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04_Approval / multiwinner — Bloc Approval

Multi-winner Approval with the same 0/1 ballot: the num_winners most-approved candidates win (bloc / at-large counting). Declared as voting_method: Approval_Multi_Winner with num_winners: ≥ 2. Ballot mockup: 3-seat council ballot.

Five cases, in teaching order — the last two swap the counting rule instead of the ballot, and are the runnable half of Satisfaction Approval Voting:

  1. approval_bloc_3seats_c6_b5.yaml — the plain mechanics: 3-seat council, six candidates, five voters; sum the columns, top three win (Adams, Brown, Clark). Matches the ballot mockup.
  2. approval_bloc_2seats_c4_b6.yaml — the majority sweep + priority tie-break (below).

  3. approval_bloc_4seats_c7_b12_lackner_skowron.yaml — the running example from Lackner & Skowron's Multi-Winner Voting with Approval Preferences (open access): AV ties {A,B,C,D} with {A,B,C,F}; PAV elects {A,B,C,F} outright — the committee that leaves fewer voters wholly unrepresented. Its "shadow STAR" companion runs the same profile through the STAR family (Bloc STAR / Allocated Score / SSS all seat D; only RRV recovers F, matching PAV): Lackner & Skowron — approval and its shadow STAR. Live on BetterVoting (BV27): jt6r76 results ↗ — BV's Approval engine seats A,B,C,F (its random draw broke the D/F tie for F, and it correctly flags tieBreakType: random): BV27 two-view case.

  4. SAV vs AV — disjoint committees (.yaml) — Brams & Kilgour's own example (Proposition 2): ten voters, four candidates, two seats. AV elects {Ada, Ben}, SAV elects {Cleo, Dev} — no candidate in common, from identical ballots. The four slate voters' marks were worth a whole vote each to AV and half a vote each to SAV. PAV splits the difference (one seat from each side, four-way tied), which is the proportional answer for a 40% bloc holding two seats. Live on BetterVoting (BV2271): 4hfwqd results ↗ — Approval, STAR and Ranked Robin races on the same ten voters, and all three elect Ada and Ben.

  5. SAV covers everyone AV leaves out (.yaml) — the same paper's Proposition 5: seventeen voters, three candidates, two seats. AV seats the two most-approved and strands the three Cole-only voters; SAV seats the smallest pair that represents all seventeen. The most-approved candidate of all wins no seat — every one of their supporters already has a second choice in the committee. Here PAV agrees with SAV but sequential PAV does not. Live on BetterVoting (BV2272): dr6fmg results ↗ — again Approval, STAR and Ranked Robin, and again all three land on Ash and Bree.

The flagship case, approval_bloc_2seats_c4_b6.yaml, teaches the one thing that matters about bloc counting: it is majoritarian. Six voters, four candidates, two seats — a 4-voter majority (all approve Amy, two also Ben) takes both seats; the 2-voter minority behind Cora and Doug gets nothing. Bonus lesson: Ben and Cora tie 2–2 for the last seat, and the engine's tie note shows candidate priority order breaking it for Ben:

The six ballots, as marked and as counted — note that the same Yes/No paper fills two seats as easily as one; only the count changes:

The ballots as marked — a filled Yes is a 1 in that candidate's column, a filled No a 0:

# Ballot as marked Amy Ben Cora Doug
1 A Yes/No Approval ballot — Voter 1 — majority faction: bullet-approves Amy: Amy Yes, Ben No, Cora No, Doug No. 1 0 0 0
2 A Yes/No Approval ballot — Voter 2 — majority faction: Amy and Ben: Amy Yes, Ben Yes, Cora No, Doug No. 1 1 0 0
3 A Yes/No Approval ballot — Voter 3 — majority faction: Amy and Ben: Amy Yes, Ben Yes, Cora No, Doug No. 1 1 0 0
4 A Yes/No Approval ballot — Voter 4 — majority faction: bullet-approves Amy: Amy Yes, Ben No, Cora No, Doug No. 1 0 0 0
5 A Yes/No Approval ballot — Voter 5 — minority faction: Cora and Doug: Amy No, Ben No, Cora Yes, Doug Yes. 0 0 1 1
6 A Yes/No Approval ballot — Voter 6 — minority faction: bullet-approves Cora: Amy No, Ben No, Cora Yes, Doug No. 0 0 1 0 
--- Approval Voting (2 winners) ---
 Tabulating 6 ballots (any non-zero score = approval).

Ballots:
   columns = Amy, Ben, Cora, Doug      (1 = approve; 0 = not approved)
     2 × 1,0,0,0
     2 × 1,1,0,0
     1 × 0,0,1,1
     1 × 0,0,1,0

   Amy  -- 4 (67%) -- Elected
   Ben  -- 2 (33%) -- Elected
   Cora -- 2 (33%)
   Doug -- 1 (17%)
  Note: Ben, Cora each have 2 approvals and tie for the last 1 seat.
        Candidate priority order (Ben > Cora) broke the tie: Ben elected, Cora not elected.

[Approval Distribution] (how many candidates each ballot approved)
   9 approvals across 6 ballots — average 1.5 of 4 (range 1–2).
     approved 1: 3 ballots
     approved 2: 3 ballots

[Co-Approval Matrix]
 Of the voters who approved the ROW candidate, the % who ALSO approved the COLUMN candidate.
         |  Amy   |  Ben   |  Cora  |  Doug  |
   -------------------------------------------
   Amy   |   --   |  50%   |   0%   |   0%   |
   Ben   |  100%  |   --   |   0%   |   0%   |
   Cora  |   0%   |   0%   |   --   |  50%   |
   Doug  |   0%   |   0%   |  100%  |   --   |

Winners — Approval Voting (2 winners)
  Amy, Ben

Run it yourself:

python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 04_Approval/02_Examples/multiwinner/cases/approval_bloc_2seats_c4_b6.yaml

The proportional Approval rules (SPAV, PAV, seq-Phragmén) run on the same file via the abcvoting engine — all of them break the sweep and give the minority its seat (Amy + Cora):

pip install abcvoting   # once
python 06_Other/abcvoting_tabulation_engine/abc_tabulation.py 04_Approval/02_Examples/multiwinner/cases/approval_bloc_2seats_c4_b6.yaml

Same trade-off, score-ballot edition: Bloc STAR sweeps too (Bloc STAR); the proportional STAR methods fix it (proportional STAR). Concepts: Approval — Multi-Winner.

file: README.md