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SOURCE FILE:     monroe_pareto_dominated_c4_b24.yaml
TABULATED FILE:  monroe_pareto_dominated_c4_b24_tabulated.txt
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election_title: Monroe elects a committee every voter would trade away

scenario_description: |-
  Example 3.1 from Lackner & Skowron. Twenty-four voters, four candidates, two
  seats. Two voters approve only A; one approves A and C; one approves A and D;
  ten approve B and C; ten approve B and D.

  Monroe's rule elects {C,D}, scoring 22 - it assigns each winner an
  equal-sized constituency, and C and D each have exactly twelve voters to
  represent. But {C,D} is DOMINATED by {A,B} in the book's sense (Definition
  3.1): every one of the 24 voters approves someone in {A,B}, while only 22
  approve someone in {C,D}. Nobody is worse off under {A,B} and two voters -
  the two who approve A alone - go from no representative at all to one. Every
  voter would weakly prefer {A,B}, and Monroe elects {C,D} anyway.

  That is what Pareto efficiency forbids, and the example shows exactly WHY
  Monroe fails it: equal-sized constituencies are a constraint, and a rule that
  insists on the constraint must sometimes refuse a committee that is better
  for everybody.

  Approval Voting - the count in this file - is the control. B leads with 20;
  C and D tie at 11 for the second seat, settled here by priority order. AV
  never elects a dominated committee (Proposition A.1), which is the "strong"
  in Table 3.1's Pareto column.

  Reproduce the Monroe side:
  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, Example 3.1.

lh_only_reason: "The lesson is what MONROE'S RULE does with these ballots, and Monroe 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, and its second seat is a C/D tie that BetterVoting would break at random."

voting_method: Approval_Multi_Winner
num_winners: 2

ballots: |-
  A,B,C,D
  1,0,0,0   # 2 voters — approve A only
  1,0,0,0
  1,0,1,0   # 1 voter — approves A and C
  1,0,0,1   # 1 voter — approves A and D
  0,1,1,0   # 10 voters — approve B and C
  0,1,1,0
  0,1,1,0
  0,1,1,0
  0,1,1,0
  0,1,1,0
  0,1,1,0
  0,1,1,0
  0,1,1,0
  0,1,1,0
  0,1,0,1   # 10 voters — approve B and D
  0,1,0,1
  0,1,0,1
  0,1,0,1
  0,1,0,1
  0,1,0,1
  0,1,0,1
  0,1,0,1
  0,1,0,1
  0,1,0,1

expected_winners:
  - B
  - C

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TABULATION RESULTS
======================================================================

--- Approval Voting (2 winners) ---
 Tabulating 24 ballots (any non-zero score = approval).

Ballots:
   columns = A, B, C, D      (1 = approve; 0 = not approved)
     2 × 1,0,0,0
     1 × 1,0,1,0
     1 × 1,0,0,1
    10 × 0,1,1,0
    10 × 0,1,0,1

   B -- 20 (83%) -- Elected
   C -- 11 (46%) -- Elected
   D -- 11 (46%)
   A -- 4 (17%)
  Note: C, D each have 11 approvals and tie for the last 1 seat.
        Candidate priority order (C > D) broke the tie: C elected, D not elected.

[Approval Distribution] (how many candidates each ballot approved)
   46 approvals across 24 ballots — average 1.9 of 4 (range 1–2).
     approved 1: 2 ballots
     approved 2: 22 ballots

[Co-Approval Matrix]
 Of the voters who approved the ROW candidate, the % who ALSO approved the COLUMN candidate.
      |   B    |   C    |   D    |   A    |
   ----------------------------------------
   B  |   --   |  50%   |  50%   |   0%   |
   C  |  91%   |   --   |   0%   |   9%   |
   D  |  91%   |   0%   |   --   |   9%   |
   A  |   0%   |  25%   |  25%   |   --   |

Winners — Approval Voting (2 winners)
  B, C