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"Majority candidate" and "minority candidate" — five senses, and which one you're using

Level: 201 → 301 · for debaters

One line: majority candidate names at least five different things, minority candidate is nearly automatic once three people are running, and almost every argument that turns on these words is really an argument about which sense is meant.

Companion to the Majority Criterion, which handles the method property. This page handles the candidate label — the thing people actually say out loud in a debate.

The problem, as Hillinger poses it

Claude Hillinger raises it as a definitional aside and then walks past it, but it is the more useful half:

With several candidates, it will be hard for anyone to win an arithmetic majority.

If a "minority candidate" is one elected without more than half the vote, then in any field of three or more the winner is a minority candidate as a matter of arithmetic, not as a matter of failure. The label describes the size of the field at least as much as it describes the winner. So the literature reaches for a repair — and its favourite repair, as Hillinger notes, is the Condorcet winner: the candidate who beats every rival head-to-head. That repair has its own hole, which Condorcet knew about: the pairwise contest can cycle, and then no such candidate exists.

Neither observation settles anything, and Hillinger's own conclusion is the right one — you do not need to resolve this to see that plurality's outcomes are bad. But you do need to resolve it before saying "minority winner!" in an argument, because your opponent is allowed to ask which sense you meant.

The five senses, side by side

Sense A candidate such that… Always exists? What it actually licenses you to say
Absolute (arithmetic) majority more than half of voters mark them first No — uncommon with 3+ real candidates The strongest claim available. This is the subject of the majority criterion
Mutual majority a majority ranks some set above all others; the winner should come from that set No A set, not a candidate — weaker than it sounds, and satisfied by IRV
Condorcet / beats-all winner beats every rival head-to-head No — cycles "A majority prefers them to any one rival." Pairwise, one at a time — not one coalition of over half
Runoff majority (STAR's) preferred by more of the decided voters between the two finalists Yes, barring a tie Only a fact about the two finalists. This is what STAR's automatic runoff delivers
Majority of consensus the smallest 50%+1 clustered where opinion actually concentrates Only a rated ballot can see it The sharper critique, worked at whose majority is it?

The trap is that rows 1 and 3 both get called "the majority winner." They are different candidates in general: a beats-all winner may hold no absolute majority at all, and a candidate with an absolute majority is always the beats-all winner but is answering a different question. This library treats majority winner as an unsafe alias for the Condorcet winner for exactly that reason — the full alias table is in the naming decoder.

What "majority" means on a score ballot

On a ranked ballot the word has one obvious reading: over half the voters put X first. On a score ballot even that isn't obvious, because there is no "first" — there is a number in every box. Four readings are available, and they are not equivalent:

Reading "X has a majority" means The problem
1. Half the scale X's average score is over half the maximum (above 2.5 on a 0–5 ballot) Several candidates can hold one at once. It's a threshold on an average, not a majority of anybody
2. Share of all points X received over half of every point awarded in the election Almost nobody ever does. With three candidates X would need more points than the other two combined; the denominator grows with the field
3. Majority favorite over half of ballots score X strictly higher than every other candidate Well-defined and the right one — but need not exist, and a coarse scale produces ties at the top
4. Pairwise / runoff over half of the voters with a preference rate X above Y Always exists between two candidates (barring a tie), and says nothing about the rest of the field

The house answer: this library uses 3 whenever it says majority about a candidate — it's the reading the majority criterion is stated in — and 4 whenever it says majority about STAR's runoff, where the phrase is always "of voters with a preference" and the runoff-percentage line prints the denominator rather than leaving it to be inferred. Readings 1 and 2 are never used, and neither should be accepted from anyone else without a definition attached.

Why it matters — one election, three answers. Take the counterfactual half of the CES case, 41 voters on a 0–5 ballot:

Alice Brian Colin
average score 2.56 4.24 2.76
reading 1 — above half the scale? ✅ ✅ ✅
reading 3 — majority favorite? ✅ (21 of 41) ❌ ❌
reading 4 — wins the runoff? not a finalist ✅ ❌

Under reading 1 all three candidates have "an absolute majority" simultaneously. Under reading 3 only Alice does. Under reading 4 it's Brian. Same 41 ballots. This is why "did the majority winner win?" is not a question a score ballot answers until you say which majority you mean.

(The convention question underneath — should a score result be reported against the ballot count or the total points awarded? — is settled the same way by both engines this repo uses: ballot count. See reading a STAR report.)

Why "minority winner" is weak on its own — and what makes it strong

The weak version. Under Choose-One, the winner's share falls roughly as the field grows, because the ballot only ever counts one favourite per voter. The repo's pineapple progression runs this deliberately: the same electorate, the same universally-liked compromise, and a winner who drops from 34% → 25% → 11% as the menu grows from 3 toppings to 11. Pineapple's fan club never grew. The menu did.

So "elected with 34%" is, by itself, a statement about how many candidates ran. It is real — 11.9% of all 5,662 US national primaries in 2022 had multiple candidates and a sub-majority winner (how to read that figure honestly) — but on its own it is not yet an indictment, and a prepared opponent will say so.

The strong version is not about the winner's share at all. It is about what the rest of the ballot would have said, and it comes in two grades:

  • The winner is the Condorcet loser — loses every head-to-head.
  • The winner is the absolute loser — an outright majority ranks them dead last. This is the strongest form, and the one that needs no theory to land: most voters said anyone but this one, and got this one.

Seven ballots are enough to show it. Hillinger's own Table 1 — 3 voters a > b > c, 2 b > c > a, 2 c > b > a — elects a on 3 first choices while 4 of the 7 rank a last. That profile is already in this library, and already live, as Felsenthal Example 1 (BV2144): same seven ballots, cast as Ana / Bo / Cal, where it demonstrates four plurality paradoxes at once.

Hillinger's next move, and where this library parts from him

Having noted that the literature repairs "majority candidate" with the Condorcet winner, Hillinger argues the score-sum winner is superior to the Condorcet winner when the two differ. That claim is asserted rather than argued in the paper, and it is precisely the question STAR answers the other way — STAR's second round exists to re-impose a majoritarian check on the sum. It is claim-checked, with his own concessions and the places he cuts against STAR, on Cardinal utility.

Lean disclosure: Hillinger is advocating his own proposal (evaluative voting) against plurality, Borda, approval and IRV. Useful for the framing above — the definitional problem is real and he states it cleanly — and partisan on the verdict.

House usage

  • Say "majority" only for over-half. If you mean the beats-all candidate, say beats-all winner or Condorcet winner.
  • Say which sense when the argument turns on it. "The majority-preferred finalist" is STAR's runoff (one pairwise result); "the majority-preferred candidate" reads as over the whole field, and swapping them silently would imply STAR is Condorcet-compliant, which it is not.
  • Don't lead with the winner's percentage. Lead with what the whole ballot says — the Condorcet loser or absolute-loser fact if you have it. The percentage is the hook, not the argument.
  • A ranked ballot's "majority" is a majority of preference, not of enthusiasm — it records order, not intensity (preference vs. support).