election_title: Committee monotonicity (1 of 2) — one seat, and the consensus candidate takes it

scenario_description: |-
  Half of a matched pair. The SAME ten ballots are counted twice, once for one
  seat and once for two: this file is the one-seat count, its companion
  abc_committee_monotonicity_2seats_c3_b10.yaml is the two-seat count.

  Ten voters, three candidates: two 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.

  For ONE seat, eleven of the thirteen rules in Lackner & Skowron's Table 3.1
  elect the consensus candidate C - Approval Voting, CC, PAV, seq-PAV, seq-CC,
  Monroe, Greedy Monroe, seq-Phragmen, leximax-Phragmen, the Method of Equal
  Shares and MAV. C is the only candidate a majority approves, and she is
  nobody's enemy. (SAV and rev-seq-PAV instead pick A or B here, because SAV
  divides each ballot's vote among its marks: A and B score 3.5 to C's 3.)

  The interesting half is the companion file. Committee monotonicity
  (Definition 3.2) says that growing the committee from k to k+1 should ADD a
  member, never reshuffle: the one-seat winner should still be seated when
  there are two seats. Approval Voting and the sequential rules honour that.
  Chamberlin-Courant, PAV, Monroe, leximax-Phragmen and MAV all elect {A,B} for
  two seats and drop C entirely - the candidate who won outright when there was
  a single seat is not even on the committee once a seat is ADDED. That is
  Proposition A.2 in the book, and it is the ✗ in Table 3.1's committee
  monotonicity column.

  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."

voting_method: Approval
num_winners: 1

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:
  - C