme Court cases. One of the challenges is how to measure gerrymandering to prove that it has occurred. The measurement must be easy to explain and understand in court.

Several ways to measure gerrymandering have been proposed. Some of the first methods used geometry to measure how compact a district is, rather than the sprawling salamander. But it has been shown that being compact does not equate with fairness. More recent methods use computer simulations to determine whether a certain map is an extreme outlier compared to other possible maps, or measure something called partisan bias

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projects.fivethirtyeight.com/partisan-gerrymandering-north-carolina

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In this book we will focus on the efficiency gap, proposed in 2015 by Nicholas Stephanopoulos and Eric McGhee. The efficiency gap is a measure of the advantage one party has over the other party due to the partisanship of the voters in each district.

Later in this section we will learn how to calculate and interpret the efficiency gap. First, let’s look at how we can tell whether a map is fair or not.

For a map to be fair, we would expect the portion of seats that a political party wins in an election to be about the same as the portion of that party in the state. This is called proportionality. For example, if a state is 50:50 Democrat to Republican, they should each get about half of the seats. There is nothing in the constitution that guarantees proportionality, but this is a basic measure of fairness. Let’s look at a couple of examples using fractions and decimals.

Now that we can tell whether a map is fair, let’s see how gerrymandering works.

There are two main ways to gerrymander a map, by packing and cracking. Packing is when all the people of one political party are packed into a district. Since a party only needs a plurality or majority to win, all those extra votes would be surplus. Cracking dilutes a party’s votes by spreading out the voters so they can’t win as many districts.

Before we draw a fair map, let’s try gerrymandering it for Democrats and then for the Republicans.

To try to measure how unfair the map is, let’s learn how to calculate the efficiency gap. First, we will tally the votes in the table below, and note how the election results would turn out by using bold numbers for the party with more voters in each district. For example, there are 4 D’s and 3 R’s in district 1, so the Democrats would theoretically win that district.

Next, we will calculate how many votes would be extra over the amount needed to win the election. These are the surplus votes. In this example, with 7 voters, the majority of a district is 4 votes, so we will subtract 4 from each winning side. From each losing side, all of the votes are considered surplus because they did not go toward electing a candidate of their party. Then we add all the surplus votes for each party.

The difference between the surplus votes as a percentage of the population is the efficiency gap. As a formula it is written:

To get a positive result, we put the larger number first and we have

\(\frac{13-2}{35}=\frac{11}{35} \approx 0.314\) or 31.4%.

The efficiency gap is 31.4%. To understand what this percentage means, we compare it with the percentage that each district represents in the state. Since our state has 5 districts, each one is 20% of the population. Here’s how we got that by dividing or as a fraction of the state.

\(100\% \div 5\) districts \(=20\%\text{,}\) or, \(\frac{1}{5}=0.20\) or \(20\%\)

In this case, the efficiency gap is worth more than one full seat, which suggests that the Democratic party has at least one extra seat than they would with proportional representation.

There are still many court cases in progress alleging gerrymandered maps so it seems like a new solution is needed. Some alternatives are appointing independent commissions to draw the lines, or changing the system altogether with proportional representation

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fairvote.org/fair_representation#what_is_fair_voting

. The statisticians at FiveThirtyEight used a web application to draw new lines in all of the states according to six different measures. You can read about it in this article

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fivethirtyeight.com/features/hating-gerrymandering-is-easy-fixing-it-is-harder

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