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BV2159 — Many IRV pathologies in one election — Brams' 21-voter sampler

Method: RCV-IRV (Instant Runoff) · 1 seat · Expected winner: B · full count →

▶ Live on BetterVoting: vote · results ↗ (election f4cjpy, Test ID BV2159; STAR is race 1, and every race matches the LH tabulation below).

Level 301 · the sampler platter. Steven Brams' famous example (Notices of the AMS, 1982; via rangevoting.org §12) packs several IRV anomalies into one 21-voter election. IRV elects B, but G is the Condorcet winner — and the same ballots demonstrate a no-show paradox, a truncation incentive, favorite-betrayal, and non-monotonicity.

A ranked-ballot case (no scores invented). → fairness test: Reading these fairly — the test for an honest "whoops" · README · (the repo's standalone monotonicity demo: monotonicity_irv_before.yaml).

The ballots (21 voters)

7 : B>G>N>F
6 : G>B>N>F
5 : N>G>B>F
3 : F>N>G>B

Source: bv2159_f4cjpy_brams_irv_pathologies.yaml.

G beats everyone — and IRV still drops it

  • G is the Condorcet winner: G beats B 14–7, beats N 13–8, beats F 18–3.
  • IRV elects B:
Round 1:  B 7 · G 6 · N 5 · F 3      → F eliminated, flows to N
Round 2:  N 8 · B 7 · G 6            → G eliminated (the Condorcet winner!), flows to B
Final:    B 13 · N 8                 → B wins

Four pathologies in one (per Brams / rangevoting.org)

  1. Condorcet failure — G would beat every opponent one-on-one, yet is eliminated.
  2. No-show paradox — if the 3 F>N>G>B voters had stayed home, G would win — an outcome they prefer to B. Voting hurt them.
  3. Truncation / favorite-betrayal incentive — those same voters could get the better (for them) result by not fully ranking, or by insincerely ranking G top — so IRV pushes voters to hide or betray their true favorite (the opposite of what it promises).
  4. Non-monotonicity — a variant where voters raise a candidate flips the result the wrong way (Brams' "raise-to-top failure").

Why it's a fair example

Sincere ballots, a published academic source, and the anomalies are logical consequences of the ballots — not artifacts of weird weights.

Reading this fairly - How common: the individual anomalies are structural (Smith's analysis: ~15% of 3-candidate IRV elections are non-monotonic); getting all four in one tiny election is a curated illustration, but each is real. - Sincere or strategic: the failures arise under sincere voting; the truncation/betrayal points are about the incentive IRV creates, clearly labeled. - What IRV does well: simple, familiar, majority-of-the-final-two. These pathologies are rarer in 2-strong-candidate races. - The symmetric whoops: see Whoops 02 (STAR misses a Condorcet winner) and Whoops 03 (no Condorcet winner exists). Tally in the balance ledger.