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Free Elections in the Free State: Ensemble Analysis of Redistricting in New Hampshire

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read New Hampshire's enacted State Senate and Executive Council maps are Republican-leaning outliers in a neutral ensemble of plans.

desk verdict Solid, useful case study; main claim credible but under-quantified, and the town-weight prior needs robustness checks. read the letter →

arxiv 2509.07328 v2 pith:EAKQWQO7 submitted 2025-09-09 cs.SI

Atticus McWhorter , Daryl DeFord This is my paper · ORCID
classification cs.SI
keywords ComputationalRedistrictingMarkovChainMonteCarloEnsembleAnalysisSpanningTreesNewHampshirepartisansymmetrygerrymanderingRECOM
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the State Senate and Executive Council maps New Hampshire enacted after the 2020 census are unlikely to be neutral products of the state's political geography. By generating millions of legally plausible districting plans with a recombination Markov chain and comparing the enacted maps against this ensemble, the authors find the enacted plans sit consistently at the Republican-favoring tail of partisan symmetry measures, even though New Hampshire's geography by itself tends to give a slight Democratic edge. The pattern is strongest in close elections, where the enacted maps would turn narrow Republican wins into seat majorities while narrow Democratic wins would not. The paper presents this as quantitative context for pending state litigation claiming Democratic-leaning towns were packed together. A careful reader should care because the result shows that New Hampshire's enacted maps are measurable outliers on standard fairness metrics across eight different statewide elections.

What carries the argument

The recombination (RECOM) Markov chain algorithm is the central instrument: it merges two districts, builds a weighted spanning tree, and cuts an edge to produce a new pair of districts, with edge weights favoring same-town connections so that towns are rarely split, matching New Hampshire's legal requirements. From this chain the authors draw an ensemble of millions of plans that defines the 'typical' range of compactness and partisan behavior for the state. The comparison metric that carries the argument is the position of the enacted plan's value relative to the ensemble distribution on vote-share, seats-won, mean-median, efficiency-gap, and partisan-bias measures, computed separately for

What would settle it

Generate new ensembles under different but still legally compliant parameter choices—for example, town-edge weights drawn from [0,2] instead of [0,4], or a Senate population tolerance of 2% instead of 5%—and see whether the enacted plan's mean-median and partisan-bias values remain outside the ensemble's central range. The paper's claim predicts they do; a neutral-baseline explanation predicts they would fall inside.

Watch

Extended reading notes

Core claim

The central discovery is that relative to an ensemble of districts sampled under New Hampshire's own rules—towns kept whole, population deviations capped at 5% for the Senate and 1% for the Executive Council—the enacted plans are Republican-leaning outliers on mean-median, efficiency gap, and partisan bias. The ensemble's typical plan carries a slight Democratic advantage from the state's political geography, so the Republican edge cannot be explained by geography alone. The specific mechanism shows in the Executive Council district containing Hanover, Keene, and Concord, which packs Democratic voters and pushes other districts' Republican shares above 50%, and in the State Senate's less-Dem

Load-bearing premise

The load-bearing premise is that the ensemble generated with the chosen parameters is a fair and neutral baseline for legally compliant New Hampshire maps; if the town-weighting, population tolerances, or election set are not representative, then showing the enacted plan is an outlier does not by itself show partisan intent.

Editorial extensions

If this is right

  • If the analysis is right, the enacted Senate and Executive Council maps are not ordinary reflections of New Hampshire's geography; they are statistical outliers with a consistent Republican tilt.
  • In close statewide elections, the enacted maps would give Republicans more seats than typical neutral plans, while in landslide elections the Democratic advantage in packed districts shows up instead—a pattern the paper describes as a Republican firewall.
  • The ensemble provides a neutral benchmark that courts and map-drawers could use to pre-check a plan for partisanship before adoption, not just after a challenge.
  • The specific packing of the Executive Council district containing Hanover, Keene, and Concord, and the weakened Democratic middle districts in the Senate, are the mechanisms that produce the outlier values.
  • The result holds across eight different elections, so it does not depend on which one statewide race a critic chooses to measure partisanship.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's outlier verdict is conditional on its ensemble being a legitimate neutral baseline; a skeptic could re-run the same analysis with different but still legal parameter choices (town-edge weights, population tolerances, or a metropolized sampler with an explicit target distribution) and check whether the enacted plans remain outliers.
  • A natural next test is to apply the same ensemble construction to New Hampshire's multi-member and floterial House districts, which the paper explicitly leaves for future work, to see whether the Republican advantage appears there as well.
  • The split-ticket volatility in New Hampshire makes it a useful test bed for the general question of which election data courts should use when evaluating fairness; the paper implies that single-election analyses can be misleading, and a cross-state comparison could sharpen that lesson.
  • Because the paper only evaluates the enacted plan against its own ensemble, a direct out-of-sample check—comparing the enacted maps' performance in the 2022 or 2024 elections with ensemble predictions—would test whether the observed bias is stable over time.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper applies the RECOM ensemble method to the New Hampshire State Senate (24 districts) and Executive Council (5 districts). Districts are built from precinct dual graphs with soft town weighting and population-deviation tolerances of 5% (Senate) and 1% (Executive Council). The enacted maps are compared to the ensemble across eight statewide elections (2012–2020) using compactness measures, seat counts, sorted district vote shares, mean-median score, efficiency gap, and partisan bias. The authors report that the enacted plans are consistently more Republican-leaning than typical nonpartisan plans, especially in close elections, and that district-level patterns suggest packing of Democratic voters in Executive Council District 2. They conclude that the analysis supports claims made in Brown v. Scanlan that the maps may have been drawn with partisan interests, while noting that the exact target distribution of the ensemble is unknown and that town-weighting was chosen to balance mixing and split-town reduction.

Significance. If the outlier findings are robust, the paper provides a useful state-specific evidentiary analysis for ongoing litigation and a clear demonstration that election-data choice affects partisan-symmetry conclusions. The study uses standard, well-established methods; reports mixing diagnostics; examines multiple metrics and elections; and includes the enacted plan as a seed. These are real strengths. The main value is empirical rather than methodological: the paper does not introduce new theory, but it operationalizes redistricting rules for New Hampshire and offers a template for similar small-state analyses. Its broader significance hinges on whether the enacted plans are indeed outliers under a defensible neutral baseline—a point that the current manuscript supports only qualitatively.

major comments (4)
  1. [§2.2, §4] The ensemble is presented as a neutral baseline, but the legal requirement that districts be composed of contiguous towns/cities/wards without splitting is implemented only as a soft 4:1 edge-weight prior. §2.2 says the upper bound was 'chosen to balance mixing time ... and reduction of split towns', and Fig. 2 anchors the plot to the enacted plan's three split towns. Since §4 admits the target distribution is unknown, the outlier conclusion is only meaningful if tail status is insensitive to reasonable modeling choices. Please report sensitivity analyses: hard no-split constraints, town-weight ratios such as 2:1 and 8:1, and population deviations of 0.5%, 2%, and 10%, with tail probabilities for each.
  2. [§3.1, §3.2] Central claims that the enacted plan is 'often' an outlier or 'outside the typical range' are supported only by visual inspection of Figs. 4–7 and 10–12. No empirical quantiles, tail probabilities, or p-values are reported for any metric or election. For the Executive Council, distributions are discrete with five seats, so exact tail probabilities are straightforward; for the State Senate, report the fraction of ensemble plans more extreme than the enacted plan for each metric and election. State how multiple comparisons across eight elections and several metrics are handled.
  3. [§3.1.2, §3.2.2] The 'firewall' claim—that the enacted plan favors Democrats in landslides but Republicans in close elections—is inferred from seat-count plots (Figs. 4a and 10a) without any statistical test. With only 5 and 24 seats, seat counts are coarse and a single district can move the result. Quantify the enacted plan's seat count as a tail event for each election, and test whether the interaction between election margin and enacted-vs-ensemble difference is significant (e.g., logistic regression or permutation test).
  4. [Appendix A] The mixing diagnostics in Tables A1–A4 are computed for sorted district Republican vote totals, but the paper's conclusions rely on nonlinear summary statistics (mean-median, efficiency gap, partisan bias, cut edges). Reporting KS distances for vote totals does not by itself establish that the chains have mixed for these metrics. Please also compute pairwise KS distances and autocorrelation lags for the actual metrics used in §3, or justify why vote-total mixing suffices for all reported comparisons.
minor comments (4)
  1. [§1, §2] Typos: 'have been been' is repeated in the Introduction; 'preformed' should be 'performed' in §2; Fig. 9 caption spells 'Polsby-Popper' as 'Polsby-Poppy'; ref. [33] spells 'Hillsborough' as 'Hilsborough'.
  2. [General] The manuscript provides no data or code availability statement. Since exact preprocessing of MGGG shapefiles and the GerryChain version are described, a repository link or appendix with replication code would strengthen reproducibility.
  3. [§3.1.2] The sentence 'The candidate up for election appears to have a greater effect than in other states' is vague; specify the comparison set or remove it.
  4. [§4] The conclusion uses 'may have been drawn with partisan interests,' which is appropriately hedged, but then states 'Our result validates some of the claims made in Brown v. Scanlan.' 'Validates' overstates what an ensemble-based outlier analysis can establish; rephrase to 'is consistent with' or 'supports.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ensemble baseline is constructed from external legal constraints, public election data, and independently published recombination methods; the enacted plan's outlier status is the measured output, not an input.

full rationale

The paper's derivation chain is linear and self-contained: (1) construct a dual graph from public NH GIS/census data; (2) run the RECOM chain with town weighting and fixed population deviations; (3) overlay eight pre-existing statewide elections; (4) compare the enacted plan's compactness and partisan-symmetry metrics to the ensemble. No parameter is fitted to the enacted plan's partisan outcomes, so the conclusion in Section 4 that the enacted plans are 'often outliers' is a measured comparison, not a constructed identity. The only self-citations are to the RECOM algorithm [34] and to the county-weighting idea in [26]; both are published, code-implemented (GerryChain), and make no assumption about New Hampshire's partisan outcome, so they are independent support that does not raise the circularity score. The town-weight upper bound of 4 is admittedly chosen to balance mixing and split-town reduction, and Figure 2 displays the enacted plan's three split towns; this is a modeling-choice sensitivity concern (a different prior could shift the tail), not circularity, because split towns are not the target partisan metrics. Section 4's admission that RECOM's target distribution is unknown is a stated limitation of the sampling method, but it does not make the comparison definitional. No circular step could be exhibited by quoting an equation where an input equals the output.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim depends on the representativeness of the RECOM ensemble and on the choice of the two main algorithmic parameters. These are not fitted to reproduce the enacted plan, so the circularity burden is low. No new physical or conceptual entities are introduced.

free parameters (2)
  • town_weight_upper_bound = 4 (same-town edges drawn from U[0,4], cross-town from U[0,1])
    Chosen by hand in Section 2.2 to balance mixing time against reducing split towns. This ratio shapes the ensemble's spatial structure and therefore its partisan distribution.
  • population_deviation_tolerance = 5% for State Senate, 1% for Executive Council
    Set in Section 2.2 as the maximum allowed deviation. This determines which maps are in the legal space and directly affects the ensemble baseline.
assumptions (3)
  • domain assumption The RECOM Markov chain, after the chosen number of steps, samples from a distribution that is representative of legally compliant plans.
    Invoked in Section 2.2 and Appendix A via autocovariance and KS heuristics. The true stationary distribution is not known, as the paper acknowledges in Section 4 citing [42].
  • domain assumption A district plan generated with no partisan objective is a valid neutral baseline for comparison.
    Central to the ensemble method described in Section 1. The normative claim that the RECOM ensemble represents 'typical' apolitical maps is assumed throughout.
  • domain assumption Historical statewide election results aggregated by precinct are a valid proxy for how voters in those precincts would vote in district elections.
    Stated in Section 2.1 as standard practice, but particularly uncertain in New Hampshire due to split-ticket voting, which motivates the multi-election analysis.

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Pith. "Pith review of Free Elections in the Free State: Ensemble Analysis of Redistricting in New Hampshire." pith.science (2026). https://pith.science/paper/EAKQWQO7

@misc{pith2026250907328,
  author       = {Pith},
  title        = {Pith review of: Free Elections in the Free State: Ensemble Analysis of Redistricting in New Hampshire},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EAKQWQO7}},
  note         = {Machine review of arXiv:2509.07328}
}
read the original abstract

The process of legislative redistricting in New Hampshire, along with many other states across the country, was particularly contentious during the 2020 census cycle. In this paper we present an ensemble analysis of the enacted districts to provide mathematical context for claims made about these maps in litigation. Operationalizing the New Hampshire redistricting rules and algorithmically generating a large collection of districting plans allows us to construct a baseline for expected behavior of districting plans in the state and evaluate non-partisan justifications and geographic tradeoffs between districting criteria and partisan outcomes. In addition, our results demonstrate the impact of selection and aggregation of election data for analyzing partisan symmetry measures.

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.