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Multi-member plurality — Block vs. Limited vs. SNTV (same electorate, three outcomes)

The three multi-member plurality methods differ in exactly one thing — how many votes each voter casts — and that one dial slides a 60/40 electorate from "majority sweeps everything" to "minority tops the poll." A clean demonstration that our engine's multi-winner Plurality (run_plurality_multi) tabulates all three correctly (it auto-detects which is which from the ballots).

The electorate

10 voters, 3 seats: a 6-voter Home majority (60%) and a 4-voter Away minority (40%). Same voters in all three races — only the number of votes per voter changes.

Method Votes / voter Winners Home : Away Read · run
Block Voting (plurality-at-large) 3 (= seats) Ada, Ben, Cal 3 : 0 — majority sweeps page · yaml
Limited Voting 2 (< seats) Ada, Ben, Uma 2 : 1 page · yaml
SNTV 1 Uma, Ada, Ben 2 : 1 — minority tops the poll page · yaml

Each method name links to its page — the ballots and the engine's full count, written up; the yaml is the same election as raw source. The fourth case in this family, the majority ceiling (yaml), is why a multi-seat winner's "%" can never reach 50 — see the folder README.

What the dial does

  • Block voting (3 votes = seats). Each Home voter votes the full Home slate, so Ada/Ben/Cal each get all 6 majority votes and beat every Away candidate (4). The 60% majority takes all three seats; the 40% minority is shut out. Majoritarian.
  • Limited voting (2 votes < seats). A party can no longer fill every seat with its own votes, so over-nominating splits the vote; disciplined parties run about as many as they can win. Home runs 2 (Ada, Ben = 6), Away runs 2 (Uma, Val = 4) → 2 : 1. Reducing votes-per-voter below the seat count is what opens the door.
  • SNTV (1 vote). Pure vote-management: the majority spreads its 6 votes (2/2/2) while the minority concentrates all 4 on Uma — so Uma finishes first and the majority's split leaves its candidates on 2 each. 2 : 1, and the minority candidate is the overall leader. Roughly proportional — the reason SNTV was used for multi-member districts (Japan, Taiwan) before STV.

The whole family is one tally — most marks fill the N seats — so the same engine handles all three; it reads votes-per-voter and labels the output Block Voting, Limited Voting, or SNTV accordingly.

Can these run on BetterVoting? Yes — as bloc Approval.

BV has no method named "Block Voting" or "Limited Voting" (its Plurality is choose-one), but that's a naming point, not a capability gap. All three are "mark k candidates, top N win" — which is exactly what BV's multi-winner Approval (Approval + num_winners) does. So each is reproducible on BV with the same 0/1 ballots and the same winners:

  • Block voting = bloc Approval where each voter approves their full N-candidate slate.
  • Limited voting = bloc Approval where each voter approves k < N.
  • SNTV = bloc Approval (or Plurality) where each voter approves exactly 1 — already BV-confirmed as the BV2134 governance Bloc Plurality race.

This set's point is the Plurality-family framing (votes-per-voter), and our engine labels the tally Block / Limited / SNTV accordingly — but the arithmetic is BV-confirmed end-to-end. Block and Limited are backed by BV election BV2135 (3x4vrv), which runs both as bloc Approval and elects exactly the same seats (Block → Ada, Ben, Cal; Limited → Ada, Ben, Uma). SNTV is BV-backed separately as the governance Bloc Plurality race. Full recipe: running_on_bettervoting.md.

▶ Block & Limited live on BetterVoting: results ↗ (election 3x4vrv, Test ID BV2135) · frozen export mmp_bv2135_3x4vrv_bv_export.json.

References

See also