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BV2157 — A Condorcet cycle — rock, paper, scissors: no pairwise winner exists

Method: STAR (single winner) · 1 seat · Expected winner: Rock · full count →

▶ Live on BetterVoting: vote · results ↗ (election mmcmpy, Test ID BV2157; STAR is race 1, and every race matches the LH tabulation below).

Level 301 · the paradox of voting itself. The most famous anomaly in the field, and the one that humbles the Condorcet ideal: a majority can prefer Rock over Paper, Paper over Scissors, and Scissors over Rock — a loop with no top. When that happens there is no Condorcet winner at all, so Condorcet / Ranked Robin can't name one without an extra cycle-breaking rule. This is their whoops.

→ fairness test: Reading these fairly — the test for an honest "whoops" · GLOSSARY (Condorcet cycle) · the set: README · how Condorcet methods break cycles: Cycle resolution. ↔ BV QA tracker: BV147 / BV188 (Condorcet cycle / paradox — BV: w7y3wv, 6mbvhg).

The ballots (100 voters)

Rock, Paper, Scissors
35 × 5, 3, 0      # Rock > Paper > Scissors
33 × 0, 5, 3      # Paper > Scissors > Rock
32 × 3, 0, 5      # Scissors > Rock > Paper

The cast is the lesson: it really is rock-paper-scissors. Source: bv2157_mmcmpy_condorcet_cycle_rps.yaml.

Majority rule eats its own tail

Tally the head-to-heads and you get a loop:

  • Rock beats Paper (67–33) · Paper beats Scissors (68–32) · Scissors beats Rock (65–35).

There is no "beats everyone" candidate — majority preference is intransitive. So the Condorcet question simply has no answer here:

[Condorcet Winner] No Condorcet winner (majority cycle: Rock > Paper > Scissors > Rock)
[Divergence from STAR]  STAR = Rock   ·   Approval = Paper

Scoring Round:    Rock 271 · Paper 270 · Scissors 259   (Rock & Paper advance)
Automatic Runoff: Rock beats Paper  → Rock wins (by a single point upstream)

The score methods still finish — STAR elects Rock (271 vs Paper's 270, razor thin), Approval picks Paper — but notice they don't agree either, and the Condorcet method can't even start. Full audit copy: _tabulated.

The teaching moment

"Let the majority decide" sounds airtight until you meet a cycle: every option loses to something by a majority. It's not a bug in any one method — it's a fact about aggregating preferences (the root of Arrow's theorem). Methods differ in how they cope: Condorcet/Ranked Robin need a documented cycle-breaker; STAR and Approval always return a winner, but which one now depends on intensities and thresholds, not on a clean majority.

Reading this fairly - How common: rare with large, varied electorates, but foundational — this is the Condorcet paradox itself, not a contrived weighting. Cycles get likelier with sharply divided three-way races. - Sincere or strategic: fully sincere — the cycle is in the honest preferences. - What Condorcet/Ranked Robin do well: when a Condorcet winner exists (the usual case), electing them is a very strong guarantee. Cycles are the price of asking for it. - The symmetric whoops: STAR doesn't escape clean here either — its 271-vs-270 finish is essentially a coin-flip margin, and it disagrees with Approval. No method makes a cycle not be a cycle.