election_title: Committee monotonicity (2 of 2) — add a seat, and five rules drop the winner

scenario_description: |-
  The other half of a matched pair: the SAME ten ballots as
  abc_committee_monotonicity_1seat_c3_b10.yaml, counted for TWO seats instead
  of one.

  Two voters approve only A, three approve A and C, three approve B and C, two
  approve only B. Approval counts: C 6, A 5, B 5.

  With one seat, eleven of the thirteen rules in Lackner & Skowron's Table 3.1
  elect the consensus candidate C (SAV and rev-seq-PAV pick A or B instead).
  Add a seat and the rules split:

  - Approval Voting (the count in this file) keeps C and adds a second - so the
    one-seat winner is still seated. AV is committee monotone.
  - Chamberlin-Courant, PAV, Monroe, leximax-Phragmen and MAV all elect {A,B}
    and DROP C. The candidate who won outright when there was one seat is not
    on the committee once a seat is added.

  That is what committee monotonicity (Definition 3.2) forbids, and the reason
  it matters is practical rather than aesthetic: a body that expects to grow -
  a hiring round that may fund one more post, a purchase list that may afford
  one more item - cannot use a rule whose answer to "who else?" is "start over".

  A and B tie at 5 for the second seat, so AV's own answer here is a tie the
  engine settles by priority order; both tied committees contain C, which is
  the only part the lesson rests on.

  Reproduce it: python 06_Other/abcvoting_tabulation_engine/abc_axiom_check.py

  Source: Lackner, M. & Skowron, P. (2023), "Multi-Winner Voting with Approval
  Preferences", SpringerBriefs, doi:10.1007/978-3-031-09016-5, Proposition A.2.

lh_only_reason: "The lesson is what CC / PAV / Monroe / leximax-Phragmen / MAV do across the two seat counts, and none of those rules exists on BetterVoting or in the LH engine - they are counted here by Lackner's own abcvoting library. The Approval count in this file is the control, and its second seat is a tie that BetterVoting would break at random."

voting_method: Approval_Multi_Winner
num_winners: 2

ballots: |-
  A,B,C
  1,0,0   # 2 voters — approve A only
  1,0,0
  1,0,1   # 3 voters — approve A and C
  1,0,1
  1,0,1
  0,1,1   # 3 voters — approve B and C
  0,1,1
  0,1,1
  0,1,0   # 2 voters — approve B only
  0,1,0

expected_winners:
  - A
  - C