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Generalized Sequential Monte Carlo Sampling for Redistricting Simulation

Philip O'Sullivan, Kosuke Imai, Cory McCartan

Abstract

Simulation methods have become important tools for quantifying partisan and racial bias in redistricting plans. We generalize the Sequential Monte Carlo (SMC) algorithm of McCartan and Imai (2023), one of the commonly used approaches. First, our generalized SMC (gSMC) algorithm can split off regions of arbitrary size, rather than a single district as in the original SMC framework, enabling the sampling of multi-member districts. Second, the gSMC algorithm can operate over various sampling spaces, providing additional computational flexibility. Third, we derive optimal-variance incremental weights and show how to compute them efficiently for each sampling space. Finally, we incorporate Markov chain Monte Carlo (MCMC) steps, creating a hybrid gSMC-MCMC algorithm that can be used for large-scale redistricting applications. We demonstrate the effectiveness of the proposed methodology through analyses of the Irish Parliament, which uses multi-member districts, and the Pennsylvania House of Representatives, which has more than 200 single-member districts.

Generalized Sequential Monte Carlo Sampling for Redistricting Simulation

Abstract

Simulation methods have become important tools for quantifying partisan and racial bias in redistricting plans. We generalize the Sequential Monte Carlo (SMC) algorithm of McCartan and Imai (2023), one of the commonly used approaches. First, our generalized SMC (gSMC) algorithm can split off regions of arbitrary size, rather than a single district as in the original SMC framework, enabling the sampling of multi-member districts. Second, the gSMC algorithm can operate over various sampling spaces, providing additional computational flexibility. Third, we derive optimal-variance incremental weights and show how to compute them efficiently for each sampling space. Finally, we incorporate Markov chain Monte Carlo (MCMC) steps, creating a hybrid gSMC-MCMC algorithm that can be used for large-scale redistricting applications. We demonstrate the effectiveness of the proposed methodology through analyses of the Irish Parliament, which uses multi-member districts, and the Pennsylvania House of Representatives, which has more than 200 single-member districts.
Paper Structure (54 sections, 48 theorems, 182 equations, 28 figures, 4 algorithms)

This paper contains 54 sections, 48 theorems, 182 equations, 28 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The Problem Formulation
  3. Setup
  4. Target Distribution
  5. The Proposed Algorithm
  6. The Overview of the gSMC Algorithm
  7. The Splitting Procedure
  8. Optimal Weights
  9. Practical Implementation Details
  10. Choosing the multidistrict selection probability
  11. Choosing a Splitting Schedule
  12. Choosing \\ mathcal{K} r
  13. Diagnostics
  14. Extensions
  15. Different Sampling Spaces
  16. ...and 39 more sections

Key Result

Proposition 3.1

Let $\pi_N=\sum_{i=1}^N W^{(i)} \delta_{\xi^{(i)}}(\cdot)$ be the weighted particle approximation generated by Algorithm alg-gsmcs. Then, for all measurable $h$ on unlabeled plans, as $N \to\infty$, we have, for some asymptotic variance $V_\text{SMC}(h)$.

Figures (28)

  • Figure 1: Iowa as a graph
  • Figure 2: Two regions of size two
  • Figure 3: One district and one region of size three
  • Figure 5: Partial plan $\xi_1$
  • Figure 6: Partial plan $\xi_2$
  • ...and 23 more figures

Theorems & Definitions (139)

  • Proposition 3.1: Central Limit Theorem for the gSMC Algorithm
  • Proposition 3.2: Optimal weights
  • Definition A.1: General Notation and Terminology
  • Definition A.2
  • Definition A.3: Districting Scheme
  • Definition A.4: Plan Operations
  • Definition A.5: Adjacent Regions
  • Definition A.6: Population Tolerance Function
  • Definition A.7: Splittable Score Function
  • Definition A.8: Region Deviance
  • ...and 129 more