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Moon Duchin

Publications and source records attributed to Moon Duchin.

At least 19 recordsLinked to original sources
arXiv2026

Metric geometry for ranking-based voting: Tools for learning electoral structure

In this paper, we develop the metric geometry of ranking statistics, proving that the two major permutation distances in the statistics literature -- Kendall tau and Spearman footrule -- extend naturally to incomplete rankings with both coordinate embeddings and graph realizations. This gives us a unifying framework that allows us to connect popular topics in computational social choice: metric preferences (and metric distortion), polarization, and proportionality. As an important application, the metric structure enables efficient identification of blocs of voters and slates of their preferred candidates. Since the definitions work for partial ballots, we can execute the methods not only on synthetic elections, but on a suite of real-world elections. This gives us robust clustering methods that often produce an identical grouping of voters -- even though one family of methods is based on a Condorcet-consistent ranking rule while the other is not.

math.MG↗
arXiv2025

Quantitative Relaxations of Arrow's Axioms

In this paper we develop a novel approach to relaxing Arrow's axioms for voting rules, addressing a long-standing critique in social choice theory. Classical axioms (often styled as fairness axioms or fairness criteria) are assessed in a binary manner, so that a voting rule fails the axiom if it fails in even one corner case. Many authors have proposed a probabilistic framework to soften the axiomatic approach. Instead of immediately passing to random preference profiles, we begin by measuring the degree to which an axiom is upheld or violated on a given profile. We focus on two foundational axioms-Independence of Irrelevant Alternatives (IIA) and Unanimity (U)-and extend them to take values in $[0,1]$. Our $\sigma_{IIA}$ measures the stability of a voting rule when candidates are removed from consideration, while $\sigma_{U}$ captures the degree to which the outcome respects majority preferences. Together, these metrics quantify how a voting rule navigates the fundamental trade-off highlighted by Arrow's Theorem. We show that $\sigma_{IIA}\equiv 1$ recovers classical IIA, and $\sigma_{U}>0$ recovers classical Unanimity, allowing a quantitative restatement of Arrow's Theorem. In the empirical part of the paper, we test these metrics on two kinds of data: a set of over 1000 ranked choice preference profiles from Scottish local elections, and a batch of synthetic preference profiles generated with a Bradley-Terry-type model. We use those to investigate four positional voting rules-Plurality, 2-Approval, 3-Approval, and the Borda rule-as well as the iterative rule known as Single Transferable Vote (STV). The Borda rule consistently receives the highest $\sigma_{IIA}$ and $\sigma_{U}$ scores across observed and synthetic elections. This compares interestingly with a recent result of Maskin showing that weakening IIA to include voter preference intensity uniquely selects Borda.

cs.GT↗
arXiv2024

Models of random spanning trees

There are numerous randomized algorithms to generate spanning trees in a given ambient graph; several target the uniform distribution on trees (UST), while in practice the fastest and most frequently used draw random weights on the edges and then employ a greedy algorithm to choose the minimum-weight spanning tree (MST). Though MST is a workhorse in applications, the mathematical properties of random MST are far less explored than those of UST. In this paper we develop tools for the quantitative study of random MST. We consider the standard case that the weights are drawn i.i.d. from a single distribution on the real numbers, as well as successive generalizations that lead to \emph{product measures}, where the weights are independently drawn from arbitrary distributions.

cs.DM↗
arXiv2023

Redistricting for Proportionality

American democracy is currently heavily reliant on plurality in single-member districts, or PSMD, as a system of election. But public perceptions of fairness are often keyed to partisan proportionality, or the degree of congruence between each party's share of the the vote and its share of representation. PSMD has not tended to secure proportional outcomes historically, partially due to gerrymandering, where line-drawers intentionally extract more advantage for their side. But it is now increasingly clear that even blind PSMD is frequently disproportional, and in unpredictable ways that depend on local political geography. In this paper we consider whether it is feasible to bring PSMD into alignment with a proportionality norm by targeting proportional outcomes in the design and selection of districts. We do this mainly through a close examination of the "Freedom to Vote Test," a redistricting reform proposed in draft legislation in 2021. We find that applying the test with a proportionality target makes for sound policy: it performs well in legal battleground states and has a workable exception to handle edge cases where proportionality is out of reach.

cs.CY↗
arXiv2022

Spanning Trees and Redistricting: New Methods for Sampling and Validation

Deciding whether a political districting plan was distorted by a hidden agenda, or whether it dilutes the voting power of some group, requires a neutral baseline for comparison. Remarkably, all nine U.S. Supreme Court justices have now signed on to decisions that find that computational methods can provide key evidence. Today, the leading approaches for benchmarking districting plans are based on the use of spanning trees for sampling graph partitions. We present a new *reversible recombination* algorithm and rigorously prove its fundamental properties. Furthermore, we argue for a canonical sampling distribution called the *spanning tree distribution* that is well adapted to redistricting and provides a principled foundation for comparing and validating methods. Together with a highly efficient (and open-source) implementation that can generate and handle large datasets, this work provides the most powerful null model to date for the gerrymandering problem, meeting an urgent democratic challenge with sound scientific methodology.

physics.soc-ph↗
arXiv2022

Census TopDown: The Impacts of Differential Privacy on Redistricting

The 2020 Decennial Census will be released with a new disclosure avoidance system in place, putting differential privacy in the spotlight for a wide range of data users. We consider several key applications of Census data in redistricting, developing tools and demonstrations for practitioners who are concerned about the impacts of this new noising algorithm called TopDown. Based on a close look at reconstructed Texas data, we find reassuring evidence that TopDown will not threaten the ability to produce districts with tolerable population balance or to detect signals of racial polarization for Voting Rights Act enforcement.

cs.CY↗
arXiv2021

Measuring Segregation via Analysis on Graphs

In this paper, we use analysis on graphs to study quantitative measures of segregation. We focus on a classical statistic from the geography and urban sociology literature known as Moran's I, which in our language is a score associated to a real-valued function on a graph, computed with respect to a spatial weight matrix such as the adjacency matrix associated to the geographic units that tile a city. Our results characterizing the extremal behavior of I illustrate the important role of the underlying graph structure, especially the degree distribution, in interpreting the score. In addition to the standard spatial weight matrices encoding unit adjacency, we consider the Laplacian L and a doubly-stochastic approximation M. These alternatives allow us to connect I to ideas from Fourier analysis and random walks. We offer illustrations of our theoretical results with a mix of stylized synthetic examples and real geographic/demographic data.

cs.SI↗
arXiv2020

Implementing partisan symmetry: Problems and paradoxes

We consider the measures of partisan symmetry proposed for practical use in the political science literature, as clarified and developed in Katz, King, and Rosenblatt (2020). Elementary mathematical manipulation shows the symmetry metrics to have surprising properties that call their meaningfulness into question. To accompany the general analysis, we study measures of partisan symmetry with respect to recent voting patterns in Utah, Texas, and North Carolina, flagging problems in each case. Taken together, these observations should raise major concerns about the available techniques for quantitative scores of partisan symmetry -- including the mean-median score, the partisan bias score, and the more general "partisan symmetry standard" -- as the decennial redistricting begins.

physics.soc-ph↗
arXiv2020

The (homological) persistence of gerrymandering

We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. Our method combines the geographic information from a political districting plan with election data to produce a persistence diagram. We are then able to visualize and analyze large ensembles of computer-generated districting plans of the type commonly used in modern redistricting research (and court challenges). We set out three applications: zoning a state at each scale of districting, comparing elections, and seeking signals of gerrymandering. Our case studies focus on redistricting in Pennsylvania and North Carolina, two states whose legal challenges to enacted plans have raised considerable public interest in the last few years. To address the question of robustness of the persistence diagrams to perturbations in vote data and in district boundaries, we translate the classical stability theorem of Cohen--Steiner et al. into our setting and find that it can be phrased in a manner that is easy to interpret. We accompany the theoretical bound with an empirical demonstration to illustrate diagram stability in practice.

math.AT↗
arXiv2020

A Computational Approach to Measuring Vote Elasticity and Competitiveness

The recent wave of attention to partisan gerrymandering has come with a push to refine or replace the laws that govern political redistricting around the country. A common element in several states' reform efforts has been the inclusion of competitiveness metrics, or scores that evaluate a districting plan based on the extent to which district-level outcomes are in play or are likely to be closely contested. In this paper, we examine several classes of competitiveness metrics motivated by recent reform proposals and then evaluate their potential outcomes across large ensembles of districting plans at the Congressional and state Senate levels. This is part of a growing literature using MCMC techniques from applied statistics to situate plans and criteria in the context of valid redistricting alternatives. Our empirical analysis focuses on five states---Utah, Georgia, Wisconsin, Virginia, and Massachusetts---chosen to represent a range of partisan attributes. We highlight situation-specific difficulties in creating good competitiveness metrics and show that optimizing competitiveness can produce unintended consequences on other partisan metrics. These results demonstrate the importance of (1) avoiding writing detailed metric constraints into long-lasting constitutional reform and (2) carrying out careful mathematical modeling on real geo-electoral data in each redistricting cycle.

cs.CY↗
arXiv2020

Mathematics of Nested Districts: The Case of Alaska

In eight states, a "nesting rule" requires that each state Senate district be exactly composed of two adjacent state House districts. In this paper we investigate the potential impacts of these nesting rules with a focus on Alaska, where Republicans have a 2/3 majority in the Senate while a Democratic-led coalition controls the House. Treating the current House plan as fixed and considering all possible pairings, we find that the choice of pairings alone can create a swing of 4-5 seats out of 20 against recent voting patterns, which is similar to the range observed when using a Markov chain procedure to generate plans without the nesting constraint. The analysis enables other insights into Alaska districting, including the partisan latitude available to districters with and without strong rules about nesting and contiguity.

cs.SI↗
arXiv2019

Recombination: A family of Markov chains for redistricting

Redistricting is the problem of partitioning a set of geographical units into a fixed number of districts, subject to a list of often-vague rules and priorities. In recent years, the use of randomized methods to sample from the vast space of districting plans has been gaining traction in courts of law for identifying partisan gerrymanders, and it is now emerging as a possible analytical tool for legislatures and independent commissions. In this paper, we set up redistricting as a graph partition problem and introduce a new family of Markov chains called Recombination (or ReCom) on the space of graph partitions. The main point of comparison will be the commonly used Flip walk, which randomly changes the assignment label of a single node at a time. We present evidence that ReCom mixes efficiently, especially in contrast to the slow-mixing Flip, and provide experiments that demonstrate its qualitative behavior. We demonstrate the advantages of ReCom on real-world data and explain both the challenges of the Markov chain approach and the analytical tools that it enables. We close with a short case study involving the Virginia House of Delegates.

cs.CY↗
arXiv2018

Locating the representational baseline: Republicans in Massachusetts

Republican candidates often receive between 30 and 40 percent of the two-way vote share in statewide elections in Massachusetts. For the last three Census cycles, MA has held 9-10 seats in the House of Representatives, which means that a district can be won with as little as 6 percent of the statewide vote. Putting these two facts together, one may be surprised to learn that a Massachusetts Republican has not won a seat in the U.S. House of Representatives since 1994. We argue that the underperformance of Republicans in Massachusetts is not attributable to gerrymandering, nor to the failure of Republicans to field House candidates, but is a structural mathematical feature of the distribution of votes. For several of the elections studied here, there are more ways of building a valid districting plan than there are particles in the galaxy, and every one of them will produce a 9-0 Democratic delegation.

physics.soc-ph↗
arXiv2018

Discrete geometry for electoral geography

"Compactness," or the use of shape as a proxy for fairness, has been a long-running theme in the scrutiny of electoral districts; badly-shaped districts are often flagged as examples of the abuse of power known as gerrymandering. The most popular compactness metrics in the redistricting literature belong to a class of scores that we call contour-based, making heavy use of area and perimeter. This entire class of district scores has some common drawbacks, outlined here. We make the case for discrete shape scores and offer two promising ideas: a cut score and a spanning tree score. We use recent United States redistricting history as a source of examples. No shape metric can work alone as a seal of fairness, but we argue that discrete metrics are better aligned both with the grounding of the redistricting problem in geography and with the computational tools that have recently gained significant traction in the courtroom.

physics.soc-ph↗
arXiv2018

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville current.

math.GT↗