BV2160 — Fishburn's Borda truncation electorate: three counts, three winners — and Borda's flips on paper

▶ Live on BetterVoting: vote · results ↗ (election r6qc8h).

7 voters, four candidates, a cyclic profile. The live races: STAR → B, Choose-One → A. The paradox this example exists for — Borda's Truncation paradox — is worked below on paper, because neither BetterVoting nor the LH engine has a Borda tabulator (Borda counts are cross-checkable with pref_voting).

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 §A5 ("Demonstrating Paradoxes Afflicting Borda's procedure"), Example 14 — adapted from Fishburn (1974: 543).

The election

Voters     Preference ordering
V1–V3      A > B > C > D
V4         B > C > A > D
V5         B > C > D > A
V6–V7      C > D > A > B

Borda, worked (the paper paradox)

Borda with k points for a top rank … 0 for unranked: A 19, B 19, C 20, D 12 → C elected. Now let V1–V3 — unhappy with C — truncate C from their ballots: A 16, B 16, C 14, D 12 — C falls to third and A/B tie for the win. Revealing less of their ballots got the truncators a result they prefer: the Truncation paradox under Borda. (Borda's vulnerability comes from unranked candidates scoring 0 — truncation is a weapon aimed at whoever you leave off.)

The live races

Pairwise, this profile is a cycle (A>B 5–2, B>C 5–2, C>A 4–3) — no Condorcet winner, and the reason there's no Ranked Robin race (BV would resolve a 3-way Copeland tie at random) and no IRV race (its first elimination is a random B/C tie at 2 first-choices each).

Race	BetterVoting	LH engine	Agree?
STAR (5/4/2/1 map: 22/24/24/14; B and C take both finalist seats)	B (runoff 5–2)	B	✓
Choose-One (Plurality)	A (3 of 7)	A	✓
Borda (paper only)	—	C 20 → truncation → A/B 16 tie	n/a

Three counts, three answers (A, B, C) from seven ballots. Files: star yaml · plurality yaml · frozen export · mirrors: star, plurality.