Map of Elections Framework

• Map of Elections Framework is a methodological paradigm that represents elections as high-dimensional vectors then embeds them into low-dimensional, interpretable maps.
• It employs efficient distance metrics like positionwise Earth-Mover’s Distance and approval-wise ℓ1 distance to compare elections even when exact isomorphic matching is computationally infeasible.
• The framework facilitates practical analysis of synthetic models and real-world elections by providing canonical compass points and robust embedding techniques for model benchmarking and visualization.

A map of elections framework is a methodological paradigm and visualization tool for representing, analyzing, and comparing elections (or election data) by embedding them into a metric or feature space, and constructing interpretable low-dimensional representations—often two-dimensional maps—where geometric proximity reflects high-dimensional similarities. These frameworks enable both abstract exploration of "election spaces" constructed from synthetic or real-world data and the systematic evaluation of model, computational, or social-choice properties associated with different types of elections (Szufa et al., 2022, Szufa, 2024, Szufa et al., 4 Apr 2025).

1. Representing Election Spaces

The foundational element is to formalize a space of elections. In the standard variant, an (ordinal) election $E=(C,V)$ is specified by a fixed set of $m$ candidates $C$ and a multiset $V$ of $n$ voters, each casting a linear order over $C$. For approval ballots, each voter submits a subset $A(v)\subseteq C$ instead of a ranking. The totality of such elections (with given $m$ and $n$) forms a discrete but very high-dimensional "election space".

Each election $E$ can be transformed into a high-dimensional vector (or matrix), such as the position-frequency matrix $F(E) = (f_{i,j})$ with $f_{i,j}$ equal to the fraction of voters ranking candidate $c_j$ in position $i$ (ordinal), or the approval-wise vector $\mathrm{av}(E)=(x_1,\ldots,x_m)$ with $x_j$ being the normalized approval fraction for $c_j$.

This vectorization is the basis for the subsequent definition of inter-election distances and the construction of low-dimensional maps (Szufa, 2024, Szufa et al., 4 Apr 2025, Szufa et al., 2022).

2. Election Distances and Metrics

To compare elections quantitatively, the framework defines a distance $d(E,F)$ between any two elections $E$ and $F$. The gold-standard isomorphic distance—minimizing, e.g., swap (Kendall-tau) distance under all voter/candidate permutations to align elections—is computationally infeasible for realistic dataset sizes, as it is NP-hard (GI-hard by Kemeny-score reduction) (Boehmer et al., 2022, Szufa et al., 4 Apr 2025).

As a tractable yet expressive alternative, Szufa et al. [AAMAS 2020] introduce the positionwise (EMD) distance $d_{\mathrm{pos}}$, defined as follows: For each pair of elections, the distance is the minimum (over candidate bijections) of the sum (over candidates) of Earth-Mover's Distances (EMD) between their frequency vectors. For approval elections, the approval-wise $\ell_1$ distance between sorted score vectors is employed (Szufa et al., 2022). These distances are true metrics (or pseudometrics), admit efficient computation via the Hungarian algorithm or prefix-sum methods, and exhibit high empirical correlation with the intractable isomorphic metrics (Boehmer et al., 2022, Szufa et al., 4 Apr 2025).

Summary of primary metrics:

Metric	Type	Complexity
Isomorphic swap	NP-hard	impractical
Positionwise (EMD)	Polynomial	recommended
Approval-wise $\ell_1$	Polynomial	fast

Computing $d_{\mathrm{pos}}$ typically involves: constructing the frequency matrices, deriving a cost matrix of vector distances between candidates (via EMD), and solving a min-cost matching (Boehmer et al., 2022, Szufa et al., 4 Apr 2025).

3. Low-Dimensional Embedding: Map Construction

Given a collection $\{E_1,\ldots,E_k\}$ and their pairwise distances, the next step is to embed them as points $\{x_{E_1},\ldots,x_{E_k}\}\subset\mathbb{R}^2$ such that $\|x_{E_i}-x_{E_j}\|\approx d(E_i,E_j)$.

Common dimensionality reduction and graph-drawing methods include:

• Kamada–Kawai (spring energy minimization; preferred for monotonicity and low distortion)
• Fruchterman–Reingold (force-directed; more evenly spreads points)
• Metric Multidimensional Scaling (MDS; stress minimization)
• Others: PCA, t-SNE, LLE (mainly for exploratory use)

Empirical and theoretical evaluations establish that e.g., Kamada–Kawai embeddings achieve $\sim$0.97 Pearson correlation between original and map distances, and average distortion ratios of $\sim$1.2, ensuring that map geometry is a faithful summary of intrinsic inter-election distances (Szufa, 2024, Szufa et al., 4 Apr 2025).

4. Canonical Structure and Interpretability: Compass and Skeleton Maps

A key innovation for interpretability is the inclusion of extremal ("compass") elections as canonical corner points:

• Identity (ID): all voters perfectly agree
• Uniformity (UN): all possible orders equally present
• Stratification (ST): two internally cohesive camps, each with block preference
• Antagonism (AN): two camps with orders exactly reversed

By analytically computing inter-compass distances (e.g., $d_{\rm pos}(\mathrm{ID},\mathrm{UN}) = (m^2-1)/3$), convex-combination paths between them structure the map's backbone quadrilateral (the "skeleton"), and any election can be barycentrically projected onto the compass axes for interpretable “concentration” and “polarity” coordinates (Boehmer et al., 2021, Boehmer et al., 2022, Szufa et al., 4 Apr 2025).

The frequency-matrix “skeleton map” enables fast, geometric assessment of any dataset or model's placement relative to these archetypes (Boehmer et al., 2022).

5. Applications: Synthetic Models, Real Data, and Visualization

Synthetic election cultures (e.g., Impartial Culture, Mallows, urn, Euclidean, single-peaked) and real-world elections (political, sports, surveys) can be rendered and positioned on the map. Extensive benchmarks show:

• Real elections typically cluster near the Mallows path with normalized dispersion parameters ($\theta\in[0.25,0.4]$) (Boehmer et al., 2021, Boehmer et al., 2022).
• Sporting events and indicator-based surveys spread from ID through UN, with varying degree of consensus.
• Synthetic models often define distinctive map regions—e.g., IC near UN, low-dispersion Mallows near ID, Euclidean and urn models as intermediary points (Szufa, 2024, Szufa et al., 4 Apr 2025, Boehmer et al., 2022).

Map colorings facilitate further empirical and algorithmic studies, such as mapping computational hardness of winner determination, cohesiveness (e.g., $\ell$-cohesive groups in approval maps), Condorcet winner frequency, or approximation ratios in multiwinner rule heuristics (Szufa et al., 2022, Szufa et al., 4 Apr 2025).

6. Extensions: Varying Sizes, Incomplete Ballots, and Feature Maps

Faliszewski et al. extend the standard framework to handle varying numbers of candidates, voters, and top-truncated ballots. The extended positionwise distance $\hat{d}_{\rm pos}$ employs matrix "stretching" for comparison, and the DAP (Diversity, Agreement, Polarization) feature metric provides a fast, robust alternative summary (Faliszewski et al., 25 Jan 2026, Faliszewski et al., 2023).

The framework also supports applications to approval ballots (using approval-wise $\ell_1$ distance with corresponding convex hulls and clustering interpretation) (Szufa et al., 2022), and construction of “maps of preference orders” at the voter-ballot level for fine-grained internal structure analysis (Faliszewski et al., 2023).

7. Theoretical Guarantees and Practical Guidance

Rigorous propositions validate the metric properties, embedding accuracy, and nearly lossless “backlifting” from frequency-matrix skeletons back to valid elections (Szufa et al., 4 Apr 2025, Boehmer et al., 2022). The framework provides robust empirical reliability across variable dataset sizes, candidate counts, and sampling errors (Boehmer et al., 2022).

From a design perspective, the map of elections guides experimental regime selection (e.g., sampling from a Mallows model with realistic $\theta$ per survey and political data analysis), benchmarking of algorithmic performance, and diagnosis of structural “gaps” in the space for further model innovation.

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References:

• (Szufa, 2024) "Map of Elections"
• (Szufa et al., 4 Apr 2025) "Drawing a Map of Elections"
• (Szufa et al., 2022) "How to Sample Approval Elections?"
• (Boehmer et al., 2021) "Putting a Compass on the Map of Elections"
• (Boehmer et al., 2022) "Collecting, Classifying, Analyzing, and Using Real-World Elections"
• (Boehmer et al., 2022) "Understanding Distance Measures Among Elections"
• (Boehmer et al., 2022) "Expected Frequency Matrices of Elections: Computation, Geometry, and Preference Learning"
• (Faliszewski et al., 25 Jan 2026) "Distances Between Top-Truncated Elections of Different Sizes"
• (Faliszewski et al., 2023) "Diversity, Agreement, and Polarization in Elections"