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Election simulation models — how voters and ballots are generated for testing

You can't read voters' minds, and you can't run millions of real elections under a dozen methods. So to compare methods (the VSE / Bayesian-Regret studies), you simulate: generate synthetic electorates, cast ballots, tabulate, and score the outcome — thousands of times. This page is the neutral technical menu of the models used to generate those electorates, and a standing caution: every conclusion is conditional on the model.**

→ Level: 301 · deep dive — Curriculum 301.10. Read first: Simulate utilities, not ballots — why you sample preferences (the models below) and derive ballots from them, rather than drawing random ballots directly. Companion: What makes a good winner? · What makes a voting method good?.

The prerequisites (what the math is built from)

Reading the wider literature rather than building a model? Start with the statistics you actually need — mean vs median as a method-defining choice, variance as the meaning of "divisive", and why correlated electorates make impartial culture a stress test rather than a prediction.

  • Combinatorics — N! strict rankings of N candidates (5 → 120); combinations for candidate subsets.
  • Probability distributions — Uniform (Impartial Culture), Normal/Gaussian (spatial clustering around a center), Dirichlet (random bloc mixtures that sum to 1).
  • Distance metrics — Euclidean √Σ(xᵢ−yᵢ)² (the heart of spatial models), and Kendall-Tau (number of pairwise disagreements / adjacent swaps between two rankings — used by Mallows).

Two families of model

A. "Noise" / statistical models (no geometry)

This page is the library's home for these five, so the term itself isn't a link — the deeper treatment is why you sample preferences at all, the standing caveat below, and what IC does to a method comparison. The outward citation at the end of each line is the defining source, for readers who want the original.

  • Impartial Culture (IC) — every one of the N! rankings is equally likely (each voter is an independent die-roll, prob 1/N!). Simplest and most common in the literature; "nothing up my sleeve." Caveat below on cycles — and a harder one on distortion. (Wikipedia)
  • Impartial Anonymous Culture (IAC) — every anonymous tally (vote-count profile) is equally likely, via "stars-and-bars" combinatorics. It weights unusual/close configurations more heavily, so it's favored for stress-testing edge cases and paradox rates. (Wikipedia)
  • Mallows (φ-model) — a reference ("true") ranking plus noise: the probability of a ballot decays exponentially in its Kendall-Tau distance from the reference. φ=0 → everyone votes the reference; φ=1 → pure IC; 0<φ<1 → clustered-with-noise. Good for "polarized but correlated" electorates. (Mallows 1957 — no Wikipedia article exists)
  • Plackett-Luce — each candidate has a "strength" γ; P(A ranked 1st) = γ_A / Σγ, then repeat for 2nd place among the rest. Common in machine-learning "learning-to-rank." (Wikipedia)
  • Pólya-Eggenberger (urn) models — draw a ballot for A, return it plus an extra A: "the rich get richer." Produces heavy bloc/clustering correlation naturally, modeling social influence — no geometry needed. (Wikipedia)

You don't have to write any of these samplers: all five ship in pref_voting's profile generators — the same library this repo already leans on for its independent Copeland cross-check, so it's already a dependency. (docs)

B. Spatial (geometric / ideological) models — the realistic "gold standard"

New to the idea? Start with the concept page: The spatial model — voters and candidates as points on a map (the political spectrum, the median voter theorem, and why it predicts center-squeeze). This subsection is the simulation-recipe version.

Voters and candidates are points in an N-dimensional space (1-D left–right, 2-D adds e.g. libertarian–authoritarian). A voter prefers the candidate closest to them (Euclidean distance); for scored methods, distance is converted to a utility (e.g. utility = 100 − distance × factor, or a Gaussian/decreasing function). Voters are typically drawn from a multivariate Normal (a bell curve centered on the median voter); candidates uniform or clustered. This is what the Ka-Ping Yee diagrams visualize, and what the more sophisticated VSE "hierarchical clusters" model elaborates (issue clusters, identity clusters, varying salience).

The standing caveat: results are conditional on the model

Different generators make different scenarios common or rare, which changes what a study concludes:

  • Impartial Culture produces many near-ties (all candidates ~equal quality), which some theorists (e.g. Regenwetter) call unrealistic — methods never get to show their skill at "ferreting out the best candidate." That objection is now a theorem, not just a worry: under IC, every voting rule — deterministic or randomized — has average distortion Ω(m), while drawing a winner uniformly at random and ignoring the ballots achieves ≤ m. So on this model there is no skill to show, by proof (Caragiannis & Fehrs 2024; worked through here). Treat an IC-computed method comparison as close to information-free.
  • Spatial / n-dimensional models make central candidates genuinely stronger (more realistic), but can make hard cases like Condorcet cycles nearly impossible, so cycle-resolving methods never get tested.
  • Hierarchical-cluster models sit in between and produce cycles at a plausible ~5–15% rate — which is why the headline VSE numbers use them.

So a method's measured score is only as trustworthy as the voter/strategy model behind it. This is the same evenhandedness point from the method-quality page: VSE is a strong tool, but it is not model-independent.

A correction worth flagging

A common summary says "IC makes ties and cycles vanishingly rare in large elections." Exact ties do vanish as the electorate grows — but cycles do not: under IC the probability of no Condorcet winner converges to a positive limit (≈ 8.8% for 3 candidates, rising with more candidates). And IC vs. IAC runs the opposite way often assumed: for 3 candidates in the large-electorate limit, IC ≈ 8.77% vs. IAC ≈ 6.25% (Gehrlein) — IC yields more cycles, not fewer. The right takeaway is just that the model choice materially changes paradox rates, so it must be stated.

In this repo

Our test cases are hand-crafted (small, legible elections designed to isolate one behavior), not simulation-generated — the opposite end from VSE. The two are complementary: simulations answer "how often does a method go wrong across many electorates?", while our worked cases answer "how does it go wrong, concretely, on this ballot set?" (see What makes a good winner? and the test-case catalog).

References