multi_member_plurality — the block voting family (LH-only)¶
Two small electorates, eight cases. The first runs Block / Limited / SNTV on one 60/40 electorate (6-voter Home majority, 4-voter Away minority; 3 seats): they differ only in votes per voter, and that alone slides the result from majority sweep to minority-tops-the-poll. The second adds the rest of the family — plurality block, block approval and two-round majority block — on an electorate where the bloc that sweeps is a minority.
All three are majoritarian at-large methods — whether you want that at all, or a proportional body instead, is the decision that comes first: Electing more than one, simply. The expressive majoritarian alternative to these is Bloc STAR.
Each row links the page — the readable write-up with the ballots and the engine's full count — and the yaml you feed the engine to run it yourself.
| Method | Votes/voter | Winners | Home : Away | Read · run |
|---|---|---|---|---|
| Block Voting (plurality-at-large) | 3 | Ada, Ben, Cal | 3 : 0 | page · yaml |
| Limited Voting | 2 | Ada, Ben, Uma | 2 : 1 | page · yaml |
| SNTV | 1 | Uma, Ada, Ben | 2 : 1 (Uma leads) | page · yaml |
Plus a fourth case on the same family — the majority ceiling (yaml): with k marks per voter no candidate can exceed 1/k of the votes cast, so a unanimous candidate still shows 33.3% at 3 seats. That kills the common "the winner got under 50%, so the vote was split" reading in any multi-seat race — see How often does vote splitting actually happen?.
Second electorate — when the sweeping bloc is a minority¶
Above, a 60% majority sweeps. The sharper version of the same defect is a bloc that most voters voted against taking every seat, which happens as soon as the opposition is split. 9 voters, 3 seats, 9 candidates: Party Oak has 4 voters, Party Pine 3, and 2 voters prefer the independents. Same voters in all three rows — only the ballot rule changes.
| Rule | Marks per voter | Winners | Result | Read · run |
|---|---|---|---|---|
| Plurality block voting | 3 | Alma, Bram, Cleo | Oak sweeps 3 : 0 on 44% | page · yaml |
| Block approval voting | uncapped | Dev, Enzo, Finn | Pine sweeps — one round | page · yaml |
| Majority block voting, round 2 | 3 | Dev, Enzo, Finn | Pine sweeps with a real 5-of-9 majority | page · yaml |
Three counts of one unchanged set of opinions, two opposite clean sweeps. The round-2 case's round 1 is the plurality-block case — Oak leads on 4, nobody reaches 5, the independents are eliminated, and their voters' marks move to Pine. Uncapping the marks reaches the same place in a single round, because it lets those voters say "independents, and Pine over Oak" the first time they are asked.
And the floor: the smallest possible minority sweep (yaml) — 5 voters, two marking a full slate and three each bullet-voting a different name. 40% of the voters take 100% of the seats, and no smaller electorate can do it.
Confirms our engine's multi-winner Plurality (run_plurality_multi) tabulates the whole family (it auto-detects Block / Limited / SNTV from votes-per-voter), and its multi-winner Approval covers the uncapped member.
The first electorate is BV-confirmed (not exceptions). BV has no method named Block/Limited Voting, but all three are "mark k, top N win" = BV's multi-winner Approval with each voter approving k (full slate = block, k<N = limited, 1 = SNTV). Block & Limited are backed by BV election BV2135 (3x4vrv) — same ballots, same winners; SNTV by the BV2134 governance Bloc Plurality race. The LH yamls carry the Plurality-family teaching label; BV runs them as Approval. Full recipe: running it on BetterVoting.
The second electorate's four cases are LH-only so far, but reproducible on BV by the same recipe — each is 0/1 ballots and top-N-wins, and the two-round case is simply its two rounds run as two races.
→ Full lesson (with Wikipedia + electowiki links): Block, Limited & SNTV — the multi-member plurality family