BV2152 — Felsenthal Example 5: Approval misses the Condorcet winner¶

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47 voters. Bert beats both rivals head-to-head — yet the approval count elects Anna. The paradox is the approval cutoff: 31 voters rank Bert second, but 31 ballots draw the approve/don't-approve line above him.

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 ("Demonstrating the Paradoxes Afflicting the Approval Voting Procedure"), Example 5 — due to Felsenthal & Maoz (1988: 123, Example 2).

The election¶

Rankings (approved candidates in braces, per the text's parentheses):

No. of voters    Ranking                     Approves
     18          Anna  > Bert > Carla        {Anna}
      6          Bert  > Carla > Anna        {Bert, Carla}
      8          Bert  > Anna  > Carla       {Bert, Anna}
      2          Carla > Anna  > Bert        {Carla, Anna}
     13          Carla > Bert  > Anna        {Carla}

Pairwise: Bert beats Anna 27–20 and Carla 32–15 — the Condorcet winner (social ordering Bert > Anna > Carla). Approval totals: Anna 28, Bert 14, Carla 21 → Anna wins. The 18 Anna-bullet and 13 Carla-bullet voters rank Bert second but approve only their favorite, so the pairwise favorite finishes last on approvals — the Condorcet winner paradox under Approval.

The races¶

Race	BetterVoting	LH engine	Agree?
Approval (the text's sets)	Anna (28/14/21)	Anna	✓
Ranked Robin (same voters' rankings)	Bert	Bert	✓

Files: approval yaml · rr yaml · frozen export · mirrors: approval, rr. Related LH-only case: Ex.6 — Approval can elect a Pareto-dominated candidate.