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