Redistricting: Drawing the Line

Abstract

We develop methods to evaluate whether a political districting accurately represents the will of the people. To explore and showcase our ideas, we concentrate on the congressional districts for the U.S. House of Representatives and use the state of North Carolina and its redistrictings since the 2010 census. Using a Monte Carlo algorithm, we randomly generate over 24,000 redistrictings that are non-partisan and adhere to criteria from proposed legislation. Applying historical voting data to these random redistrictings, we find that the number of democratic and republican representatives elected varies drastically depending on how districts are drawn. Some results are more common, and we gain a clear range of expected election outcomes. Using the statistics of our generated redistrictings, we critique the particular congressional districtings used in the 2012 and 2016 NC elections as well as a districting proposed by a bipartisan redistricting commission. We find that the 2012 and 2016 districtings are highly atypical and not representative of the will of the people. On the other hand, our results indicate that a plan produced by a bipartisan panel of retired judges is highly typical and representative. Since our analyses are based on an ensemble of reasonable redistrictings of North Carolina, they provide a baseline for a given election which incorporates the geometry of the state's population distribution.

Locally Fair Partitioning

“…If we restrict points to deviate only when the interval they create has exactly σ points, this sharp threshold holds for all ε ≤ 1/3. To interpret this result, when ε = 1/3, this means there is a fair partition where all intervals have size in the range 2 3 σ, 4 3 σ , and no subset of unhappy points can create an interval with σ points, where they form 2/3-majority. Furthermore, there is an instance where the bound of 2/3 on the majority cannot be reduced any further.…”

“…There has been extensive work on redistricting algorithms, going back to 1960s [19], for constructing contiguous, compact, and balanced districts. Many different approaches, including integer programming [17], simulated annealing [1], evolutionary algorithms [22], Voronoi diagram based methods [12,29], MCMC methods [4,13], have been proposed; see [5] for a recent survey. A line of work on redistricting algorithms focuses on combating manipulation such as gerrymandering: when district plans have been engineered to provide advantage to individual candidates or to parties [6].…”

“…Let X be a set of n points in R 1 , each colored red or blue, and let σ ∈ [n] be a parameter called ideal population size. 4 We wish to construct a locally fair partition of X into intervals so that all intervals have roughly σ points. Only ordering of points in X really matters, so we describe the input as a (binary) sequence X = x 1 , .…”

Reconfiguration of Connected Graph Partitions

“…Research has focused on exploring the space of all possible district maps that meet certain basic criteria. Since this space is computationally intractable, even for relatively small instances, randomized algorithms play an important role in finding "average" district maps under suitable distributions [3]. Being an outlier may indicate that gerrymandering has been applied in the drawing of a given map [20].…”

Utility Ghost: Gamified redistricting with partisan symmetry

“…In 2018, the Pennsylvania Supreme Court struck down the 2011 congressional map as an unconstitutional partisan gerrymander; here, the plantiffs leveraged Markov Chain-based techniques to sample the space of admissible maps and make informative comparisons with the map in question [5]. This trend of policing partisan gerrymandering in the courts has led to a flurry of relevant mathematical research, primarily in Markov Chain Monte Carlo sampling methods [6,3,9,8,16] and in evaluating various fairness criteria [4,14,1,2,10]. However, in light of recent changes to the U.S. Supreme Court bench, a new approach might be necessary to effectively combat partisan gerrymandering in the future.…”