BV2151 — Felsenthal Example 4 (2 of 2): two supporters stay home, and their side does better

▶ Live on BetterVoting: vote · results ↗ (election 97hbpw).

The same election as BV2150 with one change, ceteris paribus: two of the four Andy>Beth>Carl voters don't participate. Result: the runoff procedure now elects Beth — the abstainers' second choice — instead of Carl, their last. Not voting served them better than voting. That is the No-Show paradox, 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 4 (continued).

The one changed datum

No. of voters    BV2150 (all vote)       BV2151 (two stay home)
      4 → 2      Andy > Beth > Carl      Andy > Beth > Carl   (two abstain)
      3          Beth > Carl > Andy      Beth > Carl > Andy
      1          Carl > Andy > Beth      Carl > Andy > Beth
      3          Carl > Beth > Andy      Carl > Beth > Andy

First choices go from Andy 4, Beth 3, Carl 4 to Andy 2, Beth 3, Carl 4. The runoff procedure (= IRV for three candidates) now deletes Andy instead of Beth — and Beth beats Carl 5–4. In BV2150 the same four voters' full turnout deleted Beth and elected Carl 7–4.

Two paradoxes, one pair

No-Show (BV2150 → BV2151): removing two sincere Andy>Beth>Carl ballots improves the outcome for exactly those voters — from their last choice to their second. Their ballots' first-choice weight propped Andy up just enough to get Beth (their real fallback, and the Condorcet winner) eliminated.

Twin, weak form (BV2151 → BV2150): read the other direction. Start here, with two Andy>Beth>Carl voters. Two twins — voters with the identical ordering — join the electorate. One would expect reinforcement of their shared preference; instead the twins' arrival elects Carl, the group's common worst choice.

Both are elimination-order diseases, and both vanish on the same ballots under pairwise or score counting: Ranked Robin and STAR elect Beth in both electorates (Beth is the Condorcet winner both times — 6–5/7–4 with 11 voters, 6–3/5–4 with 9). Teaching page: no_show.md. (Condorcet methods are not participation-proof in general — Moulin's theorem — but this electorate doesn't trigger it.)

View 1 — BetterVoting

Live results: bettervoting.com/97hbpw/results ↗. BV elects Beth in all three races — matching LH, no tiebreaks.

View 2 — LH engine

Runoff/IRV race (bv2151_97hbpw_irv.yaml):

 Tabulating 9 ballots (ranked ballots).

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Carl               4  Hopeful
Beth               3  Hopeful
Andy               2  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Beth               5  Elected
Carl               4  Rejected
Andy               0  Rejected

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

Ranked Robin (bv2151_97hbpw_ranked_robin.yaml) → Beth (6–3, 5–4). STAR (bv2151_97hbpw_star.yaml): Andy 19, Beth 31, Carl 31 — Beth and Carl advance over Andy, Beth wins the runoff 5–4. Full detail: IRV mirror · RR mirror · STAR mirror.

Agreement

Election	Race	BetterVoting	LH engine	Agree?
BV2150 all vote	Runoff (IRV)	Carl	Carl	✓
BV2151 two no-shows	Runoff (IRV)	Beth	Beth	✓
BV2150 / BV2151	Ranked Robin	Beth / Beth	Beth / Beth	✓
BV2150 / BV2151	STAR	Beth / Beth	Beth / Beth	✓

Frozen export: bv2151_97hbpw_bv_export.json · Part 1: BV2150 — everyone votes.