Felsenthal Ex.6 — Approval can elect a Pareto-dominated candidate (LH-only)¶

Generated from felsenthal_ex6_pareto_approval.yaml — do not edit by hand. Regenerate: python STARVote_LH_tabulation_engine/tools_adam/scripts/build_yaml_pages.py.

Method: Approval Voting · 1 seat · Expected winner: Beau

Official tie-break (lot) order: Beau > Aria > Cole > Dean — consulted only if every deterministic tiebreaker stays tied (how the ladder works).

Scenario¶

LH-only reference (no BetterVoting election ON PURPOSE: the paradox turns on a RANDOM Aria/Beau tie, and a random BV result can't be frozen — compare the RR tiebreak study). 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 A3, Example 6 — due to Felsenthal & Maoz (1988: 123, Example 4). 3 voters, four candidates, rankings: Aria>Beau>Cole>Dean; Cole>Aria>Beau>Dean; Dean>Aria>Beau>Cole. EVERY voter prefers Aria to Beau (Aria Pareto-dominates Beau), and Aria is the Condorcet winner. But if each voter approves their top THREE, the approval totals are Aria 3, Beau 3, Cole 2, Dean 1 — a tie between Aria and Beau. Broken randomly, there is a 0.5 chance BEAU is elected: a Pareto-dominated candidate can win under Approval. This file pins the lot order adversarially (Beau first) to exhibit exactly that branch; the companion felsenthal_ex6_ranked_robin.yaml shows the same voters' rankings electing Aria.

Ballots¶

Row 1 = candidate names; each later row is one voter's approvals (1 = approve, 0/blank = not approved).

Aria,Beau,Cole,Dean
1,1,1,0
1,1,1,0
1,1,0,1

What the engine says¶

Full report from the _tabulated mirror (regenerated on every run; every analysis forced on):

--- Approval Voting (single winner) ---
 Tabulating 3 ballots (any non-zero score = approval).

Ballots:
   columns = Aria, Beau, Cole, Dean      (1 = approve; 0 = not approved)
     2 × 1,1,1,0
     1 × 1,1,0,1

   Beau -- 3 (100%) -- Elected
   Aria -- 3 (100%)
   Cole -- 2 (67%)
   Dean -- 1 (33%)
  Note: Beau, Aria each have 3 approvals and tie for the last 1 seat.
        Candidate priority order (Beau > Aria) broke the tie: Beau elected, Aria not elected.

[Approval Distribution] (how many candidates each ballot approved)
   9 approvals across 3 ballots — average 3.0 of 4 (range 3–3).
     approved 3: 3 ballots

[Co-Approval Matrix]
 Of the voters who approved the ROW candidate, the % who ALSO approved the COLUMN candidate.
         |  Beau  |  Aria  |  Cole  |  Dean  |
   -------------------------------------------
   Beau  |   --   |  100%  |  67%   |  33%   |
   Aria  |  100%  |   --   |  67%   |  33%   |
   Cole  |  100%  |  100%  |   --   |   0%   |
   Dean  |  100%  |  100%  |   0%   |   --   |

Winner — Approval Voting (single winner)
  Beau

Run it yourself:

python STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/felsenthal_paradoxes/cases/felsenthal_ex6_pareto_approval.yaml

See also¶