Condorcet implies majority — why STAR's two famous failures are one failure¶

Two criteria get listed side by side on every comparison chart, as though a method could fail them independently: the majority criterion and the Condorcet criterion. They are not independent. One implies the other, which means failing the first guarantees failing the second — so STAR's two most-quoted criterion failures are a single failure counted twice. This page proves it in one line, shows it on five ballots, and marks the one precondition that does all the work.

Level: 301 · for debaters → Companions: the majority criterion hub · majority & minority candidates · the Condorcet hub · criteria at a glance (the chart this page is a warning about) · STAR's honest limits. Curriculum: 301.

The implication¶

Condorcet criterion ⟹ majority criterion. Any method that always elects the beats-all winner also always elects a majority's favorite.

The proof is one sentence. Suppose more than half of voters rank X first. Then for every other candidate Y, that same majority ranks X above Y — so X beats Y head-to-head. X therefore beats everyone, X is the Condorcet winner, and a Condorcet-consistent method elects X. ∎

This is Wikipedia's own formulation and it is correct as stated. It is also the single most useful sentence on that page, for reasons the article doesn't draw out.

The converse is false — and that's where people invert it¶

Satisfying the majority criterion does not make a method Condorcet-consistent. Two methods prove it:

Method	Majority criterion	Condorcet criterion
Choose-One / Plurality	✅	❌
RCV-IRV	✅	❌
Ranked Robin, Ranked Pairs, Schulze	✅	✅
STAR	❌	❌

IRV satisfies the majority criterion trivially — a candidate with over half the first choices wins in round one, before any elimination happens. And it fails Condorcet spectacularly, which is the whole of center squeeze and of Alaska 2022.

So "satisfies the majority criterion" is a weak badge — the weakest ranked methods in common use carry it. When a comparison chart gives it a green check, that check is doing far less work than its neighbours.

The contrapositive is the payload¶

Flip the implication and you get the version that matters in an argument:

A method that fails the majority criterion must also fail the Condorcet criterion.

STAR fails the majority criterion. Therefore STAR's Condorcet failure is not a second, independent demerit — it is entailed by the first. There was never a world in which STAR failed one and passed the other.

Two consequences, pointing in opposite directions, which is why the fact is worth holding:

Don't defend them separately. They are one design decision — STAR picks its finalists by score total rather than by beating everyone — with two names in the criteria literature. A defence that answers the majority-criterion charge has already answered the Condorcet one.
Don't let an opponent bill them as two. A chart showing STAR with two red X's beside a Condorcet method's two green checks is showing you one difference twice. That is not dishonest — the criteria really are both defined, and both really do fail — but the rhetorical weight of "fails two criteria" overstates the underlying disagreement.

Watch it happen on five ballots¶

BV95b is the smallest election that shows both failures at once. Three of five voters make Ada their unique favorite — a strict majority — and Ada also beats both rivals head-to-head:

[Divergence from STAR]
  STAR                   = Bruno
  Choose-One (Plurality) = Ada   (differs from STAR)
  RCV-IRV                = Ada   (differs from STAR)
  RCV-RR (Condorcet)     = Ada   (differs from STAR)
  Note: 2 of 5 ballots (40%) 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) sides with RCV-IRV, so STAR is the outlier
        here — STAR need not elect the Condorcet candidate.
  Full round-by-round reports (generated for review):
  RCV-IRV rounds: cases_tabulated/bv95b_7pdq3r_favorite_loses_two_rivals_RCV-IRV_tabulated.txt
  RCV-RR round-robin: cases_tabulated/bv95b_7pdq3r_favorite_loses_two_rivals_RCV-RR_tabulated.txt

--- STAR Voting Method (single winner) ---

[STAR Voting]
 Tabulating 5 ballots.
Count × Ada,Bruno,Cleo
    3 ×   5,    4,   3
    2 ×   0,    5,   5

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Bruno         -- 22 -- First place
   Cleo          -- 19 -- Second place
   Ada           -- 15
 Bruno and Cleo advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Bruno         -- 3 -- First place
   Cleo          -- 0
   Equal Support -- 2
 Bruno wins.
   Runoff math:
     5  ballots cast
   − 2  Equal Support (no preference between the two finalists)
     ─
     3  voters with a preference  (majority = 2)
           Bruno 3 (100%)  ·  Cleo 0 (0%)

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Bruno

Read the two failures off the same five rows. Majority: Ada is scored strictly highest by 3 of 5 voters and does not win. Condorcet: Ada beats Bruno 3–2 and Cleo 3–2, and does not win. Same ballots, same losing candidate, same cause — Ada's 15 points are beaten by Bruno's 22 in the scoring round, so Ada never reaches the runoff she would have won. One mechanism, two criterion names.

The precondition that does all the work¶

The proof needs "ranked first by a majority" to mean strictly above every other candidate. On a ranked ballot that is automatic. On a score ballot it is not — a voter may give 5 to two candidates at once, and then that voter's "first choice" is shared and contributes no pairwise preference between them.

This is not a technicality. It is the difference between the two cases in this folder:

	Majority's top rating	Strict majority favorite?	STAR elects the Condorcet winner?
BV95b	3 of 5 rate Ada 5, everyone else lower	Yes — Ada	❌ No — Condorcet winner Ada, STAR elects Bruno
51/49	51 of 100 rate Alma 5 — but 3 of them also rate Celia 5	No — only 48 rank Alma strictly first	✅ Yes — Condorcet winner Celia, STAR elects Celia

In the 51/49 election the "majority" is a majority only in the shared-top-rating sense. Strip out the three voters who rate Alma and Celia equally and just 48 of 100 strictly prefer Alma — not a majority at all. So the strict majority criterion is never violated there, the implication is never triggered, and STAR duly elects the Condorcet winner. The engine confirms it: Condorcet Winner: Celia — matches the STAR winner.

That contrast is the whole argument over the Relaxed Majority Criterion, compressed into two elections. Equal Vote's position is that the strict criterion over-weights a bare 50.1% of first choices while ignoring how the other half feels; the counter-position is that a majority's favorite should win, full stop. What the pair shows is that the two sides are partly arguing about which reading of "first choice" a rated ballot licenses — a definitional question — and only partly about values. Both halves are real; keep them apart.

What this does not show¶

Keep the claim inside its bounds:

It is not an argument that STAR is fine. BV95b is a genuine double failure and Ada genuinely should have a strong claim on that seat. "It's only one failure" changes the count, not the severity.
It says nothing about frequency. The implication is a logical fact about criteria, not a statement about how often either failure occurs. On that, see Condorcet efficiency measured — STAR runs 74–99% depending on the electorate model.
It does not collapse every criterion pair. Most criteria on a comparison chart genuinely are independent. This one implication is worth knowing precisely because it is the exception that a chart's grid layout hides.
It runs the other way too. Any method advertising Condorcet-consistency gets the majority criterion for free and should not be credited twice for it either. The rule is symmetric; apply it to Ranked Robin's column as readily as to STAR's.

The short version¶

Condorcet ⟹ majority. One line: a majority's strict favorite beats everyone pairwise.
The converse fails — Plurality and IRV both satisfy the majority criterion. It's a weak badge.
Contrapositive: fail majority ⟹ fail Condorcet. STAR's two headline criterion failures are one.
Precondition: "first choice" must be strict. On a score ballot it can be shared, which is why the 51/49 election fails the loose majority criterion and satisfies both the strict one and Condorcet.
Symmetric discipline: don't double-count STAR's failure, and don't double-count Ranked Robin's pass.

file: condorcet_implies_majority.md¶