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Finding Pareto Efficient Redistricting Plans with Short Bursts

Cory McCartan

TL;DR

The paper addresses the problem of redistricting under multiple criteria by extending the short bursts optimization method to approximate the Pareto frontier $P(f)$ for a multi-criteria score $f:\Xi\to\mathbb{R}^J$. It introduces a Pareto-aware variant of short bursts that maintains a non-dominated set and samples bursts from this frontier to progressively uncover $P(f)$, with theoretical remarks on convergence under certain Markov chain conditions. Empirically, it applies the method to Iowa's congressional redistricting with population deviation and compactness as the two criteria, showing that the estimated frontier expands with more bursts and is robust to burst size, yielding insights into tradeoffs between compactness and equality. The work provides an open-source tool and demonstrates practical utility for researchers and practitioners to explore multi-criteria tradeoffs in redistricting, with future work comparing to other multiobjective approaches and studying scalability.

Abstract

Redistricting practitioners must balance many competing constraints and criteria when drawing district boundaries. To aid in this process, researchers have developed many methods for optimizing districting plans according to one or more criteria. This research note extends a recently-proposed single-criterion optimization method, short bursts (Cannon et al., 2023), to handle the multi-criterion case, and in doing so approximate the Pareto frontier for any set of constraints. We study the empirical performance of the method in a realistic setting and find it behaves as expected and is not very sensitive to algorithmic parameters. The proposed approach, which is implemented in open-source software, should allow researchers and practitioners to better understand the tradeoffs inherent to the redistricting process.

Finding Pareto Efficient Redistricting Plans with Short Bursts

TL;DR

The paper addresses the problem of redistricting under multiple criteria by extending the short bursts optimization method to approximate the Pareto frontier for a multi-criteria score . It introduces a Pareto-aware variant of short bursts that maintains a non-dominated set and samples bursts from this frontier to progressively uncover , with theoretical remarks on convergence under certain Markov chain conditions. Empirically, it applies the method to Iowa's congressional redistricting with population deviation and compactness as the two criteria, showing that the estimated frontier expands with more bursts and is robust to burst size, yielding insights into tradeoffs between compactness and equality. The work provides an open-source tool and demonstrates practical utility for researchers and practitioners to explore multi-criteria tradeoffs in redistricting, with future work comparing to other multiobjective approaches and studying scalability.

Abstract

Redistricting practitioners must balance many competing constraints and criteria when drawing district boundaries. To aid in this process, researchers have developed many methods for optimizing districting plans according to one or more criteria. This research note extends a recently-proposed single-criterion optimization method, short bursts (Cannon et al., 2023), to handle the multi-criterion case, and in doing so approximate the Pareto frontier for any set of constraints. We study the empirical performance of the method in a realistic setting and find it behaves as expected and is not very sensitive to algorithmic parameters. The proposed approach, which is implemented in open-source software, should allow researchers and practitioners to better understand the tradeoffs inherent to the redistricting process.
Paper Structure (9 sections, 4 theorems, 4 equations, 4 figures, 2 algorithms)

This paper contains 9 sections, 4 theorems, 4 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Approximating the Pareto Frontier with Short Bursts
  3. Pareto ordering and efficiency
  4. Pareto optimization by short bursts
  5. Algorithm properties
  6. Empirical Demonstration
  7. Conclusion
  8. Proofs of Propositions
  9. Scaling of the Pareto frontier size

Key Result

Proposition 2.1

Let $\mathcal{M}^b$ be the Markov chain obtained by taking $b$ steps from $\mathcal{M}$ at a time. If the Markov chain $\mathcal{M}^b$ has a strictly positive transition probability between any pair of plans, then $X_n\xrightarrow{\;\!a.s.\:\!} P(f)$ as $n\to\infty$.

Figures (4)

  • Figure 1: Pareto frontier estimated by the proposed method across a range of total bursts (colored lines), plotted against 1,000 samples from an MCMC sampling algorithm using the same Markov chain (grey points).
  • Figure 2: Plans along the Pareto frontier estimated with 10,000 bursts.
  • Figure 3: Estimated Pareto frontier across a range of burst sizes, with 10 replications of 200 bursts each.
  • Figure B1: Pareto frontier size for samples from a multivariate Normal distribution, by dimension and sample size. Twenty samples were generated for each combination of parameters.

Theorems & Definitions (6)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition A.1
  • proof
  • Proposition A.1
  • proof