Claim check — Wikipedia's "Condorcet winner criterion"¶

Wikipedia's Condorcet winner criterion article is the page most people land on first, and its "Desirable properties" section is unusually strong — these are real properties, not advocacy inventions. This page checks all of them anyway. Three hold. One is too soft on Condorcet, which is the opposite of the failure mode you expect. One defines the Smith set as the Schwartz set. And one cites a 2023 study for a finding that study does not report — we pulled the paper; the details are below.

Level: 301 · for debaters → Companions: the Condorcet hub · FairVote's article, claim-checked (the same exercise pointed the other way) · the naming decoder (where this article's vocabulary comes from, and which of it is safe) · criteria at a glance. Curriculum: 301.

Source lean, disclosed (house rule). Wikipedia is this library's neutral tier — right for notability and for criteria claims. But this specific article cluster is an exception worth naming: the Condorcet pages have seen sustained editorial reframing toward friendlier, less mathematical vocabulary ("round-robin voting" for the family, "beats-all winner" and "majority-preferred candidate" for the winner), documented in the naming decoder. Nothing below is wrong because of that. But when an article's terminology is contested, check the terminology as carefully as the claims — which is why half this page is about definitions.

Claim 1 — "Stability (no weak spoilers)"¶

"Condorcet methods are highly resistant to spoiler effects. Intuitively, this is because the only way to dislodge a Condorcet winner is by beating them, implying spoilers can exist only if there is no majority-rule winner."

Verdict: true, and it is a real advantage over STAR. Concede it cleanly.

The mechanism is exactly as described. If X beats everyone and candidate Y enters, one of three things happens: Y loses to X, and X still beats everyone (nothing changed); Y beats X and everyone else, so Y is the new Condorcet winner — Y won, Y did not spoil; or Y beats X but loses to someone else, creating a cycle, and only then does the outcome depend on the tiebreak rule. So a Condorcet method can only be spoiled by an entry that destroys the Condorcet winner altogether. That is a strong and genuine property.

Two things to add before you repeat it.

It is close to definitional. "The only way to dislodge a Condorcet winner is by beating them" follows from what a Condorcet extension is. The syllogism isn't the evidence; the empirical rarity of cycles is — and that is a separate, checkable claim. The article's own source is the right one to cite: Van Deemen, "On the empirical relevance of Condorcet's paradox," Public Choice 158(3), 2014, 311–330, putting cycles at 1–10% of races — which is the same window as this library's 90–99%-have-a-Condorcet-winner figure, independently arrived at, with the first cycle in a ranked US election found only in 2021. Lead with the frequency, not the logic. Compare the criterion-built-to-fit-the-method tell: a property that follows from a definition is worth less in debate than one that survives data.

It is not IIA, and should never be read as such. Condorcet methods do fail independence of irrelevant alternatives — Arrow guarantees that no ranked method escapes. "Stability" / "no weak spoilers" names a conditional guarantee: spoilerproof whenever a Condorcet winner exists, undefined otherwise. That is weaker than IIA and stronger than nothing, and the newer vocabulary blurs which.

Where it cuts against STAR — say this before an opponent does. STAR is not Condorcet-compliant. Its finalists are chosen by score total, not by beating everyone — so a new entrant can change which two candidates reach the runoff and flip the result with a Condorcet winner sitting right there, un-elected. Score and Approval are IIA-clean, but only on an absolute scale; real voters normalize around the field, which puts the spoiler back. The honest ordering on this one axis:

	Spoiler resistance
Score / Approval	None in principle — but only on an absolute scale; normalization reopens it
Condorcet (Ranked Robin, Ranked Pairs, Schulze)	Immune whenever a Condorcet winner exists; vulnerable only in a cycle
STAR	Removes the forced split; a narrow residual survives, and finalist selection is disturbable
RCV-IRV	Reduces the classic spoiler, adds center squeeze
Choose-One	Full

This library's spoiler page already grades it that way. Nothing to correct — but the Condorcet row is a point for Condorcet and against STAR, and pretending otherwise is the exact behaviour the fairness rule forbids.

Claim 2 — Participation¶

"One disadvantage of majority-rule methods is they can all theoretically fail the participation criterion in constructed examples. However, studies suggest this is empirically rare for modern Condorcet methods, like ranked pairs. One study surveying 306 publicly-available election datasets found no examples of participation failures for methods in the ranked pairs-minimax family."

Verdict: the theory is too soft on Condorcet, and the empirical sentence misreports its own source. Do not quote it.

This is the interesting one, and it fails in both directions at once.

"Can all theoretically fail… in constructed examples" is weaker than the truth. It is Moulin's theorem: every Condorcet extension fails the participation criterion once there are four or more candidates — the paper is literally titled "Condorcet's principle implies the no show paradox" (Hervé Moulin, Journal of Economic Theory 45(1), 1988, 53–64). Not a quirk of ingenious construction; a proven incompatibility. "Constructed examples" makes it sound like a curiosity someone had to work hard to find. → the No-Show paradox, worked.

The genuinely interesting nuance is missing. Moulin's bound is ≥4 candidates, and at exactly three the picture inverts: Brandt, Dong & Peters (2024) prove that refinements of maximin (leximin, Nanson) are uniquely immune to the no-show paradox among homogeneous Condorcet extensions. So "Condorcet methods fail participation" is true in general and false at three candidates for a specific family — sharper and more useful than what the article gives. → Condorcet-consistent choice among three candidates.

The 306-dataset sentence is wrong — we ran it down¶

The citation is Mohsin, Han, Ruan, Chen, Rossi & Xia, "Computational Complexity of Verifying the Group No-show Paradox," AAMAS 2023, pp. 2877–2879 (PDF). Its experimental paragraph says, verbatim:

"We used our algorithms on real election data on PrefLib. Out of the 306 observed preference profiles, only one profile each violates group participation for Copeland, Black's rule and STV, and we found no violations for Maximin."

Set that beside Wikipedia's gloss — "found no examples of participation failures for methods in the ranked pairs-minimax family" — and two things are wrong:

Ranked Pairs was never tested. The rules examined are Copeland, Maximin, Black's rule and STV. Ranked Pairs appears nowhere in the experiments; the "ranked pairs-minimax family" is not the study's grouping.
"No examples" is false for three of the four rules. One violating profile each for Copeland, Black's and STV. Only Maximin came back clean — so the honest sentence is "no violations for Maximin," which is a claim about one rule, not a family.

Two further caveats the gloss drops. The paper measures the group no-show paradox (a coalition abstaining, its Definition 1), not the single-voter participation criterion. And it is a three-page extended abstract from a poster session whose subject is computational complexity — the PrefLib survey is an aside, not the contribution.

What survives. The direction is still right, and the paper says so itself: group no-show paradoxes "rarely occur in real-world data," and where they did occur "there were high number of candidates and all agents had unique preference rankings" — i.e. exactly the large, strict-ranking conditions this library keeps flagging as unrepresentative of real ballots. Rare is the defensible claim; absent is not, and Moulin's theorem says the failures are always there to be found. Note too that the one Copeland violation is a violation for Ranked Robin — the Condorcet method this repo actually teaches.

Fairness both directions. STAR fails participation too. The methods that provably cannot are the score-summing ones — Score and Approval — because an added sincere ballot only ever adds support in the direction the voter wanted. So on this axis: Score / Approval > Condorcet ≈ STAR.

Fairness both directions. STAR fails participation too. The methods that provably cannot are the score-summing ones — Score and Approval — because an added sincere ballot only ever adds support in the direction the voter wanted. So on this axis: Score / Approval > Condorcet ≈ STAR, and Condorcet is not the outlier the passage's hedging implies.

Claim 3 — Smith / top cycle¶

"The Smith criterion guarantees an even stronger kind of majority rule. It says that if there is no majority-rule winner, the winner must be in the top cycle, which includes all the candidates who can beat every other candidate, either directly or indirectly. Most, but not all, Condorcet systems satisfy the top-cycle criterion."

Verdict: right conclusion, wrong definition, and it silently merges two different sets.

The definition given is not the Smith set. "All the candidates who can beat every other candidate, either directly or indirectly" describes beatpath reachability — X reaches Y if X beats Y, or beats someone who beats Y, and so on. That defines the Schwartz set (the top cycle). The Smith set is a different construction: the smallest non-empty set whose every member beats everyone outside it. The two coincide when there are no pairwise ties, and come apart when there are — Smith is the larger, more conservative set, because a tie is not a win. Using "Smith criterion" and "top-cycle criterion" as synonyms, as the passage does, is fine in casual reading and wrong the moment a tie appears.

"Most, but not all" is correct — and worth making concrete rather than leaving abstract. The split matters, because it separates two methods people treat as interchangeable:

Method	Smith-efficient?
Ranked Robin / Copeland	✅	the best win–loss records are the top of the club
Ranked Pairs, Schulze	✅
Minimax	❌	in a 4+ candidate cycle its "least bad worst loss" pick can land outside the Smith set

That is a real argument for Ranked Robin over Minimax, and it is the kind of thing "most Condorcet systems satisfy it" hides.

This one is runnable. The engine prints the Smith set on every Ranked Robin and RCV-IRV report (show_smith_set, forced on in the _tabulated mirror) — it names the set, says whether it is a lone Condorcet winner or a top cycle, and states whether the winner landed inside it. For RR the block is descriptive; for RCV-IRV it is a genuine pass/fail. → the Smith set, worked, with a four-candidate cycle where the fourth candidate is provably out of contention.

Claim 4 — Majoritarian criteria¶

"The Condorcet criterion implies the majority criterion since a candidate ranked first by a majority is clearly ranked above every other candidate by a majority."

Verdict: true, and it is the most load-bearing sentence in the article for a STAR debater — because of what follows from it by contraposition.

The implication holds, and the one-line proof is the one given. If more than half of voters rank X first, then in every head-to-head that same majority ranks X above the opponent. So X wins every pairwise, so X is the Condorcet winner, so any Condorcet extension elects X. Condorcet criterion ⟹ majority criterion.

The converse is false, and that is what people invert. Satisfying the majority criterion does not make a method Condorcet-consistent: Choose-One and RCV-IRV both satisfy the majority criterion and both fail Condorcet, IRV famously — Alaska 2022 is the textbook case. So "satisfies the majority criterion" is a much weaker badge than it sounds when a campaign puts it on a comparison chart.

Now the contraposition, which is the part worth carrying. If a method fails the majority criterion, it must fail Condorcet. STAR fails the majority criterion — therefore STAR's Condorcet failure is not an independent second demerit, it is entailed by the first. Two consequences, one in each direction: don't defend them as separate charges (they are one), and don't let an opponent bill them as two.

One caveat the sentence carries silently. It is stated for ranked ballots, where "ranked first" is unambiguous. On a score ballot it is not — a voter may give 5 to two candidates, so "ranked first by a majority" needs a convention before it means anything. That ambiguity is exactly what Equal Vote's Relaxed Majority Criterion addresses, and why the majority-criterion argument about STAR is partly a definitional dispute rather than a purely factual one.

→ Worked in full, with both failures visible on the same five ballots and the two cases that separate the strict from the shared reading: Condorcet implies majority. Also: the majority criterion hub and the runnable 51/49 electorate in electowiki's cardinal voting, claim-checked.

Claim 5 — the vocabulary itself¶

The article's framing terms — majority-rule winner, majority-preferred candidate, beats-all winner, tournament winner — all name the same candidate, and three of the four are not safe swaps. "Majority winner" collides with over-half (a beats-all winner may hold no absolute majority at all); "majority-preferred candidate" is scope-ambiguous and means something narrower throughout this library (STAR's runoff finalist); "tournament winner" collides with tournament solutions, which usually return a set. Full table and what to say instead: the naming decoder.

The short version¶

Take the stability argument — it is sound and it beats STAR. But lead with cycles are rare (the evidence) rather than only a beat dislodges a beats-all winner (the definition), and never let it be heard as IIA.
Do not quote the participation paragraph at all — rewrite it. Moulin's theorem, ≥4 candidates, provably unavoidable; then the three-candidate maximin exception, which is the part worth knowing. And the "306 datasets" sentence misreports its own source: Ranked Pairs was never tested, and Copeland, Black's and STV each had a violation. If you want the empirical point, say "one study of 306 PrefLib profiles found group no-show paradoxes in at most one profile per rule, and none for Maximin" — which is a strong finding stated accurately.
Steal Claim 4 for your own use. Condorcet ⟹ majority, so failing majority means failing Condorcet. That is why STAR's two most-cited criterion failures are one failure, not two.
Don't reuse the Smith definition. Say smallest set whose every member beats everyone outside it, and name Minimax as the family member that fails.
Concede the axis. Condorcet's spoiler resistance is better than STAR's. The STAR answer is not to dispute that — it is that the ballot carries intensity and a ranked one does not, which is a different design disagreement and the honest place to have the argument.

file: condorcet_criterion_claim_check.md¶