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SOURCE FILE:     cc_pareto_dominated_c4_b2.yaml
TABULATED FILE:  cc_pareto_dominated_c4_b2_tabulated.txt
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election_title: Chamberlin-Courant can elect a Pareto-dominated committee

scenario_description: |-
  The smallest counterexample in Lackner & Skowron's book: TWO voters, four
  candidates, two seats. Voter 1 approves A, C and D; voter 2 approves B, C
  and D.

  Committee {C,D} gives EVERY voter two approved winners. Committee {A,B}
  gives every voter exactly one. So {C,D} dominates {A,B} in the book's sense
  (Definition 3.1): nobody is worse off and somebody is better off. A rule
  that can return {A,B} is electing a committee that every single voter would
  trade away.

  Approval Voting - the count in this file - gets it right: C and D lead 2-2
  against A and B at 1-1, so AV elects {C,D} outright. Chamberlin-Courant does
  not, because CC only asks whether a voter has AT LEAST ONE approved winner.
  Both voters are covered by {A,B} and both are covered by {C,D}, so CC scores
  the two committees equally at 2 and returns both - the dominated one
  included. That is Proposition A.1 in the book, and it is why CC is marked
  only WEAKLY Pareto efficient in Table 3.1 while AV is marked strong.

  Reproduce the CC side with the axiom checker:
  python 06_Other/abcvoting_tabulation_engine/abc_axiom_check.py --verbose

  Source: Lackner, M. & Skowron, P. (2023), "Multi-Winner Voting with Approval
  Preferences", SpringerBriefs, doi:10.1007/978-3-031-09016-5, Proposition A.1.

lh_only_reason: "The lesson is what CHAMBERLIN-COURANT does with these ballots, and CC exists on neither BetterVoting nor the LH engine - it is counted here by Lackner's own abcvoting library. The Approval count in this file is the control that shows AV avoiding the trap, not a BV-backable result."

voting_method: Approval_Multi_Winner
num_winners: 2

ballots: |-
  A,B,C,D
  1,0,1,1   # voter 1 — approves A, C, D
  0,1,1,1   # voter 2 — approves B, C, D

expected_winners:
  - C
  - D

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TABULATION RESULTS
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--- Approval Voting (2 winners) ---
 Tabulating 2 ballots (any non-zero score = approval).

Ballots:
   columns = A, B, C, D      (1 = approve; 0 = not approved)
     1 × 1,0,1,1
     1 × 0,1,1,1

   C -- 2 (100%) -- Elected
   D -- 2 (100%) -- Elected
   A -- 1 (50%)
   B -- 1 (50%)

[Approval Distribution] (how many candidates each ballot approved)
   6 approvals across 2 ballots — average 3.0 of 4 (range 3–3).
     approved 3: 2 ballots

[Co-Approval Matrix]
 Of the voters who approved the ROW candidate, the % who ALSO approved the COLUMN candidate.
      |   C    |   D    |   A    |   B    |
   ----------------------------------------
   C  |   --   |  100%  |  50%   |  50%   |
   D  |  100%  |   --   |  50%   |  50%   |
   A  |  100%  |  100%  |   --   |   0%   |
   B  |  100%  |  100%  |   0%   |   --   |

Winners — Approval Voting (2 winners)
  C, D