Section 4.6 Partisan Gerrymandering
Partisan gerrymandering is the practice of drawing district lines to maximize a political party's advantage in elected representatives. This can often be seen in states where the representatives by party are vastly disproportionate from the voter affiliations. For example, consider Pennsylvania's 2012 elections, the first statewide elections following the 2011 redistricting. Almost 58% of voters registered with one of the two major parties (Democratic and Republican) were registered as Democrats [4.12.33], but Republicans (who controlled the redistricting process) won 13 of the 18 House elections, just over 72% of the seats. This extreme difference led to a 2018 challenge in court and in League of Women Voters of Pennsylvania v. the Commonwealth of Pennsylvania, the Pennsylvania Supreme Court ruled that this was a case of partisan gerrymandering that violated the state constitution and thus a new map needed to be drawn [4.12.34]. Notably, the November 2018 elections using the new map yielded 9 representatives for each party, demonstrating the impact of the previous map.
Similarly, in North Carolina, a state whose votes are nearly evenly split between the two major parties (see the North Carolina Election Results Dashboard (opens in new tab) 1 for specific information), the congressional plan was challenged due to the Republicans winning 10 of the 13 districts in the 2016 elections. The case made it all the way to the Supreme Court, who ruled in 2019 in Rucho v. Common Cause that the constitutionality of partisan gerrymandering was not a federal issue and left it to states to decide. (For more information, see [4.12.45].) Following this ruling, a state court in North Carolina ruled in Common Cause v. Lewis (2019) that their map was unconstitutional and needed to be redrawn ahead of the 2020 elections. The new map immediately changed the outcomes as Democrats won 2 additional seats (5 rather than 3) in 2020.
Although these examples of disproportionality seem unfair, how can we determine whether the outcomes are a result of partisan gerrymandering or a reflection of some other factors? We turn to mathematics to help us explore this issue.
Motivating Questions: Can we quantify partisan advantage? Can we define a threshold for “too much” partisan advantage?
We look at two different metrics to answer these questions. Keep in mind that these metrics use only party affiliation and election results and do not consider geography or voter demographics, and note that there are other measures of partisan advantage.
Subsection 4.6.1 Efficiency Gap
From our earlier examples, we know that maximizing representation requires a combination of “packing” and “cracking” strategies with the idea that one party's voters are best utilized through narrow wins and wide losses. The efficiency gap calculation is based on the idea of wasted votes:
If a party wins a district, any votes for that party above the minimum needed to win are considered “wasted.”
If a party loses a district, all votes for that party are considered “wasted.”
Following an election, we can determine the total number of wasted votes for each party (across all districts) and calculate the efficiency gap:
This formula will produce a value between 0 and 1 and can be converted to an advantage in actual number of representatives through multiplying by the number of districts.
Example 4.6.1.
| 2 | ||||||||||
| \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\clubsuit \) | ||
| 1 | \(\heartsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\clubsuit \) | \(\clubsuit \) | \(\clubsuit \) | 3 |
| \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | ||
| \(\heartsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\heartsuit \) | \(\heartsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | ||
| 5 | \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\clubsuit \) | \(\clubsuit \) | \(\heartsuit \) | \(\heartsuit \) | 4 |
| District | \(\heartsuit \) wasted | \(\clubsuit \) wasted |
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| total |
| District | \(\heartsuit \) wasted | \(\clubsuit \) wasted |
| 1 | 1 | 3 |
| 2 | 4 | 0 |
| 3 | 4 | 0 |
| 4 | 1 | 3 |
| 5 | 1 | 3 |
| total | 11 | 9 |
in favor of Clubs (\(\clubsuit \)) since the Club party has fewer wasted votes
Each district has 9 voters, so the minimum number of votes needed to win is 5. Any votes over 5 for the winning party (in that district) are considered “wasted” votes for that party. In District 1, the Heart party wins with 6 votes, thus the Heart party has 1 wasted vote in that district. All votes for the losing party in a district are classified as wasted votes, so the Club party has 3 wasted votes in District 1. In District 2, Club wins with exactly 5 votes, so Club party has no wasted votes. Heart party, then, has 4 wasted votes. The computations for the other districts are done in the same manner.
To compute the efficiency gap, find the total wasted votes for each party, then use the formula given above. Since the Club party has fewer wasted votes, we calculate the efficiency gap from their perspective (subtracting their wasted votes from the other party's in the formula):
In the example above, we calculate that the efficiency gap is approximately \(4 \% \text{.}\) But what does that mean? Does it suggest gerrymandering occurred? What does the model consider to be “fair”? If both parties waste the same number of votes, the efficiency gap would be \(0\text{.}\) This is considered “perfectly fair”. The farther the percentage from \(0\text{,}\) the more unfair the map. If we multiply the percentage by the number of districts, we get the advantage for the dominant party in terms of seats. In the above example, there are 5 districts, so the advantage for the Club party is \(4 \% \times 5\) districts \(= 0.2\) seats. In the original paper introducing the efficiency gap [4.12.47], the authors defined a threshold for partisan gerrymandering as an advantage of at least two seats for congressional plans and at least 8 percent for legislative plans. Following that guideline, the district plan and election above do not show evidence of gerrymandering.
NOTE: Since its introduction, some shortcomings of the efficiency gap have been raised (see [4.12.30] for some discussion). Importantly, it sets the \(75 \% / 25 \%\) vote split as the ideal outcome - but is that appropriate? The efficiency gap uses only wasted votes totals in its computation, without regard to how competitive elections were in each district or the geography of the region. Also, the +2 seat advantage is much more likely in larger states. How useful, then, is the calculation of the efficiency gap? As with other measures, note that it is one of many tools that could be used.
Subsection 4.6.2 Mean-Median Difference
The mean of a set of numbers is also known as the average. We find this value by adding all of the numbers and dividing by the number of values in the set.
The median of a set of numbers is found by listing all values from least to greatest and finding the middle number. If there is an even number of values in the set, the median is the mean of the middle two numbers.
The mean-median difference takes a statistical approach to measuring partisan advantage. If there was a lot of packing and cracking in a map, we would expect to see one party win many districts with just over 50% of the votes and lose a few districts with a small share of the votes. If we looked at the collection of these vote shares of districts as a set of data, we would expect the median to be higher than the mean. This skewness represents an advantage for that political party.
Example 4.6.5.
| district | Heart (\(\heartsuit \)) | Club (\(\clubsuit\)) |
| 1 | 53 | 47 |
| 2 | 55 | 45 |
| 3 | 52 | 48 |
| 4 | 54 | 46 |
| 5 | 27 | 73 |
the mean of the votes cast for the Heart party in each district
the median of the votes cast for the Heart party in each district
subtract to find the difference: mean \(-\) median
mean = \(48.2\) %
median = \(53\) %
mean \(-\) median \(= -4.8 \%\)
To compute the mean vote share, we add the vote share for Heart for each of the five districts, then divide by the total number of districts.
To find the median vote share for Heart, first we order the individual vote shares from least to greatest: \(27, 52, 53, 54, 55 \text{.}\) Then we find the number in the middle of the list - in this case, the third number: \(53 \text{.}\)
Thus mean \(-\) median \(= 48.2 - 53 = -4.8.\)
We see in the above example that the median is larger than the mean. Is this a significant difference? Does it indicate that the Heart party has a meaningful advantage over the Club party? How do we decide? In [4.12.49] (p. 1307), Wang proposes the following formula to compute the significance level, \(t\text{,}\) in a closeWindow()ly divided state:
We say that the difference between the mean and median is statistically significant and qualifies as partisan gerrymandering if \(|t|>1.75\text{.}\) (Note: this threshold follows from a significance level set to \(0.05\text{,}\) or \(5\%\text{,}\) meaning that the likelihood of a value of \(|t|\) above \(1.75\) occuring by chance is \(5\%\) or lower. The exact threshold varies depending on the number of districts; we follow Wang in using the value \(1.75 \text{.}\) For more discussion, see [4.12.49]).
Returning to our example above, we have (mean \(-\) median) = \(-4.8\) and number of districts is \(5\text{.}\) Using technology, we compute \(t\text{:}\)
Since \(|t| = 1.33 < 1.75\text{,}\) we conclude here that there is no statistically significant evidence of gerrymandering.Subsection 4.6.3 Caution
Because we could expect the mean and median to be closeWindow() in a state that has not been gerrymandered, the mean-median difference produces a measure of skew between the two. The mean-median difference can help to identify district plans that are likely to have been crafted with partisan influence. The test for significance presented here is more effective in states where the vote is closeWindow()ly divided ([4.12.49], with some discussion of this in [4.12.42]). Unlike the efficiency gap, this test does not indicate how many more seats the gerrymandering party won. We note here again that both of these measures only use party affiliation and election results, with no accounting for other factors that may be influential.
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