Coombs Ex.20 — amalgamated: both districts chose B, their union chooses A¶

Generated from coombs_ex20_amalgamated.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 amalgamation half of Felsenthal's Coombs reinforcement example. Source: Dan S. Felsenthal (2010), Appendix A7, Example 20. All 41 ballots from both districts in one election: District I's 34 (9×A>B>C, 9×B>C>A, 11×C>A>B, 5×C>B>A) plus District II's 7 (1×A>B>C, 6×B>A>C). Each district elected B on its own. Counted together the last-place tally changes hands: C is now last on 16 ballots — more than A's 14 or B's 11 — so Coombs deletes C instead of A, and the ballots C was holding lift A to a majority. A wins. Neither district wanted A; their union does. That is the reinforcement paradox, also called the inconsistency paradox, and it is the formal reason a method that cannot be summed district by district cannot be canvassed that way either. Labels are Felsenthal's own A/B/C 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 here and in District II. In District I it does not have a determinate answer at all: A and B tie on nine first places each of 34, and that arbitrary first elimination decides the winner (this engine breaks it toward B, RCTab toward C in three of six declared candidate orders). So IRV is a clean control in two of the three files, not all three — a weaker contrast than it first looked, and worth stating rather than glossing. The Coombs reinforcement failure does not lean on it either way: Coombs' own eliminations are untied in all three files, which is why the paradox is still Coombs' alone. That contrast is why the three files are worth having separately.

Ballots¶

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

9:A>B>C
9:B>C>A
11:C>A>B
5:C>B>A
1:A>B>C
6:B>A>C

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.20 — amalgamated: both districts chose B, their union chooses A
 Tabulating 41 ballots (ranked ballots).

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
C                 16  Hopeful
B                 15  Hopeful
A                 10  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
B                 25  Elected
C                 16  Rejected
A                  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 3): A, B, C
   Outside (0):        —
   More than one member ⇒ NO Condorcet winner: the top of the tournament is a
   cycle, so the strongest "candidate" is a set, not a person. Which member of
   the set should win is exactly what Minimax / Ranked Pairs / Schulze disagree
   about — see 05_Ranked_Robin/01_Learn/cycle_resolution.md.
   RCV-IRV winner B is INSIDE the Smith set. ✓
      Not guaranteed — RCV-IRV is not Smith-efficient — but it holds 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_ex20_amalgamated.yaml

Coombs Ex.20 — amalgamated: both districts chose B, their union chooses A¶

Scenario¶

Ballots¶

What the engine says¶

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

See also¶