BV2149 — Felsenthal Example 3 (III of III): Bruno won both districts; Alma wins the whole¶
▶ Live on BetterVoting: vote · results ↗ (election byk9v2).
Bruno wins District I (BV2147). Bruno wins District II (BV2148) — outright, with a first-round majority. Put the two electorates together, change nothing else, and plurality-with-runoff elects Alma. That is the Reinforcement paradox (multiple-districts / inconsistency paradox): the procedure disagrees with itself when electorates that agree are combined.
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, Example 3.
The three electorates¶
Felsenthal's a, b, c → cast as Alma, Bruno, Cora (same cast across the trio — it's one story):
District I (17) District II (15) Combined (32)
4 Alma > Bruno > Cora 6 Alma > Cora > Bruno 4 Alma > Bruno > Cora
1 Bruno > Alma > Cora 8 Bruno > Cora > Alma 6 Alma > Cora > Bruno
5 Bruno > Cora > Alma 1 Cora > Alma > Bruno 1 Bruno > Alma > Cora
6 Cora > Alma > Bruno 13 Bruno > Cora > Alma
1 Cora > Bruno > Alma 7 Cora > Alma > Bruno
1 Cora > Bruno > Alma
Under plurality-with-runoff (= IRV for three candidates): District I has no first-round majority (4/6/7) → Alma deleted → Bruno beats Cora 10–7. District II: Bruno wins round one (8 of 15, an absolute majority). Combined: no majority (Alma 10, Bruno 14, Cora 8) → Cora deleted → Alma beats Bruno 17–15.
Why this is the Reinforcement paradox¶
The Reinforcement postulate says: if two disjoint electorates each elect X, their union must elect X. Both districts elected Bruno; the union elects Alma. Nothing subjective is needed to call this wrong — the procedure contradicts its own two verdicts. The mechanism is the elimination step again: amalgamation flips who gets deleted (Alma in District I, Cora in the combined count), and the deletion decides the final pair. Teaching page: multiple_districts.md.
A subtlety the case pages flag: District I and the Combined electorate are Condorcet cycles (combined: Alma>Bruno 17–15, Bruno>Cora 18–14, Cora>Alma 21–11), so there's no "true winner" to appeal to — which is exactly why the paradox is stated as self-inconsistency, not as missing a Condorcet winner. (District II does have a Condorcet winner: Bruno. No Ranked Robin races in this trio — BV would resolve the cyclic ties at random, which can't be frozen; see the BV2142 caveat.)
STAR on the same ballots¶
Ranks mapped 5/3/1. Combined: Alma 88, Bruno 98, Cora 102 — Cora and Bruno advance, Bruno wins the runoff 18–14. With Districts I and II also electing Bruno (BV2147: 47/51/55, runoff 10–7; BV2148: 41/47/47, runoff 8–7), STAR is consistent across all three counts in this election — though runoff-bearing methods are not reinforcement-proof in general.
View 1 — BetterVoting¶
Live results: bettervoting.com/byk9v2/results ↗. BV elects Alma (IRV) and Bruno (STAR) — matching LH, no tiebreaks.
View 2 — LH engine¶
Runoff/IRV race (bv2149_byk9v2_irv.yaml):
Tabulating 32 ballots (ranked ballots).
ROUND 1
Candidate Votes Status
----------- ------- --------
Bruno 14 Hopeful
Alma 10 Hopeful
Cora 8 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Alma 17 Elected
Bruno 15 Rejected
Cora 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Alma
STAR race (bv2149_byk9v2_star.yaml) → Bruno (102/98/88; runoff 18–14). Full detail: IRV mirror · STAR mirror.
Agreement¶
| Election | Race | BetterVoting | LH engine | Agree? |
|---|---|---|---|---|
| BV2147 District I | Runoff (IRV) | Bruno | Bruno | ✓ |
| BV2147 District I | STAR | Bruno | Bruno | ✓ |
| BV2148 District II | Runoff (IRV) | Bruno | Bruno | ✓ |
| BV2148 District II | STAR | Bruno | Bruno | ✓ |
| BV2149 Combined | Runoff (IRV) | Alma | Alma | ✓ |
| BV2149 Combined | STAR | Bruno | Bruno | ✓ |
Frozen export: bv2149_byk9v2_bv_export.json · The districts: BV2147 · BV2148.