Generalized Sequential Monte Carlo Sampling for Redistricting Simulation

2026-03-23

AI Summary

This work addresses the challenge of efficiently evaluating fairness in redistricting plans involving multi-member districts or large-scale single-member configurations, where existing simulation methods often struggle. The authors propose a generalized Sequential Monte Carlo (gSMC) algorithm that enables efficient exploration of complex districting structures through flexible multi-scale spatial partitioning, compatibility with diverse sampling spaces, and integration of Markov chain Monte Carlo (MCMC) steps. Key innovations include support for sampling multi-member districts, derivation of optimal incremental variance weights, and the construction of a hybrid gSMC–MCMC framework. The method’s effectiveness and scalability are demonstrated through real-world applications to the Irish Parliament (multi-member districts) and the Pennsylvania House of Representatives (over 200 single-member districts), confirming its suitability for large and structurally intricate redistricting problems.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Game Theory and Economic Paradigms: Fair Division

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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.

Problem

• redistricting simulation
• multi-member districts
• Sequential Monte Carlo
• partisan bias
• racial bias

Innovation

• Generalized Sequential Monte Carlo
• Multi-member districts
• Optimal-variance incremental weights
• Hybrid gSMC-MCMC
• Redistricting simulation