Simulating proportional systems — how the studies are built¶

One line: almost every quantitative claim about how a proportional method "performs" comes out of a simulation, and the result usually depends more on how the voters were generated than on the method being tested — so this page is the parameter list, the voter models, and the honest sensitivity warnings you need to read one of those studies critically.

→ what the word promises: What "proportional" actually means · the method: STAR-PR · the simulations this repo actually runs: 06_Other/simulations/

Level: 401 · deep dive

Why the voter model is the real design choice¶

A simulation needs preferences before it can test a method, and there is no neutral way to invent them. The generator you pick determines how much structure exists in the electorate — and structure is exactly what voting methods differ on. Two studies can test the same method, run a million trials each, and disagree because one drew voters at random and the other placed them in ideological space.

Model	How preferences are generated	Structure it produces	Consequence for simulations
Impartial Culture (IC)	each voter's full ranking drawn uniformly and independently from all possible rankings	none — no systematic correlation between voters	inflates paradox and cycle rates; a deliberate worst case, not a realistic electorate
Impartial Anonymous Culture (IAC)	uniform over profiles (vote counts) rather than over each voter independently	mild shared-culture correlation	different paradox rates from IC despite the similar name — the two are routinely confused
Spatial	voters and candidates are points in an ideological space; preference follows proximity	correlated preferences resembling real electorates	more Condorcet structure; results depend heavily on dimensionality and dispersion
Clustered spatial	a spatial model with explicit voter clusters or a multimodal distribution	clear blocs, bimodality, geographic separation	changes the comparative ranking of methods; the best empirical fit of the four

One correction worth carrying, because the summary tables usually get it wrong: spatial models are often described as producing single-peaked preferences. That is guaranteed only in one dimension. In 2D or higher — the dimensionality most studies actually use — preferences are generally not single-peaked, majority cycles reappear, and the tidy Condorcet structure of the 1D case is not available. "Spatial" does not mean "cycle-free"; it means "correlated."

The model STAR's own research uses, and why¶

If you are simulating STAR specifically, there is a concrete answer rather than a menu. Wolk, Quinn & Ogren (2023) — the peer-reviewed STAR paper — settled on a clustered spatial model, and their reasoning is the clearest published statement of why the choice matters:

Impartial Culture produces too many Condorcet cycles (Tsetlin et al., 2003) — the electorate is too chaotic to resemble anything real.
Normally-distributed spatial models overcorrect, producing too few cycles (Tideman, 2020) — too tidy in the opposite direction.
A clustered spatial model sits between them, and because it is non-parametric — it admits an unbounded number of clusters rather than a number you fix in advance — it can reproduce real-world scenarios, cycles included, to whatever precision you want.

Worth flagging honestly: selecting a generator partly because it yields a realistic rate of cycles is a calibration choice, not a neutral one. It is well-argued and the alternatives are worse, but a reader should know the electorate was tuned to match reality on a property that also affects method comparisons. This is the sort of thing to check symmetrically — it would be a fair question to put to any paper, including one favoring a method you dislike.

How the clustered spatial model is built¶

Voters and candidates are points in a vector space, distributed via a hierarchical Dirichlet structure of Gaussian clusters — a CrossCat-style construction (Mansinghka et al., 2016). Three steps, each doing a job worth naming:

Weight the issue dimensions — stick-breaking Dirichlet process. Take a stick representing everything voters care about; break off a piece for the first issue, break a piece off what remains for the second, and so on. Issue weight decays exponentially on average, so a few issues dominate, many are marginal, and only finitely many dimensions need modeling. Dimensions are added until the remaining weight drops below a threshold.
Bundle dimensions into "views" — Chinese Restaurant process. Issues aren't independent; taxes, regulation and debt travel together. Each issue "sits" at a table, preferring tables where similar issues already sit and occasionally starting a new one. The bundles — fiscal, social, and so on — are discovered, not declared in advance.
Cluster voters separately within each view — Chinese Restaurant process again. The powerful step. A voter's group membership is not one-size-fits-all: they can sit with one crowd on fiscal questions and a different crowd on social ones. Each voter cluster gets a mean and variance per dimension, and voters' ideal points are drawn normally from them.

That third step is what makes the model realistic. A voter who is free-market and socially liberal, another who is state-control and socially liberal, a third who is free-market and socially conservative — cross-cutting identities that a single left-right axis cannot represent. And it is precisely those cross-cutting cleavages that let genuine Condorcet cycles emerge on their own, rather than being injected by randomness (IC) or smoothed away (single-cluster Gaussian).

For a STAR-PR simulation specifically, this is the electorate generator to reach for, with district magnitude and the utility function as the two sweeps that will move your results most. Note what does not yet exist here: none of this repo's simulations model proportional multi-winner races, so a STAR-PR study would be new work rather than a parameter change to an existing script.

The parameters a study has to fix¶

Any of these can move a result, and papers vary in how many they report.

Electorate — number of voters (1,000–100,000; larger stabilizes results), number of candidates (typically 5–30), number of seats, and the voter-distribution model above (uniform, Gaussian, bimodal, or empirically sampled).

Candidate placement — random within the space, placed strategically by parties, or deliberately correlated / anti-correlated with voter clusters. How many parties nominate at all is itself a parameter.

Ballot generation — how spatial distance becomes a score or a ranking. The utility function (linear, quadratic, or Gaussian decay with distance) matters more than it sounds: it is what decides whether preference intensity carries real information. Plus noise and irrationality rates, truncation rate (how many voters submit partial ballots — central for STV), and the strategic-vs-sincere mix.

Method parameters — quota type (Droop, Hare, Hagenbach-Bischoff), any electoral threshold, the tie-breaking rule, STV's transfer method (weighted inclusive Gregory vs. random sample), and for scored methods the score range (0–5, 0–10).

Structure — trial count (typically 10⁴–10⁶ per condition), a fixed seed for reproducibility, sensitivity sweeps varying one parameter at a time, and whether the scenarios are Monte Carlo or adversarial.

What gets measured¶

Proportionality indices — Gallagher, Loosemore–Hanby
Voter Satisfaction Efficiency (VSE) — the most common modern summary metric
Condorcet efficiency — how often the Condorcet winner is elected
Utilitarian efficiency — total voter utility of the outcome
Representation of minorities, wasted votes, ballot exhaustion rate, strategic manipulability

The two findings that dominate¶

District magnitude beats almost everything else. Seats per district moves PR outcomes more than any other single variable. Proportionality improves sharply from 1 → 5 seats and the gains flatten after roughly 7. This is also the lever with the clearest trade-off attached, since the same increase lowers the win threshold — see what proportionality does not promise.

Spatial model × utility function is second, and it is the one that decides score-vs-rank comparisons. Scored methods can only outperform ranked ones when preference intensity carries information the ranking would discard. That is a property of the generator, not of the method: choose a utility function where intensity is flat or meaningless and STAR-family methods lose their advantage by construction. This cuts against the repo's own subject as much as for it, and is the first thing to check when a simulation reports that scored methods do especially well — or especially badly.

Reading one of these critically¶

Methodological surveys converge on the same advice:

Match model to question. IC for a neutral baseline or theoretical worst case; spatial or clustered for any claim about practical performance.
Demand sensitivity checks. Dimensions, dispersion and cluster tightness should be varied and the variation reported. Different settings can reverse which method wins.
Prefer empirical calibration — fit the model to real summary statistics where possible.
Look for more than one culture. A robust claim survives both a structured and a neutral generator. A claim reported under only one is model-dependent until shown otherwise.

The failure mode is specific and common: a normative conclusion that holds only under an unrealistic impartial culture, or only under one narrow spatial specification.

Where this repo sits¶

This library is mostly the opposite instrument — small hand-built elections where you can see every ballot, plus real frozen results. That is a deliberate complement to simulation, not a substitute: a simulation tells you how often something happens, a worked case tells you what it looks like and proves it can. The repo's own Monte Carlo tools live in 06_Other/simulations/ (Condorcet efficiency, favorite-betrayal rates, runoff reversals, STAR-vs-Approval and STAR-vs-Ranked-Robin divergence), and none of them currently model proportional multi-winner races — the parameters above are what a PR simulation here would have to declare.

Status note: this page is a synthesis of the methodological literature listed below, assembled as a reading aid. The individual claims have not been re-verified against each source here, and the two "dominant findings" above are reported as the literature's consensus rather than as anything this repo has measured. Treat it as a map of the terrain and a checklist for reading a study — not as evidence.

References¶

M. Diss and E. Kamwa, "Simulations in Models of Preference Aggregation," Œconomia 10(2), 279–308, 2020. doi:10.4000/oeconomia.8251
N. Boehmer et al., "Guide to Numerical Experiments on Elections in Computational Social Choice," 2024. arXiv:2402.11765
H. Nurmi, "An assessment of voting system simulations," Public Choice 73(4), 459–487, 1992. doi:10.1007/BF01789562
J. R. Chamberlin and M. X. Cohen, "Toward Applicable Social Choice Theory: A Comparison of Social Choice Functions under Spatial Model Assumptions," APSR 72(4), 1341–1356, 1978. doi:10.2307/1954543
S. Merrill, "A Comparison of Efficiency of Multicandidate Electoral Systems," AJPS 28(1), 23, 1984. doi:10.2307/2110786
F. Brandt and H. G. Seedig, "On the Discriminative Power of Tournament Solutions," 2016. doi:10.1007/978-3-319-28697-6_8
M. Jankowski and M. Tepe, "Social Heterogeneity and Choice Failure Under Condorcet and Borda," 143–166, 2017. doi:10.1007/978-3-658-16714-1_6
S. Wolk, J. H. Quinn, and M. Ogren, "STAR Voting, equality of voice, and voter satisfaction: considerations for voting method reform," Constitutional Political Economy, 2023. doi:10.1007/s10602-022-09389-3 — authors are STAR advocates; the paper is peer-reviewed, the affiliation is worth knowing.
T. Tideman and F. Plassmann, "The Source of Election Results: An Empirical Analysis of Statistical Models of Voter Behavior."
C. Song, "Three Empirical Analyses of Voting"; T. Matje, "Empirical Analyses of a Spatial Model of Voter Preferences," 2016.