Ensembles and Outliers

The dictionary definition of gerrymandering, “to manipulate or change the boundaries of (an electoral constituency) so as to favor one party or class,” presupposes some conception of the expected outcome and structure of an election. Implicitly a complainant challenges a map as gerrymandered because it fails to align with this expectation in some critical way.

Measures like the efficiency gap or partisan symmetry offer conceptual, idealized alternatives against which a given map’s compliance is measured. The efficiency gap values a rather arbitrary relationship between the number of seats and the overall vote fractions. (See Bernstein and Duchin or Tapp). Partisan symmetry posits the superficially appealing idea that both parties should be treated symmetrically when each have the same global vote fraction, ignoring if symmetry is truly reflected in a geopolitical landscape. Neither takes into account the heterogeneity of population densities, the shapes of states or differences in where various political constituencies live. Requirements imposed by the Voting Rights Act or statutes requiring that counties or municipalities be left whole where possible, are not directly incorporated into the partisan symmetry measure.

There is another path, which builds on well- established methods of modern computational statistics: constructing a representative ensemble which can be used normatively to establish a baseline against which deviations can be measured.

It does not dictate political outcomes based on some idea of fairness where it is difficult to incorporate demographic ground truths, the effects of geography, or the preservation of municipal entities and communities of interest. Rather one encodes the design principles of a desirable map in an ensemble of maps and then lets the typical electoral behavior reveal itself.

The design criteria are typically nonpartisan criteria such as spatial compactness, contiguity, equal population division and the minimization of fragmented counties and municipalities. It can also include items such as compliance with the Voting Rights Act (VRA). With the possible exception of VRA compliance, these criteria do not overtly create any new partisan bias, leaving (rightly) only those inherent in our political system and the distribution of voters about a particular geographically distinct state.

It is important to separate (i) the use of such an ensemble, (ii) its generation and (iii) the choice of the design criteria. Each is intellectually independent and, as such, can be critiqued separately. Here we mainly wish to concentrate on the use of an ensemble. Though we will comment briefly on the other elements, we will largely leave the discussion of the other elements to future works or posts.

Using Normative Ensembles

Once one has fixed a collection of representative maps, they can be used to probe the structure of a given election. Among other things, the ensemble reveals typical outcomes of an election and the typical over all structure of the election. The ensemble encodes the geopolitical structure of the state as it includes the population distribution as well as the spatial alignment and connectivity of the state. If desired, it can also encode the need to keep cities or counties whole or to satisfy the Votings Rights Act. Hence any conclusions drawn on the typical structure and outcome of a collection of votes is inherently informed by the geopolitical makeup of the region and the other desired criterion encoded in the construction of the ensemble.

The conclusions from such an analysis can then be turned back on a specific map of interest to evaluate how representative of the ensemble the results produced by the specific map are. If a single election is used, the Box-Plots, discussed here and used in Common Cause v. Rucho, provide a effective tool to probe an map’s relative structure.

This analysis acknowledges that the outcome of a group of district wide elections can depend dramatically on the choice of districts even if no votes are changed. For instance the level of observed fluctuations in outcomes derived from the North Carolina 2016 House of Representatives election likely dwarf any fluctuations which have been attributed to voter fraud or suppression.

One can also leverage a number of different elections into a combined analysis such as the collection histograms employed in the firewall analysis of the Wisconsin Act 43 Map or the H-index defined in the paper Evaluating Partisan Gerrymandering in Wisconsin. Both evaluate a map by considering its behavior under a number of either historic elections or shifted elections derived from a single election.

Using a number of elections can also help mitigate the effect using votes as a partisan vote for a given party rather than for a given candidate. Mixing results form a number of elections, some statewide and some more local, can control for differences in founding levels, incumbency, and more generally weak candidates.

Designing Distributions and Generating Ensembles

To execute an analysis based on an ensemble of maps, one needs to generate an ensemble. The process of generating an ensemble should be held separate from considerations of using the ensemble in analysis. One might disagree on the value judgments made in defining the ensemble without invalidating the inferential power of ensembles more broadly. It is clear that legal and legislative guidance is desirable in designing the ensemble. It is important to separate critiques regarding the framework of ensemble analysis from the choices made in constructing a particular ensemble.

We feel that it is important to further separate the algorithmic act of generating the ensemble from the definition of the ensemble. Since the article Redistricting and the Will of the People, we have advocated placing a probability distribution on the collection of possible redistrictings and then use variants of the well established Metropolis–Hastings algorithm from Markov Chain Monte Carlo to sample the distribution. This cleanly separates the assumptions and choices used in defining the ensemble and the algorithm used generate the ensemble. Both can be critiqued separately.

The same can not be said of all methods of generating a collection of maps. Algorithms which simply construct a collection of compliant maps do not make clear their particular bias. Similarly, simply performing some optimization search and reporting all compliant maps found, does not make clear the full set of assumptions in play; in contrast, sampling from a well defined distribution does. Of course different sampling algorithms might introduce their own biases, but these can be compared and explored separately once the distribution has been fixed.

Andrew Chin  (UNC Law)
Greg Herschlag (Duke Math)
Jonathan Mattingly (Duke Math)