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Bloc STAR vs Proportional STAR — the same ballots, two councils

Two voters. Three candidates. Two seats. One ballot design. Two different answers — and you can check the whole thing in your head.

Level: 101 · for voters

Both methods here use the identical 0–5 STAR ballot. Nothing changes for the voter. What changes is the count, and this folder is the smallest possible demonstration of what that difference actually does.

→ the majoritarian method: Bloc STAR · the proportional one: STAR-PR · what "proportional" promises: what proportional actually means

The election

Two voters are filling two seats from three candidates.

Ana Ben Cleo
Voter 1 0 0 1
Voter 2 2 3 0
Total 2 3 1

Voter 1 is lukewarm about exactly one candidate, Cleo. Voter 2 prefers Ben, would accept Ana, and has no time for Cleo.

Seat 1: both methods agree

Ben has the most points (3) and wins the first seat under either count. Nothing interesting has happened yet — which is the point. The two methods only part company once somebody has already been elected.

Seat 2: this is where they split

Bloc STAR re-runs the same count on the remaining candidates, with every ballot still at full weight. Ana (2) beats Cleo (1), so Ana takes the second seat.

Bloc STAR elects: Ben and Ana.

Proportional STAR asks a different question first: who is already represented? With two seats and two voters, one seat is worth one voter — the Hare quota here is exactly 1 ballot. Ben was elected on Voter 2's support, so Voter 2 is now represented and their ballot is set aside. Only Voter 1 is left to decide seat 2, and Voter 1 wants Cleo.

Proportional STAR elects: Ben and Cleo.

What just happened

Voter 2 got both of their preferred candidates under Bloc STAR. Under Proportional STAR they got one, and Voter 1 — who got nothing at all under Bloc — got the other.

That is the entire difference between the two families, and it needs only two ballots to show:

  • Bloc STAR asks "who do the voters most want?" — once per seat, with everyone voting every time. A cohesive majority can therefore take every seat. That is a feature when you want the body to speak with one voice, and a bug when you want it to represent everybody.
  • Proportional STAR asks "who is not yet represented?" — so each seat is decided by the voters who have not yet won one.

Neither is cheating. They answer different questions, and choosing between them is a decision about what the body you are electing is for.

How quickly can they diverge?

Immediately. Two ballots is the smallest election in which a two-seat race can be meaningfully counted at all, and it is already enough. This pair was found by exhaustively searching every tie-free profile of 3 candidates × 0–5 scores in ballot-count order, so no smaller example exists.

Worth being precise about why this one diverges, because there are two different reasons Bloc and PR can disagree and only one of them is about proportionality:

  1. The runoff — Bloc STAR finishes each seat with an automatic runoff; Allocated Score elects the highest scorer outright. So they can differ even on seat 1, whenever a runoff reverses the score order. That is really a STAR-vs-Score difference wearing a multi-winner coat.
  2. The reweighting — the one above. Seat 1 agrees; the methods part company only because one of them remembers who has already been served.

This example is deliberately the second kind.

The same thing at readable scale: Left, Centre, Right

Two ballots proves it can happen. This shows what it looks like. Ten voters, nine candidates (three per camp), three seats:

Camp Voters Scores their own camp the neighbours the far camp
Left (L1 L2 L3) 6 4–5 1–2 0
Centre (C1 C2 C3) 2 4–5 1–2 —
Right (R1 R2 R3) 2 4–5 1–2 0

The Left camp is 60% of the electorate.

Bloc STAR elects: L1, L2, L3. Sixty percent of the voters take 100% of the seats. The other forty percent elect nobody. This is the majority sweep, and it is not a bug — it is what a majoritarian method is for.

Proportional STAR elects: C1, L1, L2. Left takes two seats of three — close to its 60% share — and the third goes to the centre.

Two things in that proportional result are worth slowing down for, because neither is obvious:

The Right camp wins nothing, and that is correct. With ten voters and three seats the quota is 10 ÷ 3 ≈ 3.33 voters. The Right camp has two. It is below quota — it has not earned a seat, and no proportional method owes it one. Proportionality guarantees representation to groups large enough to claim it; it does not promise everyone a seat.

The Centre camp is also below quota — and still wins one. C1 gets there not on the strength of two voters but on breadth: Left and Right voters both score the centre candidates 1–2, and those partial scores add up. That is the specific thing a 0–5 ballot can see and a choose-one ballot cannot — C1 is nobody's favourite and many people's acceptable second, which is exactly the candidate a scored proportional count is built to find.

Run it yourself

Same ballots in both files; only voting_method: differs.

.venv/bin/python STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/bloc_vs_pr/cases/min_bloc_c3_b2.yaml
Case Method Elects Files
Smallest divergence — majoritarian bloc Ben, Ana page · yaml
Smallest divergence — proportional allocated Ben, Cleo page · yaml

Where to go next

(Still to come in this folder: the same boundary drawn against RCV-IRV and Ranked Robin, and the cases where the proportional methods disagree with each other — Allocated Score vs Sequentially Spent Score vs Reweighted Range.)