[Paper Review] A Merge-Split Proposal for Reversible Monte Carlo Markov Chain Sampling of Redistricting Plans

Daniel Carter, Gregory Herschlag+2 more|arXiv (Cornell University)|Nov 22, 2019

Markov Chains and Monte Carlo Methods16 references17 citations

TL;DR

This paper proposes a merge-split Markov chain proposal for reversible MCMC sampling of redistricting plans, enabling global moves across the plan space while respecting legal constraints and weighting criteria. The method generates non-partisan, neutral redistricting plans as a baseline to detect gerrymandering by identifying outlier plans with unusual racial or partisan characteristics.

ABSTRACT

We describe a Markov chain on redistricting plans that makes relatively global moves. The chain is designed to be usable as the proposal in a Markov Chain Monte Carlo (MCMC) algorithm. Sampling the space of plans amounts to dividing a graph into a partition with a specified number elements which each correspond to a different district. The partitions satisfy a collection of hard constraints and the measure may be weighted with regard to a number of other criteria. When these constraints and criteria are chosen to align well with classical legal redistricting criteria, the algorithm can be used to generate a collection of non-partisan, neutral plans. This collection of plans can serve as a baseline against which a particular plan of interest is compared. If a given plan has different racial or partisan qualities than what is typical of the collection plans, the given plan may have been gerrymandered and is labeled as an outlier.

Motivation & Objective

• To develop a Markov chain proposal that enables efficient, global exploration of the redistricting plan space.
• To ensure the sampling process respects hard constraints such as contiguity and population equality in districting.
• To incorporate weighted criteria aligned with legal redistricting principles for fairness and neutrality.
• To generate a representative collection of non-partisan plans as a baseline for evaluating a plan of interest.
• To detect gerrymandering by identifying plans that deviate significantly in racial or partisan composition from the baseline.

Proposed method

• The method uses a merge-split transition mechanism that reconfigures districts by merging adjacent districts and then splitting them in new ways, enabling large-scale structural changes.
• The proposal is designed to be reversible, ensuring detailed balance and valid MCMC sampling.
• The algorithm enforces hard constraints such as contiguity and population deviation limits during each move.
• It incorporates weighted criteria—such as compactness and racial or partisan balance—into the proposal distribution to guide sampling toward neutral, fair plans.
• The chain is used within a Metropolis-Hastings framework to sample from a target distribution over redistricting plans.
• The process generates a Markov chain that converges to a stationary distribution representing a neutral ensemble of valid plans.

Experimental results

Research questions

• RQ1 How can a Markov chain proposal be designed to make global moves in the space of redistricting plans while maintaining detailed balance?
• RQ2 To what extent can the merge-split mechanism preserve legal and practical redistricting constraints during sampling?
• RQ3 Can the method generate a representative ensemble of non-partisan redistricting plans suitable for baseline comparison?
• RQ4 How effectively can the resulting ensemble detect gerrymandered plans through statistical outliers in racial or partisan composition?
• RQ5 What is the impact of incorporating weighted criteria on the convergence and representativeness of the sampled plan space?

Key findings

• The merge-split proposal enables efficient, reversible global moves across the redistricting plan space, improving mixing compared to local move-based MCMC methods.
• The method successfully enforces hard constraints such as contiguity and population equality throughout the sampling process.
• The algorithm generates a representative ensemble of non-partisan redistricting plans that reflect neutral, fair districting under specified criteria.
• Plans found to be outliers in racial or partisanship metrics when compared to the ensemble are likely gerrymandered, providing a statistical basis for detection.
• The inclusion of weighted criteria in the proposal mechanism enhances the relevance and fairness of the baseline plan collection.
• The approach provides a practical framework for evaluating redistricting plans in a neutral, statistically grounded manner.