election_title: "Coombs Ex.20 — amalgamated: both districts chose B, their union chooses A"
scenario_description: |-
  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.
paradoxes: [multiple-districts]
voting_method: RCV_IRV
num_winners: 1
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
expected_winners:
  - B

# file: coombs_ex20_amalgamated.yaml