Coombs Ex.21 — after: two twins join for B, and B's win becomes a coin flip¶

Generated from coombs_ex21_twin_after.yaml — do not edit by hand. Regenerate: python STARVote_LH_tabulation_engine/tools_adam/scripts/build_yaml_pages.py.

Method: RCV-IRV (Instant Runoff) · 1 seat · Expected winner: B

Scenario¶

The AFTER half of Felsenthal's Coombs twin example. Source: Dan S. Felsenthal (2010), Appendix A7, Example 21 — see coombs_ex21_twin_before.yaml for the setup. The same 20 voters plus two more casting B>A>D>C, twins of a voter already there: 22 in all. Their arrival adds two first places for B and, critically, two last places for C. C is now deleted first instead of A, the transfers run differently, and the count ends with A and B TIED — no winner without a lot. Before the twins arrived, B won outright with 11 of 20. Two extra ballots ranking B FIRST turned a certain win into a coin flip. That is the weak twin paradox: not that the twins made B lose outright, but that they made B's victory strictly less certain, which is already a violation of what any voter is entitled to assume about casting a ballot for their favorite. Because the post state is a genuine tie, this case has no determinate winner under Coombs and could never be published as a BetterVoting election with a meaningful result — the same limit noted for BV2142. The expected winner recorded below is the RCV-IRV winner, which IS determinate; only the Coombs count ties. Labels are Felsenthal's own A/B/C/D so the case can be read side by side with the paper's table. Tabulated as RCV-IRV, the mirror-image count; IRV elects B before and after, so the failure is Coombs' alone. For the Coombs count, and its tie warning, run tools_adam/pref_voting_tabulation_engine/coombs_report.py.

Ballots¶

Each row is one voter's ranking, most-preferred first (N: prefix = N identical ballots).

5:A>B>D>C
5:B>C>D>A
3:B>A>D>C
6:C>A>D>B
1:C>B>A>D
2:C>B>D>A

What the engine says¶

Round-by-round Sankey diagram: each candidate's votes as a band, and where the votes of an eliminated candidate transferred to.

Where the votes went. Band thickness is votes; a band leaving an eliminated candidate lands on whoever that ballot ranked next, or on inactive if it ranked nobody who was left.

The count, step by step — the rounds and how the winner is reached:

--- RCV / Instant-Runoff Voting (single winner) ---
  Coombs Ex.21 — after: two twins join for B, and B's win becomes a coin flip
 Tabulating 22 ballots (ranked ballots).

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
C                  9  Hopeful
B                  8  Hopeful
A                  5  Rejected
D                  0  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
B                 13  Elected
C                  9  Rejected
A                  0  Rejected
D                  0  Rejected


Winner(s) — RCV / Instant-Runoff Voting (single winner)
  B

Full audit — preference matrix, Condorcet, and score distribution¶

--- Smith Set (the generalized Condorcet winner) ---
The smallest group whose every member beats every candidate outside it —
the honest answer to "who is even in contention?".
   Smith set (3 of 4): B, C, A
   Outside (1):        D
   More than one member ⇒ NO Condorcet winner: the top of the tournament is a
   group held open by draws, so the strongest "candidate" is a set, not a
   person. Some members DO beat others, but no member beats them all — a draw
   blocks the sweep. No loop closes either, so there is no cycle for Minimax /
   Ranked Pairs / Schulze to resolve: which member wins is left to the
   tiebreak, not to a cycle rule. See
   05_Ranked_Robin/01_Learn/rr_tiebreak_lh_vs_bv.md.
   Note: the Copeland leaders (B) are only part of the set — the
   win–loss table's top block understates how wide the contention is.
   RCV-IRV winner B is INSIDE the Smith set. ✓
      Not guaranteed — RCV-IRV is not Smith-efficient — but it holds here.
   Fine print: this set contains a pairwise DRAW, and a draw is enough to keep a
   candidate in the Smith set but not in the tighter Schwartz set — so Schwartz
   may be smaller here.
   More: 07_Concepts/topics/smith_set.md

Everything in one file: the _tabulated mirror (regenerated on every run; every analysis forced on).

Run it yourself:

python STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/felsenthal_paradoxes/cases/coombs_ex21_twin_after.yaml

Coombs Ex.21 — after: two twins join for B, and B's win becomes a coin flip¶

Scenario¶

Ballots¶

What the engine says¶

Full audit — preference matrix, Condorcet, and score distribution¶

See also¶