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BV2145 — Felsenthal Example 2 (1 of 2): the runoff eliminates the Condorcet winner

▶ Live on BetterVoting: vote · results ↗ (election 6fj2kg).

17 voters, three candidates. Ada beats both rivals head-to-head — yet the plurality-with-runoff procedure eliminates her before the runoff even starts, and Ben wins. Three races on the same ranked electorate show the contrast: Runoff/IRV → Ben; Ranked Robin → Ada; STAR → Ada. Part 2 (BV2146) adds the non-monotonicity twist.

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 (Leverhulme Trust Voting Power in Practice workshop, Château du Baffy, Normandy). Appendix §A2 ("Demonstrating Paradoxes Afflicting the Plurality with Runoff Procedure"), Example 2. Felsenthal lists this procedure as vulnerable to the Condorcet winner, lack-of-monotonicity, reinforcement, no-show, twin, and SCC paradoxes; this example demonstrates the Condorcet winner, non-monotonicity (part 2), and SCC.

The election

17 voters rank three candidates (Felsenthal's a, b, c → cast as Ada, Ben, Cleo):

No. of voters    Preference ordering
      3          Ada  > Ben  > Cleo
      2          Ada  > Cleo > Ben
      4          Ben  > Ada  > Cleo
      2          Ben  > Cleo > Ada
      4          Cleo > Ada  > Ben
      2          Cleo > Ben  > Ada

Head-to-head: Ada beats Ben 9–8 and Ada beats Cleo 9–8 — Ada is the Condorcet winner, and the full social ordering is Ada > Ben > Cleo (Ben beats Cleo 9–8). But first choices are Ada 5, Ben 6, Cleo 6 — Ada is last on the first count.

With three candidates, plurality-with-runoff and RCV-IRV are the same procedure: eliminate the first-count loser, then a head-to-head between the survivors. The BV race is therefore an IRV race.

The paradoxes

The Condorcet winner paradox: the runoff eliminates Ada on first choices, and Ben beats Cleo 9–8. The electorate's pairwise favorite never reaches the runoff — the same center-squeeze mechanism as BV2137, at Felsenthal's minimal scale.

The SCC / spoiler (conditional): had Cleo withdrawn before the vote, Ada would have won the first round outright — 9 of 17 first choices, a majority. Cleo can't win, but her presence hands the election to Ben.

View 1 — BetterVoting

Live results: bettervoting.com/6fj2kg/results ↗. BV elects Ben (IRV), Ada (Ranked Robin), Ada (STAR) — matching the LH engine on all three races, no tiebreaks anywhere.

View 2 — LH engine

Runoff/IRV race (bv2145_6fj2kg_irv.yaml):

 Tabulating 17 ballots (ranked ballots).

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Ben                6  Hopeful
Cleo               6  Hopeful
Ada                5  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Ben                9  Elected
Cleo               8  Rejected
Ada                0  Rejected

Winner(s) — RCV / Instant-Runoff Voting (single winner)
  Ben

Ranked Robin (bv2145_6fj2kg_ranked_robin.yaml) elects Ada — 2 pairwise wins (9–8, 9–8), the Condorcet winner. STAR (bv2145_6fj2kg_star.yaml) with the 5/3/1 rank→score map: Ada 53, Ben 51, Cleo 49; Ada wins the automatic runoff 9–8. Full detail: IRV mirror · RR mirror · STAR mirror.

Agreement

Race BetterVoting LH engine Agree?
Runoff (IRV) Ben Ben ✓
Ranked Robin Ada Ada ✓
STAR (ranks→scores) Ada Ada ✓

Frozen export: bv2145_6fj2kg_bv_export.json · Part 2: BV2146 — the non-monotonicity flip.