BV2164 — Coombs deletes the Condorcet winner first

▶ Live on BetterVoting: vote · results ↗ (election xbqq8t).

33 voters, four candidates. Arlo beats every rival head-to-head — and Coombs' procedure eliminates him first, because the same broad electorate that ranks him high everywhere also ranks him dead last on 12 ballots, the most of anyone. Live races: STAR → Arlo, Choose-One → Bree, Ranked Robin → Arlo; Coombs → Bree, worked below.

Source

Dan S. Felsenthal, "Review of Paradoxes Afflicting Various Voting Procedures Where One Out of m Candidates (m ≥ 2) Must Be Elected", University of Haifa / LSE, revised 26 May 2010; Appendix §A7 ("Demonstrating Paradoxes Afflicting Coombs' Procedure" — eliminate whoever is ranked last by the most voters, unless someone holds a first-choice majority), Example 17. Full §A7 treatment: coombs.md.

The election

No. of voters    Preference ordering
     11          Arlo > Bree > Cole > Dana
     12          Bree > Cole > Dana > Arlo
      2          Bree > Arlo > Dana > Cole
      4          Cole > Arlo > Dana > Bree
      4          Dana > Arlo > Bree > Cole

Arlo is the Condorcet winner (beats Bree 19–14, Cole 17–16, Dana 17–16; social ordering Arlo > Bree > Cole > Dana).

Coombs, worked (the paper paradox)

Nobody holds a first-choice majority (11/14/4/4), so Coombs deletes the candidate ranked last by the most voters: last-place counts are Arlo 12, Dana 11, Cole 6, Bree 4 — the Condorcet winner is the first candidate deleted. With Arlo gone, Bree is ranked first by an absolute majority and wins. Felsenthal conjectures four candidates are the minimum for a Coombs Condorcet failure (with three, the pattern can't assemble). Also worked on coombs.md: Example 18's monotonicity twist — four voters raise Bree (Cole>Arlo>Dana>Bree → Cole>Arlo>Bree>Dana) and Bree loses under Coombs (Dana, then Cole are deleted instead, and Arlo wins); the live races are unchanged by that raise, so no separate election exists.

Agreement

Race	BetterVoting	LH engine	Agree?
STAR (5/4/2/1 map: 107/126/96/67 — Bree tops the scores; Arlo wins the runoff 19–14)	Arlo	Arlo	✓
Choose-One (Plurality)	Bree (14 of 33)	Bree	✓
Ranked Robin	Arlo	Arlo	✓
Coombs (paper only)	—	Bree (Arlo deleted first)	n/a

No IRV race: its first elimination is a random Cole/Dana 4–4 tie on BV. Files: star · plurality · rr · frozen export · mirrors: star, plurality, rr.