election_title: "Copeland picks one, composition-consistency demands all five (tournament solutions)"
scenario_description: >-
  The companion case: where Ranked Robin's simple count and the literature's
  favourite structural axiom flatly disagree. Three voters, five candidates:
  A>B>C>E>D, D>C>A>B>E, E>D>B>C>A. Again every head-to-head is decided, so this
  is a genuine tournament, and again there is no Condorcet winner.
  The graph has a shape worth seeing: A, B and C form a rock-paper-scissors
  3-cycle; D beats all three of them; E beats D; and A, B, C all beat E. So the
  whole thing is one big cycle at a higher level — {A,B,C} -> E -> D -> {A,B,C}.
  Because {A,B,C} is a COMPONENT (all three stand in the same relation to D and
  to E), the tournament decomposes into {A,B,C}, {D}, {E} with a 3-cycle summary.
  Composition-consistency — "choose the best from the best components" — then
  forces a solution to select ALL FIVE candidates, since nonemptiness and
  neutrality make a 3-cycle unsplittable. The uncovered, Banks and bipartisan
  sets duly return all five.
  Copeland does not. D has the most wins (3, over A, B and C) so the Copeland
  set is {D} alone, and Ranked Robin elects D outright with no tiebreak. That is
  not a bug in the engine — it is Copeland failing composition-consistency, a
  known and published limit of the rule (chapter Section 3.3.1), and the same
  arithmetic behind Ranked Robin's one clone-independence weakness, teaming
  (see 05_Ranked_Robin/01_Learn/rr_clone_independence.md).
  This is Figure 3.1 / 3.2 of Brandt, Brill & Harrenstein, "Tournament
  Solutions" (Handbook of Computational Social Choice, 2016, ch. 3) — their own
  three-voter profile, run through this repo's engines. Bare A-E labels match
  the figure on purpose. Verified two ways: LH's Ranked Robin below, and
  pref_voting's C1 module via tournament_solutions_report.py.
  LH-only: nothing here needs a live BetterVoting election, and the teaching
  point is the disagreement between two published rules, not a live tally.
voting_method: RankedRobin
num_winners: 1
ballots: |-
  A>B>C>E>D
  D>C>A>B>E
  E>D>B>C>A
expected_winners:
  - D

# file: copeland_vs_clones_c5_b3.yaml