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The spatial model — voters and candidates as points on a map

The single most useful mental picture in voting theory. Put every voter and every candidate at a point — on a left–right line, or a 2-D map of issues — and assume each voter prefers whoever is closer to them. That one idea explains the political spectrum, predicts who wins under each method, and is why the word "spatial" keeps showing up in this repo's simulations. This page is the concept; the simulation menu that uses it is Election simulation models.

Level: 201 → 301 · deep dive

The picture

Imagine political opinion as a line — left on one end, right on the other. Every voter sits somewhere on it (their ideal point); so does every candidate. A voter likes a candidate to the degree they're near: closeness = agreement, distance = disagreement. Formally, a voter's utility for a candidate is (minus) the distance between their points.

  • One dimension = the familiar left–right political spectrum.
  • Two dimensions adds a second axis (say economic × social, or libertarian–authoritarian) — a map instead of a line.
  • More dimensions for more independent issues. Real electorates seem to need only a few dimensions to explain most behavior.

The classic intuition is Hotelling's ice-cream vendors: two carts on a beach each edge toward the middle to be closest to the most beachgoers. That "move to the center" pressure is the spatial model in one sentence.

The one theorem to know: the median voter

Put voters on a line and ask "is there a candidate who beats every other head-to-head?" The answer, under mild assumptions, is yes, and it's the candidate nearest the median voter — the median voter theorem (Duncan Black, 1948; central to Anthony Downs's Economic Theory of Democracy, 1957).

Why: any candidate to the median's left is beaten by one just to their right (a majority — the median plus everyone to the right — is closer to the righter one), and vice versa. So in one dimension a Condorcet winner always exists, and it's the center. That single fact drives most of what follows.

Why it matters — the model predicts how methods behave

Because the spatial model tells you who the Condorcet winner is (the center), it tells you which methods will find them and which will miss:

  • Ranked Robin / Condorcet methods elect the median candidate by construction — they're built to find the beats-all winner.
  • STAR usually elects near the median too: the broadly-liked center piles up scores and wins the runoff.
  • RCV-IRV can squeeze the center out. A centrist is many voters' second choice but few voters' first, so IRV eliminates them early — even though they'd beat everyone head-to-head. That's center squeeze, and it's fundamentally a spatial phenomenon: you can only see it clearly on the map. (Alaska 2022 is the real-world case.)
  • Plurality splits one side and hands it to the other — the spoiler effect, also easiest to see spatially.

So "which method elects the compromise vs. squeezes it" isn't a matter of opinion in the spatial model — it's geometry.

Yee diagrams — the method's fingerprint

Ka-Ping Yee turned the 2-D spatial model into pictures: fix a few candidates on the map, then color each point by who would win if the voters were centered there. Each method produces a distinctive map of winner-regions — Condorcet/STAR carve clean regions around the central candidate; IRV's map has jagged, non-monotonic patches where nudging the electorate flips the winner unexpectedly. A Yee diagram is a method's visual fingerprint, and it's the most intuitive proof that methods genuinely differ.

Why simulations lean on it (the "realistic" model)

This repo's simulations (fbc_simulation.py, star_vs_approval_divergence.py, star_vs_rr_divergence.py) offer two electorate models, and spatial is the realistic one: draw voters from a bell curve on the map, candidates as points, utility = −distance. Empirical work finds spatial models explain most real voting behavior, which is why VSE / Bayesian-regret studies use them. The contrast model — impartial culture (every preference independent and random) — has no geometry, manufactures far more paradoxes than reality, and is treated as an adversarial stress test. When a simulation here reports "spatial ~12%, impartial ~23%," that gap is the model dependence — see simulate utilities, not ballots.

Making it realistic — not all axes weigh the same

A flat spatial model (every dimension equally important, voters spread uniformly) has a surprising flaw: it makes the voting methods agree too much. The winner is usually the obvious central candidate, so Plurality, IRV, STAR, and Condorcet all pick the same person — and you'd wrongly conclude the method barely matters. Two refinements, which serious simulators use, restore realistic disagreement:

  • Weight the dimensions unequally (salience). Real ideology has one dominant axis (roughly left–right) and many minor ones. Chris Smith's models scale the variance of successive dimensions down by Zipf's law — later axes matter progressively less — which brings the method disagreement back up to a realistic 35–85%, against near-total method agreement when the axes are flat.
  • Cluster the voters (communities of interest). Instead of one uniform cloud, draw voters as a mixture of Gaussians — parties, regions, identity groups, with overlap. This is the "hierarchical clusters" model VSE uses; Smith's Mixture of Zipf Gaussians (MoZG) is a worked version (100 dimensions, clustered voters).

Measured, and two popular explanations don't survive it. dimension_weighting.py runs 100 dimensions, 101 voters, 5 candidates through four methods (Plurality, Approval, STAR, Copeland) under each weighting:

flat Zipf (variance ∝ 1/k) Zipf (sd ∝ 1/k)
effective dimension 100.0 16.5 2.5
all four methods agree 85.3% 67.0% 41.8%
mean voter-pair rank correlation +0.272 +0.265 +0.252
utilitarian leader's lead over the runner-up 27.7% 21.1% 16.3%
a Condorcet winner exists 99.7% 99.3% 99.7%

The direction holds and the effect is large. Three refinements the numbers force:

  • "Every method agrees" overstates it — flat 100-D still disagrees 14.7% of the time on a five-candidate field. Strong tendency, not collapse.
  • The usual mechanism is wrong. It gets explained as the electorate becoming near-unanimous — an obvious centre everyone likes. Voters are just as divided in both models (rank correlation +0.272 vs +0.252, unchanged). A flat model doesn't make voters agree; it makes methods agree. What actually moves is candidate separation: the utilitarian leader's lead falls from 27.7% of the field's spread to 16.3%, and methods split hairs on close races and agree on blowouts. Nor does Zipf abolish the centre — a Condorcet winner still exists ~99.5% of the time in every model here. (The "hyper-dense ball at the centre" version of the story is backwards too: a high-dimensional Gaussian concentrates on a thin shell, and the middle of the cloud is nearly empty.)
  • "Scale by Zipf" has two readings and the gap between them is most of the effect. Variance ∝ 1/k leaves an effective dimension of 16.5; sd ∝ 1/k leaves 2.5. Only the second matches the prose gloss everyone attaches to it ("one dominant axis and a few minor ones"), and the first lands halfway back to flat on every row. Say which one a number came from — the same discipline the encoding step needs one layer down.

The lesson mirrors the approval-cutoff sweep: how you set up the model decides whether the methods look identical or different. So a simulation that concludes "the methods barely differ" may just have an unrealistically flat electorate — always ask how the dimensions were weighted and whether voters were clustered. (Code: cdsmith/spatial-voting.)

The honest limits

The spatial model is a lens, not reality:

  • Valence / quality is missing. Voters also care about competence, honesty, charisma — things that aren't a "position." Pure distance ignores them.
  • How many dimensions, and which? Results can hinge on the dimensionality and on how much each axis weighs (salience). One-dimensional results (a guaranteed Condorcet winner) can break in higher dimensions — where cycles reappear.
  • It can suppress the hard cases. Precisely because a 1-D spatial model guarantees a Condorcet winner, it makes cycles nearly impossible — so a purely spatial simulation never stress-tests cycle-resolution. (Impartial culture over-produces them; the truth is between.)
  • Real ballots aren't utilities. Distance gives a utility; turning it into a ballot (min-max scores, an approval cutoff, a ranking) is a separate modeling step — the one that actually drives method differences.

So: use the map to build intuition and to predict, then check the prediction against countable elections and against a second, adversarial model. The geometry is a guide, not a verdict.