BV2167 — Minimax elects the absolute loser: losing to everyone, narrowly¶

▶ Live on BetterVoting: vote · results ↗ (election f3dxq9).

11 voters, four candidates. D loses every single head-to-head 5–6 and a majority rank D dead last — yet the Condorcet/Minimax procedure elects D, because D's worst loss is the smallest. The sharpest "wrong winner" in Felsenthal's whole appendix. Live races: STAR → B, Choose-One → D (agreeing with Minimax, for opposite reasons).

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 §A10 ("Demonstrating Paradoxes Afflicting the Condorcet (aka Minimax or Simpson-Kramer) Procedure"), Example 29. Full §A10 treatment: minimax.md.

The election¶

No. of voters    Preference ordering
      2          D > A > C > B
      3          D > B > A > C
      3          C > B > A > D
      1          B > A > C > D
      2          A > C > B > D

Pairwise: B>A 7–4, A>C 8–3, C>B 7–4 — a top cycle among A, B, C — and D loses to each of them 5–6. D is the Condorcet loser and the absolute loser (ranked last by 6 of 11).

Minimax, worked (the paper paradox)¶

With no Condorcet winner, Minimax elects the candidate whose worst pairwise loss margin is smallest: A's worst is 7 (vs B), B's is 7 (vs C), C's is 8 (vs A) — and D's is 6 (all three losses). D elected. Losing to everyone narrowly beats winning twice and losing once badly — the Condorcet loser and absolute loser paradoxes inside a genuine Condorcet method. Contrast Copeland (Ranked Robin's rule, most pairwise wins): D has zero wins and can never top a Copeland count — the two cycle-breakers differ in kind, not taste (see cycle_resolution.md).

Agreement¶

Race	BetterVoting	LH engine	Agree?
STAR (5/4/2/1 map: 34/35/32/31; runoff B beats A 7–4)	B	B	✓
Choose-One (Plurality)	D (5 of 11)	D	✓
Minimax (paper only)	—	D (worst loss 6 vs 7/7/8)	n/a

Choose-One and Minimax converge on the absolute loser from opposite directions — one counts only D's five committed fans, the other only D's narrow defeats. STAR elects a top-cycle member. No Ranked Robin race (the A/B/C Copeland tie → BV random) and no IRV race (a random transfer tie), per the freezability rule.

Files: star yaml · plurality yaml · frozen export · mirrors: star, plurality.