The concept of partisan symmetry explicated above is important in and of itself, but even a clear concept that everyone accepts as relevant does not immediately suggest a specific method of measuring that concept, or any rule that would necessarily apply in the legal setting. These two additional goals of measurement and legal application require, first, a statistical measure of the deviation of an electoral system from partisan symmetry; in other words, we need a measure of partisan bias. (For some purposes we describe below, we may also desire a measure of electoral responsiveness, which distinguishes different types of fair electoral systems.) Second, the Court will then also need to determine how to make use of such measures, but that is a separate issue we defer until Section III.
Over the many years in which scholars have worked on defining fairness as partisan symmetry, they have also developed a sequence of steadily improving statistical approaches designed to measure the degree of partisan bias in elections and in proposed legislative redistricting plans. These now highly mature statistical methods rely on well-tested and well-accepted statistical procedures.
Estimating partisan bias and electoral responsiveness both first require studying how the statewide average district vote for Democratic candidates (which in our simple two-party running example is 100% minus the fraction for the Republican candidates) will translate into the expected statewide fraction of seats for the Democratic Party. The relationship between these two variables is typically summarized with the seats-votes curve, which traces out the expected statewide seats division as a function of each possible value of the average district vote (i.e., for each average district vote percent for the Democrats between 0% and 100%, or at least near the middle of that scale where real results occur more commonly). Once we have traced out the seats-votes curve, we can compute partisan bias by directly examining how each party would fare in obtaining seats for any given vote fraction. For example, we could literally read off the expected seat proportion the Democrats would likely receive if they won 55% of the vote in the average of the districts in the state, and the seat proportion for the Republicans if they received 55% of the vote. Similarly, the seats-votes curve also reveals the level of electoral responsiveness by the slope of the line near the middle of the curve where most elections take place: steeper slopes mean that a small change in the average district vote (for either party) would yield a larger change in the seat division between the parties than if the seats-votes curve were flatter. It is important not to confuse electoral responsiveness, which refers to the slope of seats-votes curves and helps to distinguish different types of fair systems, with partisan bias, which refers to the degree to which an electoral system deviates from partisan symmetry. The two are totally distinct concepts.
Because we can measure both partisan bias and electoral responsiveness directly from the seats-votes curve, all that remains is to (a) identify a method of estimating the seats-votes curve from the data we observe, and (b) specifying a statistical methodology that allows us to estimate the margin of error that exists in the measurements we derive from the seats-vote curve.
Historically, four general categories of methods have been used to measure seats-votes curves, each one better than the previous.39 To offer intuition about how one can estimate the relationship from real data, and also to give a sense of the real scientific progress made in this field, we now briefly describe each approach.
The first method developed to measure a seats-votes curve was to take a number of election results and to plot the actual statewide average district vote by the statewide seat proportions, with one point representing each election. The strategy is then to fit some type of linear or nonlinear regression to these points and use that estimated regression line as the seats-votes curve. (The margin of error in the seats-votes curve measured this way is estimated from how closely the points fit the curve.) This approach works fine in principle, except that there are usually five or fewer elections between redistrictings, which is too few to pin down the seats-votes curve with much certainty. More importantly, this approach cannot be applied directly to evaluate redistricting plans before they are put into effect or even before the next redistricting is about to take place, and so it is useful only for historical and comparative purposes.40
A second strategy for measuring the seats-votes curve is to use a key relationship evident in the vast majority of district election data to construct the hypothetical relationships between votes and seats from the district-level votes in only one election. The idea is to plot first the one point representing the observed proportion of seats and of votes in the one actual election. Then one assumes that, if the swing in votes for the Democratic Party statewide increased by (say) one percentage point, the same uniform swing would occur in every district within the state. We can use this “uniform partisan swing” assumption by adding one percentage point to each district in the state and then declaring the candidates “winners” in each district based on these new hypothetical vote results; this produces one additional point on the seats-votes plot. The same procedure is repeated by adding (and subtracting) a large range of values uniformly one at a time to all districts and recomputing the statewide seat totals. In this way, we can reconstruct an entire seats-votes curve based on this one assumption.41
This uniform partisan swing strategy is an improvement since by marshaling district-level data, it productively uses much more information than the first cross-election approach, and yet it requires only a single actual election. Unfortunately, the approach still has three serious flaws for use in evaluating redistricting. First, it does require this one election, and so we could not evaluate the consequences of redistricting plans with this approach until after the first election held under the new plan. Second, although it is remarkable that the uniform partisan swing assumption does hold approximately in a vast array of democratic elections in the U.S., worldwide, and throughout history,42 the assumption (which requires uniform swing to hold exactly) is violated to a degree by almost all actual election data. And finally, the assumption of exact uniform partisan swing implies a zero margin of error in terms of predictive accuracy, which is always unrealistic in social science analyses. Newer methods address one or all of these three disadvantages.
The third approach to estimating the seats-votes curve eliminates the need to wait until after the first election. The idea is to create hypothetical votes in districts under the new redistricting plan by using the actual votes cast in a previous election for some statewide race (often a low visibility race, such as state treasurer or board of regents) and breaking them down into the new districts.43 The assumption here is not that the votes in the statewide race are the same as those that would be received by the legislative candidate in the district election, but rather only that the relationship between votes and seats can be estimated in this way. This assumption corresponds to the idea that, if you ranked the degree to which districts were Republican based on legislative elections or elections to a statewide office, the rank order, and not necessarily the actual vote, would be approximately the same. This assumption is often accurate, but never exactly of course. For example, the incumbency status of the legislators, and their typical electoral advantage, is ignored, as are many other important political differences in each legislative district election. The lack of a realistic (nonzero) margin of error is also not fixed by this approach. Thus, even though this method sometimes provides a reasonable measure of the seats-votes curve, and in turn the degree and direction of partisan bias and the extent of electoral responsiveness, this method can be improved on,
The fourth and current state of the art approach, developed in a sequence of articles by Gary King and Andrew Gelman,44 builds on the insights of the above earlier methods in three key ways. First, instead of assuming that uniform partisan swing holds exactly, it only requires the statistical assumption of approximate uniform partisan swing. This more realistic assumption has been shown to fit electoral data very closely in a vast array of elections, and so is appropriate to evaluate almost all American legislative electoral systems. In fact, the same pattern holds for elections of all kinds in the U.S., and even elections in other countries. For example, we may not have any idea how the next presidential elections will turn out but, whatever the exact results for elections in this decade, we know with a high degree of certainty that the overall vote in Utah will be more Republican than that in Massachusetts. In Republican years, both will typically become more Republican, and in Democratic years they will both usually become more Democratic but, whatever the nationwide swing, the ordering of and distance between the two will remain roughly the same.
This insight is a key empirical generalization that applies to all elections in the U.S. and most other democracies: the statewide or nationwide swing in elections is highly variable and difficult to predict, but the approximate rank order of districts is highly regular and stable. The rank order is not perfectly fixed, and local political changes can and regularly do affect them, but this uncertainty is reflected in the statistical assumption of approximate uniform partisan swing, and the changes in the ranking of different areas is usually relatively small and within predictable margins of error.45 Fortunately, the methodology necessary to estimate partisan bias requires no predictions about the swing, and indeed is not affected by whether it is a Democratic or Republican year or whether one will be more prevalent than the other. It is instead based only on this regular feature of elections that helps establish the relationship between any particular vote outcome and the likely resulting seat division.
The second advantage of this new approach is that it does not require assuming that votes in statewide elections for statewide candidates have any particular ex ante relationship with votes for legislative candidates. Instead, this methodology has adapted, incorporated, and extended standard statistical approaches (based on linear regression, one of the most commonly used methodologies in the social sciences) to measure what seat outcomes would be like given particular average district vote proportions by estimating outcomes from the available historical data. Estimating whether such a relationship exists, and what it is, is a strategy in stark contrast to previous methods, which had merely assumed that the vote for a statewide office would be a perfect predictor of results in legislative districts. In fact, the new method allows the use of any available information about the determinants of partisan voting strength in the new districts, including recent election results, the presence of an incumbent in the district, and whether the race is contested. Other factors may include party registration data, prior party control of the district, incumbency status, candidate quality, or demographic characteristics of the voting age population.
The first three approaches to estimating seats-votes curves also have the disadvantage of being sensitive to the choice of election data used and other inputs to the calculations, because, for example, you must choose which statewide office to use for the elections under study. These methods will sometimes produce very different estimates of partisan bias with a different choice of statewide office. The state of the art fourth approach does not have this disadvantage, because all available data may be used, we do not need to assume that people the same way for statewide offices as in legislative elections, actual election data from the legislature under study are used, and special statistical procedures are introduced that make the method work even when highly predictive variables are not available. Implemented properly, estimates of partisan bias for a particular redistricting plan tend to be quite similar, and within the margin of error, even if available explanatory variables change to a large degree.46
The final advantage of the new approach is that allowing approximate uniform partisan swing also turns out to provide accurate statistical measures of the “margin of error,” so that we can know how confident we can be in the values we get for partisan bias. In this way, courts can be aided in determining the weight to give an expert’s testimony about the magnitude of partisan bias.
In its present form, this standard methodology for measuring symmetry and partisan bias is well-established, widely accepted, peer-reviewed, and highly reliable.