Skip to content

The Majority Criterion (and the Relaxed Majority Criterion)

Plain-language first, jargon and references at the very bottom. This is the "a majority scored them highest but they still lost?!" idea — the objection FairVote raises most against STAR — and Equal Vote's answer to it.

Part of the Concepts by topic hub. Closely tied to Later-No-Harm and STAR's honest limits #8.

This page is about the criterion — a property of a method. For the words — what makes someone a "majority candidate" or a "minority winner," which of the five senses you're using, what "majority" even means on a score ballot, and why the bare winning percentage is the weakest form of the argument — see Majority & minority candidates. For the Approval camp's tour of the same ground, tested section by section, see "The Majority Illusion," claim-checked — whose centerpiece example turns out to hold an absolute majority that STAR elects and Score doesn't.

And before you read STAR's ✗ here next to its ✗ on Condorcet: those are one failure, not two. The Condorcet criterion implies the majority criterion (a majority's strict favorite beats everyone head-to-head), so failing this one guarantees failing that one — there was never a version of STAR that failed one and passed the other. The proof, the five-ballot election showing both at once, and the score-ballot precondition that decides which of this folder's two cases triggers it: Condorcet implies majority.

The plain-English version

A majority just means more than half the voters. The Majority Criterion is a simple fairness rule people expect from an election:

If more than half of voters like candidate X best, X should win.

Choose-One (plurality) and RCV-IRV always obey this rule. STAR does not always obey it — and that's the thing critics point at. It sounds damning until you see when it happens, because it turns out STAR only breaks the rule in a very specific, arguably-healthy situation.

The whole idea in two tiny elections

Five voters. Three of them (a 60% majority) love Ada — they give her the top score, 5. The other two can't stand Ada (they give her 0) but love Bruno and Cleo (5 each). The only thing we change between the two elections is how generous Ada's own supporters are to everyone else.

Election A — Ada's majority backs only ONE other candidate (bv95a). They give Bruno a 4 and Cleo a 0:

Count × Ada,Bruno,Cleo
    3 × 5,4,0   ← the majority bloc
    2 × 0,5,5   ← the minority bloc

Totals: Ada 15, Bruno 22, Cleo 10
Finalists (top two by score): Bruno & Ada  →  runoff: Ada beats Bruno  →  ADA WINS ✓

Ada's majority got their favorite. Supporting one compromise candidate cost them nothing.

Election B — the same majority also backs a SECOND candidate (bv95b). Everything identical except they now give Cleo a 3 instead of 0:

Count × Ada,Bruno,Cleo
    3 × 5,4,3   ← the majority bloc
    2 × 0,5,5   ← the minority bloc

Totals: Ada 15, Bruno 22, Cleo 19
Finalists (top two by score): Bruno & Cleo  →  Ada never reaches the runoff  →  BRUNO WINS ✗

Now Ada — the top choice of a clear majority — loses. That's the Majority-Criterion failure in one move.

The point everyone misses

Look at what it took to make Ada lose: her own majority had to spread real support across two other candidates. If they'd backed only one (Election A), Ada was safe. STAR's failure needs the majority to be generous to a second rival — which is a much narrower, harder-to-trigger situation than it first sounds.

And notice why Ada loses in Election B: almost everyone likes Bruno (the majority gave him a 4, the minority a 5), while 40% of voters actively reject Ada. STAR is deciding that a broadly-liked candidate nobody hates (Bruno) is a better representative than a divisive one that 40% score at rock bottom (Ada). Whether that's a bug or a feature is the whole debate. FairVote sees "the majority was overruled." Equal Vote sees "a polarizing candidate was correctly passed over for a consensus one."

"Polarizing" in that sentence is not a mood — it is variance, and it can be computed from the ballots rather than asserted. For the arithmetic behind the word, the smallest election where spread is the only difference between two candidates, and the reason raw variance is a worse divisiveness score than it looks on a bounded ballot: Variance — the statistical name for "divisive".

A sharper version of the objection — whose majority is it?

There's a deeper critique worth knowing, because it cuts under the whole argument rather than picking a side in it. It runs like this:

A ranked ballot only ever records pairwise side-taking — A over B. So the dividing line that produces a "majority" is drawn by which candidates happen to be running, not by any pre-existing split among the voters. Change the candidates and the majority changes, even though nobody's opinions moved. On that reading, the ordinal majority is a property of the candidates; a majority of consensus — the smallest group of 50%+1 voters clustered around where opinion actually concentrates — would be a property of the voters, and only a rated ballot can see it, because it measures every candidate on one common scale instead of slicing the field two at a time.

A related trap the same argument exposes: consensus is not the same as "centrism." Most real consensus positions are extreme — that murder should be illegal, that children should be educated — not moderate. And under genuine polarization the "centre" is where almost nobody is. So "the consensus candidate" and "the moderate candidate" are different claims, and conflating them muddles this whole debate.

Three honest caveats before deploying it:

  • It concedes the practical point. The same argument grants that majority-of-preference is "a very good rule of thumb" — a candidate near the consensus will usually be on the majority's side anyway. So the critique is conceptual, not a claim that ranked methods routinely elect the wrong person. Anyone citing it as a knockout is overreading it.
  • It is an argument for Score, not for STAR. STAR's automatic runoff is itself a pairwise majority step — precisely the mechanism the critique targets. A STAR advocate cannot adopt this wholesale without arguing against STAR's own second round. (It fits pure Score, and to a degree Approval.)
  • Source and lean. The clearest write-up is Lucasvb's essay on electowiki — a personal user page, not a community-reviewed article, by a cardinal-methods advocate. Its geometric argument is genuinely illuminating; its behavioural claim (that ranked ballots make voters become more factionalist) is asserted, not evidenced. Treat the first as an idea worth thinking with and the second as a hypothesis.

The Relaxed Majority Criterion — Equal Vote's answer

Equal Vote argues the strict rule is the wrong yardstick, and proposes a gentler one that captures what actually matters. Their Relaxed Majority Criterion (RMC) asks: can a majority safely give their second choice a high-but-not-top score without accidentally sinking their favorite?

"A voting system satisfies the Relaxed Majority Criterion if a majority faction of voters can express a non-zero 'maximum support − 1' to a second choice candidate, and still guarantee that the majority faction's 'maximum supported' first choice wins." — Mark Frohnmayer, Equal Vote, 2017

In our example: Ada's majority gave Bruno a 4 (that's "max − 1", one below the top). RMC asks whether that alone can cost them Ada. It can't — it took a second supported candidate. So STAR passes the Relaxed Majority Criterion. Score and Approval voting fail it — with those methods, supporting a single second candidate can already sink your favorite. RMC is thus the line Equal Vote draws between "acceptably mild" and "genuinely broken."

Methods that pass RMC: RCV-IRV, STAR, Unified Primary, 3-2-1. Methods that fail RMC: Score, Approval.

The same fork as Later-No-Harm

Here's the deep connection — the Majority-Criterion failure and STAR's Later-No-Harm failure (honest limit #3) are the same event, described two ways.

  • Later-No-Harm (a voter's-eye view): "scoring my later choices should never hurt my favorite." STAR breaks this.
  • Majority Criterion (the electorate's view): "a majority's favorite should win." STAR can break this.

In Election B, the majority's honest 3 for Cleo (a later preference) is exactly what pushed Ada out of the finals. Their later support harmed their favorite — a Later-No-Harm failure — and because it did, the majority's favorite lost — a Majority-Criterion failure. They are one phenomenon. If Ada's voters had "bullet voted" (Ada 5, everyone else 0), both rules would hold — Ada would win. STAR's failures here happen only because voters honestly expressed support they were told was safe to give.

This is a deliberate design fork, and it also explains why STAR isn't Condorcet-compliant (honest limit #1):

keeps Later-No-Harm + Majority Criterion rewards broad / consensus support
RCV-IRV (only counts first choices) ✅ ❌ (center squeeze)
STAR (runoff weighs strength & breadth) ❌ ✅ (resists center squeeze)

You cannot have both columns at once — that's a theorem, not a preference (see references). STAR picks the right column; IRV picks the left. Everything else follows.

Which methods, and where

Method Majority Criterion Relaxed Majority Criterion Detailed treatment
Choose-One (plurality) ✅ ✅ plurality
RCV-IRV ✅ ✅ rcv_irv_vs_star
STAR ❌ (needs 2 rivals) ✅ STAR honest limits #8
Score ❌ (1 rival) ❌ range voting
Approval ❌ (1 rival) ❌ approval — worked in the Approval camp's own example

Worked demonstrations in this repo — each with its own per-election page: BV95a — favorite survives / BV95b — favorite loses, plus the Black Curtain set (a polarizing "winner" vs a hidden consensus).

And the Approval side of the table, worked too. The row above says Approval fails on one supported rival; the demonstration comes from the Approval camp itself — Hamlin & Hua's academic case for approval voting concedes a majority-criterion failure in its §4.1 and argues it's harmless. Counted, that electorate turns out to have a Condorcet winner (the majority's favorite), and the paper's own "the utility gap is negligible" defence is correct and still doesn't save the result: the case set · the claim-check. It is the cleanest side-by-side of what "1 rival" versus "2 rivals" actually costs. Both BV95 elections were also reproduced live on BetterVoting — 9m6rxr elects Ada, 7pdq3r elects Bruno — so the demo is verified on the real platform, not just the LH engine.

Formal definitions, jargon & literature

Now the precise, technical statements (for readers who want them):

Majority criterion (formal). If a strict majority of voters rank a single candidate first (or, for cardinal ballots, give one candidate a strictly higher score than every other), that candidate must be elected. For rated ballots it's often stated as the majority-favorite criterion. STAR, Score, and Approval fail it; plurality, IRV, and all Condorcet methods pass it.

Bullet voting / factional bullet voting. Casting a ballot that supports only one candidate (max for your favorite, zero for all others), forgoing honest support for compromise candidates — the strategic response the strict criterion is hypothesized to encourage.

Relaxed Majority Criterion (RMC). Frohnmayer's weakening (above): a majority may give a second candidate a score of max − 1 and still be guaranteed their max-scored favorite wins. Passed by IRV, STAR, Unified Primary, 3-2-1; failed by Score and Approval. It formalizes "how severe is the majority-criterion failure" rather than the binary "does it ever occur."

"Weak majority criterion" — a different weakening, and a weaker one. The Center for Election Science uses this phrase for the claim that Approval and Score effectively satisfy the criterion because a majority can always force their favorite by bullet voting (defined below) — max for their first choice, zero for everyone else. The premise is true. The label oversells it, and the difference from RMC is the whole point: RMC asks whether the majority can support a second choice honestly and still win; "weak" majority compliance only says they can win if they stop doing so. That is a statement about a strategy the voters have available, not about a property the method has — by the same move, almost any method "weakly" satisfies almost anything, since coordinated voters can usually force a result. It also concedes what its own camp elsewhere disputes: that Score and Approval give a majority a live reason to withhold honest support from their second choice. Distinguish the two whenever the phrase appears, and ask which one is being claimed. Worked, on the article that popularized it: "The Majority Illusion," claim-checked.

Later-No-Harm (LNH). Adding a lower preference to your ballot must not decrease the probability that a candidate you rank higher wins. Satisfied by IRV; failed by STAR/Score/Approval by construction (the runoff/total can promote your second choice over your first).

The incompatibility. No monotonic method can satisfy both Later-No-Harm and the Condorcet criterion (Woodall); more broadly, Arrow's and Gibbard–Satterthwaite's theorems guarantee every deterministic method with ≥3 candidates trades away some desirable property and has some scenario rewarding insincere voting. So "STAR fails the majority criterion" is not a unique defect — it is one face of a universal set of trade-offs.

References.

Balance. FairVote's objection is real: STAR genuinely fails a criterion many find intuitive, and "we prefer the relaxed version" is a value judgment, not a proof. The counter is equally real: the failure is mild (needs two supported rivals), the resulting winner has broad support, and no method escapes the underlying impossibility results. Judge the whole board, not one square.