Math To Detect Partisan Gerrymandering

Posted: Apr 11, 2018 11:37 AM
Math To Detect Partisan Gerrymandering

Last fall, the U.S. Supreme Court heard a case involving how the boundaries of voting districts are drawn by state legislatures, Gill v Whitford, in which a relatively new mathematical measure called the efficiency gap is playing a prominent role for its potential to determine the extent to which the borders of voting districts may have been artificially contorted to provide the members of a political party with a partisan advantage through gerrymandering.

The efficiency gap was first put forward by Eric McGhee in 2013 and more fully developed in 2014 by McGhee and Nicholas Stephanopoulous. The measure identifies the number of "wasted votes" in a given voting district, which is the sum of the number of votes cast for the losing political party and the number of votes cast in excess of a simple majority for the winning political party, which is then divided by the total number of votes cast in the district.

As such, the measure is capable of detecting if a voting district has been "packed", where a disproportionate number of voters favoring one particular party are captured within its boundaries, or "cracked", where a geographic concentration of voters favoring one political party are divided among multiple voting districts in which the majority of other voters favor another political party. In both situations, a disproportionate amount of votes would be wasted.

Unfortunately, it doesn't always work:

In some cases, it leads to unintuitive conclusions. For example, you’d think that a state where one party wins 60 percent of the vote and 60 percent of the seats did things right. Not so, according to the efficiency gap. If you do the math, that state would get flagged for extreme partisan gerrymandering—in favor of the losing party. Perversely, then, the easiest remedy might to be rig things so that the minority party gets even fewer seats.

Another problem is that the efficiency gap takes no account of political geography. In Wisconsin, most Democrats are concentrated in cities like Milwaukee, producing lopsided races there. To the efficiency gap, that could look like nefarious packing, when in reality it’s simple demographics. Similarly, if several nearby districts all swung toward one party in a close election year, that completely natural outcome could get flagged as cracking.

Other critiques of the efficiency gap get more technical. (Many were first posted on, a preprint server where mathematicians and physicists share new work.) But they all boil down to the same thing: Elections are complicated and volatile, and no one number can capture all that. As Duchin writes, “gerrymandering is a fundamentally multidimensional problem, so it is manifestly impossible to convert that into a single number without a loss of information that is bound to produce many false positives or false negatives for gerrymandering.”

Last month, one of those critiques was published in the Election Law Journal: Rules, Politics and Policy, where its author, Gregory Warrington, described the technical problems of the efficiency gap:

The efficiency gap, through its role in Whitford v. Gill, has already proven its utility. And while its simplicity is an asset, it also has several drawbacks. First, by requiring that there be proportional representation for an election to be considered fair, it conflicts with constitutional law [Vieth v. Jubelirer, 541 U.S. 267 (2004) 287–288]. Second, it is simple to construct hypothetical examples for which a natural district plan leads to a constant of proportionality different from 2 (see, for example, Figs. 2.C,D). While the proportionality asserted by the efficiency gap may hold in some overall sense, for the instances for which it doesn’t we are left in the difficult position of determining whether any deviation from this average law is due to gerrymandering or natural deviation. And, as illustrated by our definition of the t-gap function, the proportionality with a constant of 2 depends on a subjective decision on how to weight various wasted votes. Third, there are historical examples in which the proportionality holds, but the shape of the vote distribution suggests significant partisan asymmetry (see, in particular, the 1974 Texas congressional election shown in Fig. 2.A as well as the 2012–2016 Tennessee congressional elections, the last of which is displayed in Fig. 8).

Warrington did more than than however, in that he developed an alternative measure for detecting the effects of partisan gerrymandering that avoids the deficiencies of the efficiency gap. Called declination, the method is likewise simple to apply and starts with ranking all the voting districts in a state from lowest-to-highest share for one political party. If no partisan gerrymandering is present, the result will be a straight line when graphed on a chart, but if the district boundary lines have been drawn to pack and crack the population to benefit a particular political party, the result will visibly deviate from the non-gerrymandered partisan result.

Like the declination on a compass that shows the angle between magnetic north and true north, Warrington’s declination is also a simple-to-compute angle. It can reveal when a voting district plan treats the 50% threshold of votes—which is the difference between winning and losing, of course—as unusually important. If a state’s voting districts have been drawn without considering whether they will place a party over or under the 50% boundary, a plot of the districts from least Democratic voters to most (or vice versa for Republicans), should make a nice straight line. However, if the line takes a sudden turn at 50%, “watch out,” says Warrington, that can be a signal that districts were drawn unfairly, to claim more seats for one party than the other. 

Click here to view the chart.

In one example, Warrington has plotted out the results of the 2014 congressional election in North Carolina, above. The ten districts that were won by Republicans all hover in a close-to-flat patch ranging from above 30% to less than 45% Democratic votes, while the three seats that were won by Democrats were each captured by districts with well above 70% Democratic voters. The line to the “center of mass” of the Republican seats below the 50% line is shallow; above 50%, on the Democratic side, the line is steep. In other words, the strongly positive declination suggests that the districts in North Carolina were gerrymandered to favor Republicans. 

Eric McGhee, who created the efficiency gap measure, recently recognized the value of the declination measure while recently assessing California's Redistricting Commission, and describes how it might be used in conjunction with the efficiency gap:

The efficiency gap measures only the partisan advantage gained by winning seats more efficiently. It does not say much by itself about the redistricting authority’s intent. This is where the second metric, declination, comes in. Declination attempts to capture discrepancies in district outcomes that suggest the redistricting authority was especially concerned about the 50 percent threshold and how seats were situated in relation to it....

Like the efficiency gap, declination can yield either positive or negative values, and values closer to 0 are fairer than values at the extremes. Declination values range between -1 and 1 and the units are not as intuitive: they are expressed as fractions of 90 degrees. Thus, declination can give a good sense of a plan’s intended advantage and the strength of that intent, while the efficiency gap can measure the magnitude of the actual advantage.

That was an intriguing application for us, and since we were able to successfully replicate Warrington's results for North Carolina from his paper, we decided to test out the declination method further and apply it to the results of the 2016 California General Assembly election for the lower chamber of that legislative body.

The chart below shows our results, with the election results for California's 80 General Assembly districts ranked from least-to-highest vote share for the Democrat party candidate. The red dots below the 50% line indicate seats won by Republican party candidate, and the blue dots above the 50% line indicate seats won by Democrat party candidates. Overall, Republicans won 26 seats and Democrats won 54, enough to claim a supermajority (more on that shortly).

Seventy-four of these races represent contests between Republican and Democrat party candidates, while the remaining six were either uncontested or were, thanks to California's "Top-Two" primary system, a contest between two candidates of the same political party. For these six districts, where possible, we substituted the last competitive election results for a previous general or primary election (Districts 23, 53, and 71 = 2012 General election vote shares, District 47 = 2016 Primary election vote shares) during the period since California's legislative district map was redrawn in 2011. For the remaining two districts, which haven't seen a contested election between Republican and Democrat candidates since the state's 2011 redistricting, we instead substituted the results for the most-extreme margin of victory in the 2016 election for the party of the winning candidate (District 59 duplicates District 15's vote share outcome, District 76 copies District 34's outcome). Here's the resulting chart:

The results are such that we find that the declination score for the California General Assembly's voting districts is -0.05, which indicates that the state's 'non-partisan' Citizen Redistricting Commission (CRC) favored the Democrat party to a slight extent, with 2 more seats tilted in the political party's favor than would result if the state's legislative districts fully avoided any partisan gerrymandering.

McGhee confirms that CRC's state legislative plan favors the Democrats, but he doesn't consider it to be an outlier (see Figure 2, which shows a near 2-seat edge for the General Assembly and a 7-seat advantage in the state Senate).

But the declination measure tells us something about the intent of California's CRC in drawing the boundaries to produce that result. Because California's state constitution requires a two-thirds supermajority to pass tax and fee increases, the state's majority party has a strong incentive to tip the partisan balance in its favor to get it above that threshold, unlocking the ability to implement its political objectives and eliminating the need to compromise with members of the Assembly's minority party.

Since California Democrats already had a considerable majority in the state legislature, they did not require much in the way of partisan gerrymandering to be able to clear the supermajority threshold, which it won in 2012lost in 2014, and regained in 2016.

McGhee comments on the impact of the CRC's 2011 redistricting:

This evaluation of the California Redistricting Commission maps suggests that the CRC plans have been more favorable to Democrats, on average, than the plans drawn by the California Legislature in 2001. However, much of the difference appears to be driven by outlier elections. There is some inconsistency, but the typical election year features a small advantage under the CRC plans.

The CRC plans are also more competitive than the 2001 legislative plans, which were among the least competitive in the country over the last two redistricting cycles. The CRC plans have produced more competitive races across all three legislative bodies, though both chambers of the state legislature are still at the lower end of the national range of competitiveness.

The evidence presented here suggests that the CRC’s decision not to examine partisan data produced a set of plans that was generally fair and competitive. However, the new maps favor Democrats more than the old ones, particularly in 2014. And, as we’ve seen, races for seats in the California Legislature remain significantly less competitive than those in other state legislatures.

There is no reason to believe that the commissioners intended to tilt their maps in favor of the Democrats or to limit competitiveness. But the decision not to use available data to assess either possibility made it easier for the plans to do both by accident. In fact, some observers have suggested that the tilt in favor of Democrats is at least partly attributable to Democratic operatives who presented themselves as nonpartisan community advocates and testified in favor of decisions that would have pro-Democratic effects (Pierce and Larson 2011).

It is important not to overstate the point. If advocates did try to lead the commission toward biased maps, they had limited success. The partisan advantage is small and inconsistent, and it pales in comparison to the size and durability of the bias in plans that are commonly considered gerrymanders.

They indeed had limited success, but perhaps being able to have complete political control over the California state government, even on an intermittent basis with a small edge that would often be dismissed in legal proceedings as being too small to worry about, provides outsized benefits, above and beyond that from simply obtaining a legislative majority. As such, the promise of obtaining a supermajority in the most-populous state of the U.S. would provide a tremendous incentive for the majority party to continue dressing up its advocates in non-partisan sheep's clothing whenever the state's Citizen Redistricting Commission redraws the state's political map to ensure that tilt continues in its favor.

It's like economics. Most of all the meaningful activity takes place at the margins!