BV2146 — Felsenthal Example 2 (2 of 2): more support makes the winner lose¶

▶ Live on BetterVoting: vote · results ↗ (election krk2px).

The same election as BV2145, with exactly one change: two voters raise Ben from Cleo>Ben>Ada to Ben>Cleo>Ada. Strictly more support for Ben, everything else held constant — and Ben, who won part 1, now loses. That is the lack-of-monotonicity ("more-is-less") paradox, Felsenthal's model conditional paradox, confirmed live on BetterVoting.

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 §A2, Example 2 (continued): "Now suppose that, ceteris paribus, the two voters whose preference ordering is c > b > a change it to b > c > a thereby increasing b's support."

The one changed datum¶

No. of voters    BV2145 (before)         BV2146 (after)
      3          Ada  > Ben  > Cleo      Ada  > Ben  > Cleo
      2          Ada  > Cleo > Ben       Ada  > Cleo > Ben
      4          Ben  > Ada  > Cleo      Ben  > Ada  > Cleo
      2          Ben  > Cleo > Ada       Ben  > Cleo > Ada
      4          Cleo > Ada  > Ben       Cleo > Ada  > Ben
      2          Cleo > Ben  > Ada  →→→  Ben  > Cleo > Ada   (the raise)

First choices go from Ada 5, Ben 6, Cleo 6 to Ada 5, Ben 8, Cleo 4. So the runoff procedure now eliminates Cleo instead of Ada — and Ada, the Condorcet winner, beats Ben head-to-head 9–8. Ben's reward for gaining two first-choice votes is losing the election he'd won.

Why this is the monotonicity failure¶

A method is monotone if ranking a candidate higher can never hurt them. Here the raise changed the elimination order — the eliminated candidate's transfers decide the final pair, so who gets eliminated is the election. This is IRV-specific machinery: the same raise leaves Ranked Robin and STAR unmoved (both elect Ada before and after), because neither depends on an elimination order. Compare the repo's standalone pair (monotonicity set) — same paradox, different construction.

Note the composite lesson of the pair: in BV2145 the runoff robbed the Condorcet winner; here it punishes Ben for being supported more. The procedure fails both its own winner and its own logic.

View 1 — BetterVoting¶

Live results: bettervoting.com/krk2px/results ↗. BV elects Ada (IRV), Ada (Ranked Robin), Ada (STAR) — matching LH on all three, no tiebreaks.

View 2 — LH engine¶

Runoff/IRV race (bv2146_krk2px_irv.yaml):

 Tabulating 17 ballots (ranked ballots).

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Ben                8  Hopeful
Ada                5  Hopeful
Cleo               4  Rejected

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

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

Ranked Robin (bv2146_krk2px_ranked_robin.yaml): the raise only swaps Ben/Cleo on two ballots, so Ada still beats Ben 9–8 and Cleo 9–8 — Ada, unchanged. STAR (bv2146_krk2px_star.yaml): Ben rises 51 → 55, Cleo falls 49 → 45, Ada stays 53; the finalist pair changes to Ben/Ada but Ada still wins the runoff 9–8 — no more-is-less. Full detail: IRV mirror · RR mirror · STAR mirror.

Agreement¶

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

Frozen export: bv2146_krk2px_bv_export.json · Part 1: BV2145 — the Condorcet winner eliminated · Teaching page: non_monotonicity.md.