electowiki's "Cardinal voting," claim-checked¶
electowiki's cardinal-voting article is the best short map of the score-ballot family anywhere — the taxonomy tables alone are worth the visit. It also contains two outright logical errors and one comparison that quietly reads two different studies off the same ruler. This page separates the three.
Level: 301 · for debaters Companions: scores vs. ranks · scale granularity can flip the winner · what makes a good winner? · criteria at a glance
Source lean, disclosed (house rule): electowiki is the election-reform community's own encyclopedia — excellent for branded method mechanics where Wikipedia is thin, advocacy-adjacent for verdicts. This article argues for cardinal methods, which is the same side this library is on. That is exactly why it's worth auditing: a source that flatters your conclusion is the one you check hardest.
What it gets right¶
- The "pure" vs. "semi-cardinal" split. Approval and Score are pure — each candidate's total depends only on the scores given to that candidate. STAR is not, because the runoff reintroduces a comparison between candidates. This is the cleanest one-line explanation of why Score and Approval satisfy the favorite betrayal criterion and STAR does not. Worth knowing — though note "semi-cardinal" is electowiki's own coinage, not standard vocabulary, so don't expect a political scientist to recognize it.
- Scale invariance. For sum, average, or median aggregation, 0–5 vs. 0–100 vs. −42–7 changes nothing; only the psychology of the voter changes. Correct.
- Gradation is what matters, not range. To carry strictly more information than a ranking, the number of score levels must exceed the number of candidates. Correct as stated — n levels are exactly enough to express any strict ranking of n candidates, so beating a ranking takes more than n. The practical consequence for this library: STAR's 0–5 is six levels, so it can express any strict ranking of up to six candidates exactly; from seven candidates on, some voters are forced into ties. That's the ceiling behind scale granularity can flip the winner.
- The majority-criterion critique, including the arithmetic — see the runnable case below.
- Its own Criticism section, which supplies the strongest rebuttal to the article's earlier IIA boast: if voters normalize (stretch their scores to fill the scale around whoever is running), then a candidate entering or leaving changes the ballots themselves, and nominal IIA compliance doesn't survive contact with real voters. This library flags the same caveat on the spoiler effect — score methods are spoilerproof on an absolute scale.
Error 1 — monotonicity is not "stricter than" IIA¶
The article argues that increasing a candidate's score can only help them, calls that monotonicity, and then says it is a stricter requirement than independence of irrelevant alternatives, so IIA "is satisfied as well."
That implication is invalid. Monotonicity and IIA are logically independent — neither entails the other. The counterexample is already in this repo: the Borda count is monotone and fails IIA spectacularly, which is the whole dark-horse story. A method can also satisfy IIA and fail monotonicity; the article itself later cites Ebert's method as a cardinal rule that fails monotonicity, which contradicts its own reasoning two sections earlier.
The conclusion is fine — Score and Approval do satisfy both. The reason is different: in a pure cardinal method each candidate's total is computed from the scores given to that candidate alone. That single property (separability) yields monotonicity and IIA independently. Neither is derived from the other, and the derivation matters, because it tells you exactly where the guarantee stops: at STAR's runoff, which compares two candidates and is therefore not separable.
Error 2 — the Bayesian Regret sign is backwards¶
The article's own argument is that the majority criterion forces a polarizing majority favorite over a consensus candidate. It then concludes that satisfying the majority criterion "reduces incentive for compromise and lowers Bayesian Regret."
Bayesian Regret is avoidable unhappiness — lower is better. If the criterion blocks the consensus winner, it raises regret. The sentence contradicts the paragraph it closes; "lowers" should read "raises."
This isn't pedantry, because the sign is the entire utilitarian case for cardinal voting. Run the article's own scenario and the direction is unmistakable — see below.
Error 3 — Bayesian Regret and VSE are not one ruler¶
The article states that Score voting has the lowest Bayesian Regret of any common single-winner method tested, notes parenthetically that STAR was never included in those studies, then reports that VSE is "an inverse of Bayesian Regret" and that STAR scores highest on VSE.
Read literally, those say two different methods are best on the same scale. What's actually going on: Bayesian Regret is Warren Smith's simulation program (rangevoting.org) and VSE is Jameson Quinn's — different electorate models, different strategy models, different candidate-generation assumptions. VSE is a normalized rescaling of regret, not its inverse, and cross-study rankings are not comparable. Both programs are also run by cardinal-method advocates, which the article doesn't say.
This library's honest version: under VSE the ordering is consistently roughly STAR ≳ Approval > RCV-IRV > Plurality (expert consensus and IRV), stated as one simulation program's result rather than a fact about the world.
Value judgments wearing lab coats¶
Two sentences deserve a flag, not a correction:
- Polarization is undesirable, therefore "forcing the electorate towards a moderate candidate should be in the general good." That's a normative claim stated as a finding — and "forcing" concedes rather more than the argument intends. This library treats the same question as genuinely contested: does a better ballot end polarization?
- Score voting is "unbiased relative to polarization if the gradation is sufficiently large." No threshold is given, so the claim can't fail.
The runnable part: the 51/49 electorate¶
The article's central argument against the majority criterion is a polarized electorate: 51% love one candidate and hate the other, 49% the reverse, and a third candidate everyone would be content with loses anyway under any majority-criterion method. It adds a specific arithmetic claim — that a Condorcet method would elect the consensus candidate, provided 4% or more of the majority expresses no preference between their favorite and the consensus choice.
We built it and ran it. The claim holds, and the threshold is tight.
100 voters. Alma is the 51% bloc's champion, Bruno the 49% bloc's, Celia is nobody's favorite and everybody's good outcome — with 3 of Alma's 51 honestly rating Alma and Celia equally.
[Divergence from STAR]
STAR = Celia
Choose-One (Plurality) = Alma (differs from STAR)
RCV-IRV = Alma (differs from STAR)
Note: 3 of 100 ballots (3%) had equal non-zero scores, so their ranks were
decided by candidate priority order. The RCV-IRV result may be an
artifact of score-to-rank tie-breaking rather than a deep
difference.
Note: Ranked Robin (RCV-RR) agrees with STAR, so RCV-IRV is the lone
outlier — the classic center-squeeze signature.
Full round-by-round reports (generated for review):
RCV-IRV rounds: cases_tabulated/majority_vs_consensus_51_49_RCV-IRV_tabulated.txt
--- STAR Voting Method (single winner) ---
[STAR Voting]
Tabulating 100 ballots.
Count × Alma,Bruno,Celia
49 × 0, 5, 4
48 × 5, 0, 4
3 × 5, 0, 5
[STAR Voting: Scoring Round]
The two highest-scoring candidates advance to the next round.
Celia -- 403 -- First place
Alma -- 255 -- Second place
Bruno -- 245
Celia and Alma advance.
[STAR Voting: Automatic Runoff Round]
The candidate preferred in the most head-to-head matchups wins.
Celia -- 49 -- First place
Alma -- 48
Equal Support -- 3
Celia wins.
Runoff math:
100 ballots cast
− 3 Equal Support (no preference between the two finalists)
───
97 voters with a preference (majority = 49)
Celia 49 (51%) · Alma 48 (49%)
[STAR Voting: Winner — STAR Voting Method (single winner)]
Celia
Alma is max-scored by an outright majority, so Choose-One and RCV-IRV — both of which satisfy the majority criterion — elect Alma, IRV in the first round. Score, STAR and Ranked Robin all elect Celia.
Drop the indifferent voters from three to two and Alma–Celia becomes a 49–49 pairwise tie, Celia is no longer the Condorcet winner, and Ranked Robin flips to Alma. So the crossover sits just above 2 voters in 100 — above 3.92% of the 51-voter majority bloc, which is what "4% or more of the majority" claims. Verified.
And Error 2 is visible in the same numbers: Celia's expressed support totals 403 against Alma's 255. Electing Alma to honor the majority criterion forfeits 148 points of stated satisfaction. That's regret going up, not down.
→ Run it: majority_vs_consensus_51_49.yaml · full report: page · _tabulated
Keep it fair¶
The case above is the utilitarian argument at its strongest, and it should not be oversold. Alma really is the sincere favorite of more than half the room, and "a majority should be able to elect its choice" is a serious democratic principle, not a bug — it is why the majority criterion exists and why May's theorem makes majority rule provably optimal at two candidates. What the case shows is the price of that principle in a three-way race, not a refutation of it. The article states the tradeoff honestly in the same paragraph where it gets the sign backwards; that's a copy-editing failure, not a bad-faith one.
Note also that this scenario is constructed to be maximally polarized — severity is high, frequency is a separate question, and real electorates that split 51/49 with a universally-liked third option are not the common case.
The short version¶
Borrow the taxonomy, the pure/semi-cardinal distinction, the gradation rule, and the 51/49 argument. Don't repeat the monotonicity-implies-IIA reasoning, don't quote the Bayesian Regret sentence, and never put "Score has the lowest BR" and "STAR has the highest VSE" in the same breath as though they settled a ranking.
See also: the majority criterion set · STAR's honest limits · criteria at a glance · how to learn about voting methods — the source-tier policy this page applies