BV2144 — Felsenthal Example 1: four plurality paradoxes in one 7-voter election¶

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

One tiny election — 7 voters, 3 candidates — demonstrates four paradoxes of Choose-One (Plurality) voting simultaneously, then shows STAR resolving all four. Two races on the same electorate: a Choose-One race (Ana wins) and a STAR race with the rankings mapped to 0–5 scores (Bo wins).

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 / Centre for Philosophy of Natural and Social Science, London School of Economics; revised 26 May 2010. Presented at the Leverhulme Trust Voting Power in Practice workshop, Château du Baffy, Normandy, 30 July – 2 August 2010. Appendix "Exemplifying the Various Paradoxes that Afflict the Various Procedures", §A1 ("Demonstrating Paradoxes Afflicting the Plurality Procedure"), Example 1.

The election¶

7 voters rank three candidates (Felsenthal's a, b, c → cast as Ana, Bo, Cal):

No. of voters    Preference ordering
      3          Ana > Bo > Cal
      2          Bo  > Cal > Ana
      2          Cal > Bo  > Ana

The head-to-head facts: Bo is the Condorcet winner (beats Ana 4–3 and Cal 5–2). Ana is the Condorcet loser (loses to both) and the absolute loser — a majority, 4 of 7 voters, rank Ana dead last.

The four paradoxes (Choose-One race)¶

All four are simple paradoxes in Felsenthal's classification — the surprising outcome follows from the ballots as cast — except the spoiler, which is conditional (it appears when one datum, Cal's presence, changes). Each links to its teaching page:

Paradox	What happens here	Teaching page
Condorcet winner paradox	Bo beats everyone head-to-head, yet loses	condorcet_winner_paradox.md
Condorcet loser paradox	Ana loses to everyone head-to-head, yet wins	condorcet_loser_paradox.md
Absolute loser paradox	A majority ranks Ana last, yet Ana wins	absolute_loser_paradox.md
SCC / spoiler	If Cal drops out, Bo beats Ana 4–3	spoiler_scc.md

Choose-One counts only first choices: Ana 3, Bo 2, Cal 2 → Ana wins — the candidate a majority ranked last.

The STAR race — same voters, different tabulation¶

Ranks map to scores with the house map (N=3: top=5, mid=3, bottom=1). Scores: Bo 25, Ana 19, Cal 19 — a scoring-round tie for the second finalist slot, broken by head-to-head preference (Cal beats Ana 4–3, so Cal advances). Bo wins the automatic runoff 5–2. STAR elects the Condorcet winner; all four paradoxes above vanish under the same voters' preferences.

View 1 — BetterVoting¶

Live results: bettervoting.com/mxfmhm/results ↗. BV elects Ana in the Choose-One race and Bo in the STAR race — matching the LH engine on both (BV's tiebreak perm is pinned as lot_numbers in the Plurality YAML; it only touches the moot Bo/Cal second-place tie).

View 2 — LH engine¶

Choose-One race (bv2144_mxfmhm_plurality.yaml):

[Divergence from STAR]
  STAR                   = Bo
  Choose-One (Plurality) = Ana   (differs from STAR)

--- STAR Voting Method (single winner) ---

[STAR Voting]
 Tabulating 7 ballots.
Count × Ana,Bo,Cal
    3 ×   5, 3,  1
    2 ×   1, 5,  3
    2 ×   1, 3,  5

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Bo            -- 25 -- First place
   Ana           -- 19 -- Tied for second place
   Cal           -- 19 -- Tied for second place
 Bo advances, but there's a two-way tie for second.

[STAR Voting: Scoring Round: First tiebreaker]
 The candidate preferred in the most head-to-head matchups advances.
   Cal           -- 4 -- Second place
   Ana           -- 3
   Equal Support -- 0
 Bo and Cal advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Bo            -- 5 -- First place
   Cal           -- 2
   Equal Support -- 0
 Bo wins.
   Runoff math:
     7  ballots cast
   − 0  Equal Support (no preference between the two finalists)
     ─
     7  voters with a preference  (majority = 4)
           Bo 5 (71%)  ·  Cal 2 (29%)

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Bo

(The Bo/Cal tie for the second finalist slot runs down LH's tie ladder to the pre-published lot — irrelevant to the winner: Ana beats either finalist. Full ladder in the tabulated mirror.)

STAR race (bv2144_mxfmhm_star.yaml):

[Divergence from STAR]
  STAR                   = Bo
  Choose-One (Plurality) = Ana   (differs from STAR)

--- STAR Voting Method (single winner) ---

[STAR Voting]
 Tabulating 7 ballots.
Count × Ana,Bo,Cal
    3 ×   5, 3,  1
    2 ×   1, 5,  3
    2 ×   1, 3,  5

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Bo            -- 25 -- First place
   Ana           -- 19 -- Tied for second place
   Cal           -- 19 -- Tied for second place
 Bo advances, but there's a two-way tie for second.

[STAR Voting: Scoring Round: First tiebreaker]
 The candidate preferred in the most head-to-head matchups advances.
   Cal           -- 4 -- Second place
   Ana           -- 3
   Equal Support -- 0
 Bo and Cal advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Bo            -- 5 -- First place
   Cal           -- 2
   Equal Support -- 0
 Bo wins.
   Runoff math:
     7  ballots cast
   − 0  Equal Support (no preference between the two finalists)
     ─
     7  voters with a preference  (majority = 4)
           Bo 5 (71%)  ·  Cal 2 (29%)

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Bo

Agreement¶

Race	BetterVoting	LH engine	Agree?
Choose-One (Plurality)	Ana	Ana	✓
STAR (ranks→scores)	Bo	Bo	✓

Frozen export: bv2144_mxfmhm_bv_export.json. Full engine detail: plurality mirror · STAR mirror.