BV2153 — Felsenthal Example 7: a 51% first-choice majority loses the approval count¶

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100 voters. Amos is ranked first by an absolute majority — 51 of 100 — and is the Condorcet winner. Everyone approves their top two, and Approval elects Bella (approved by all 100). Felsenthal's Absolute Majority paradox.

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 7.

The election¶

No. of voters    Ranking                    Approves (top two)
     51          Amos  > Bella > Chad       {Amos, Bella}
     48          Bella > Chad  > Amos       {Bella, Chad}
      1          Chad  > Bella > Amos       {Chad, Bella}

Approval totals: Amos 51, Bella 100, Chad 49 → Bella. The top-two cutoff makes a 51-voter majority's first-vs-second distinction invisible: their ballots support Bella exactly as strongly as Amos. Under any count that reads that distinction, Amos wins — IRV seats him in round one (51 is a majority) and Ranked Robin elects him as Condorcet winner (beats Bella 51–49, Chad 51–49). Tag: majority-failure; note this is a strategy-profile paradox — it needs every voter to approve two — where Ex.5's cutoffs were mixed.

Borda, worked (Felsenthal Ex.13 — the same electorate)¶

Felsenthal reuses exactly this 51/48/1 profile as Example 13 (Appendix §A5) to show Borda's Absolute-Majority failure: Borda points (2/1/0) give Amos 102, Bella 148, Chad 50 → Bella elected — Borda agrees with the top-two Approval count and misses the 51-vote absolute winner for the same underlying reason: both counts weigh a voter's second preference nearly (Approval: exactly) as much as their first. No Borda tabulator exists on BetterVoting or in the LH engine, so this count lives here on paper (cross-checkable with pref_voting).

The races¶

Race	BetterVoting	LH engine	Agree?
Approval (everyone approves top two)	Bella (51/100/49)	Bella	✓
IRV	Amos (round-1 majority)	Amos	✓
Ranked Robin	Amos	Amos	✓
Borda (paper only, Ex.13)	—	Bella (102/148/50)	n/a

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