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Lackner & Skowron's steering committee — approval, and its shadow STAR

Example 2.1 from Lackner & Skowron, Multi-Winner Voting with Approval Preferences (SpringerBriefs, 2023, open access). An academic society elects a k = 4 steering committee from 7 candidates {a,b,c,d,e,f,g} with 12 approval ballots. This page runs that exact profile through the approval rules and its "shadow STAR" (approve = 5, else 0) to show where the STAR family lands versus the approval family.

The profile

3 × {a, b}     3 × {a, c}     2 × {a, d}     1 × {b, c, f}
1 × {e}        1 × {f}        1 × {g}

Approval counts: a = 8, b = 4, c = 4, d = 2, f = 2, e = 1, g = 1. Candidate a is on 8 of 12 ballots; the small factions ({e}, {f}, {g}, and the lone {b,c,f} voter) are the ones a majoritarian committee tends to ignore.

The committees — same voters, six rules

Rule Family Character Committee (k = 4) 4th seat
AV (Bloc Approval) approval majoritarian a, b, c, d d
PAV (book, Ex. 2.4) approval proportional a, b, c, f f
Bloc STAR STAR majoritarian A, B, C, D D
Allocated Score (STAR-PR) STAR proportional-ish A, B, C, D D
SSS (Sequentially Spent Score) STAR proportional-ish A, B, C, D D
RRV (Reweighted Range Voting) STAR proportional A, B, C, F F

Every rule agrees on the first three seats (a, b, c). The whole election turns on seat 4 — d vs f — and that single seat sorts the rules into two camps:

  • Seat D — the majoritarian rules (AV, Bloc STAR) plus the loosely proportional STAR rules (Allocated Score, SSS). D is the 4th-strongest candidate by raw support (the two {a,d} voters), so once a/b/c are in, D is the "next most popular."
  • Seat F — the fully proportional rules: the book's PAV, and in the STAR family, RRV. Seating F gives the two {*,f} voters a representative they otherwise completely lack; PAV's and RRV's objectives reward covering an unrepresented faction over adding a fourth member for the already-well-served majority.

What the shadow STAR teaches

Reading the approval ballots as STAR ballots (approve = 5, else 0) and re-running is a clean way to see that "STAR" and "proportional" are two different knobs:

  1. Bloc STAR ≈ AV. With binary ballots, STAR's runoff adds nothing over approval counting — Bloc STAR reproduces the majoritarian committee A, B, C, D exactly (with the same b/c and d/f ties, here broken by the published lot).
  2. STAR's default proportional rule is not as proportional as PAV here. Allocated Score (BetterVoting's STAR_PR) and SSS still seat D — they don't reproduce PAV. Only RRV among the STAR variants recovers A, B, C, F, matching PAV.

So the takeaway isn't "STAR = proportional" or "STAR = majoritarian" — it's that the proportional guarantee depends on the specific rule, and this tiny profile pulls the STAR-family rules apart at exactly the seat where proportionality bites.

Run them

View File
Approval original (Bloc Approval → a,b,c,d) page · approval_bloc_4seats_c7_b12_lackner_skowron.yaml
Shadow Bloc STAR → A,B,C,D page · lackner_skowron_shadow_bloc_star_c7_b12.yaml
Shadow Allocated Score (STAR-PR) → A,B,C,D page · lackner_skowron_shadow_star_pr_c7_b12.yaml
Shadow RRV → A,B,C,F (matches PAV) page · lackner_skowron_shadow_star_pr_rrv_c7_b12.yaml

The approval PAV / seqPAV / Phragmén results come from Lackner's own abcvoting — run the approval file through 06_Other/abcvoting_tabulation_engine/ (guarded on the optional abcvoting install) to reproduce the book's proportional committees.

See also