Generalized Sequential Monte Carlo Sampling for Redistricting Simulation
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 with a varying number of representatives. 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, leading to more efficient sampling. Finally, we propose a hybrid gSMC-MCMC algorithm by incorporating Markov chain Monte Carlo (MCMC) steps to handle large-scale redistricting applications without changing the target distribution. We demonstrate the effectiveness of the proposed methodology through analyses of the Irish Parliament, which uses multi-member districts of varying sizes, and the Pennsylvania House of Representatives, which has more than 200 single-member districts.