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Exercise 3 — One electorate, five verdicts

Method: STAR (single winner) · 1 seat · Expected winner: Cherry · full count →

An office votes on the snack to stock: Apple, Banana, or Cherry. Nine people fill in one score ballot each — and then the count is run five different ways. How many different winners can one small electorate produce? Commit to a number before you start.

▶ Live on BetterVoting: vote · results ↗ (election ywqhq4, Test ID BV2191 — one election, five races, all five verdicts live; BV agrees with the LH engine on every race).

You practice: reading one electorate through five tabulations — Choose-One, RCV-IRV, Approval, Score, and STAR — and the core lesson that the method, not the ballot, picks the winner (ballot vs count).

Work each part on paper before opening its solution. The YAML at the bottom is runnable; its expected_winners key is regression-tested, and the _tabulated mirror is the full audit report.

The ballots

Nine voters, three snacks, scores 0–5 (rows are candidates, columns are voter blocs):

×4 voters ×3 voters ×2 voters
Apple 5 0 0
Banana 0 5 1
Cherry 3 4 5

For the ranked methods, read each column top-to-bottom by score (the ×4 bloc ranks Apple > Cherry > Banana, and so on). For Approval, use the bridge rule approve = score 3 or higher.

Your task

  • (a) Choose-One: each voter marks only their favorite. Who wins?
  • (b) RCV-IRV: rank by scores, eliminate the last-place candidate, transfer, recount. Who wins?
  • (c) Approval (approve = 3+): who wins, and with what share of the 9 voters?
  • (d) Score: add the points. Who wins?
  • (e) STAR: two finalists, then the automatic runoff. Who wins, and by what margin?
  • (f) Tally your verdicts. What is the strongest thing you can say against the Choose-One winner from these same ballots?

Solutions

(a) Choose-One — Apple First choices: Apple 4, Banana 3, Cherry 2. **Apple wins** with 44% — and a majority of the office (5 of 9) scored Apple a flat zero. This is the [wasted-votes](../../07_Concepts/topics/wasted_votes.md) baseline every other method is reacting to.
(b) RCV-IRV — Banana First-choice ranks: Apple 4, Banana 3, Cherry 2 → **Cherry is eliminated first** (fewest firsts). Cherry's two voters transfer to Banana (they scored Banana 1 > Apple 0), so the final round is Apple 4 vs Banana 5. **Banana wins.** Note what just happened to Cherry — scored 3+ by all nine voters, eliminated before anyone looked: the same squeeze mechanism as [exercise 5](ex05_center_squeeze.md).
(c) Approval — Cherry, unanimously Approve = 3+: the ×4 bloc approves {Apple, Cherry}, the ×3 bloc {Banana, Cherry}, the ×2 bloc {Cherry}. Totals: **Cherry 9 of 9**, Apple 4, Banana 3. The candidate with the *fewest* first choices carries *every single voter* — that's the information Choose-One throws away. (More on the bridge rule and its lean: [Approval Voting](../../04_Approval/01_Learn/README.md).)
(d) Score — Cherry Apple 4×5 = **20**, Banana 3×5 + 2×1 = **17**, Cherry 4×3 + 3×4 + 2×5 = **34** of a possible 45. **Cherry wins** going away.
(e) STAR — Cherry, 5–4 
[Divergence from STAR]
  STAR                   = Cherry
  Choose-One (Plurality) = Apple   (differs from STAR)
  RCV-IRV                = Banana   (differs from STAR)
  Note: no ballots had tied scores, so RCV-IRV vs STAR here is a genuine
        method difference, not a tie-breaking artifact.
  Note: Ranked Robin (RCV-RR) agrees with STAR, so RCV-IRV is the lone
        outlier — the classic center-squeeze signature.
  Full round-by-round reports (generated for review):
  RCV-IRV rounds: cases_tabulated/ex03_five_verdicts_RCV-IRV_tabulated.txt

--- STAR Voting Method (single winner) ---

[STAR Voting]
 Tabulating 9 ballots.
Count × Apple,Banana,Cherry
    4 ×     5,     0,     3
    3 ×     0,     5,     4
    2 ×     0,     1,     5

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Cherry        -- 34 -- First place
   Apple         -- 20 -- Second place
   Banana        -- 17
 Cherry and Apple advance.

[STAR Voting: Automatic Runoff Round]
 The candidate preferred in the most head-to-head matchups wins.
   Cherry        -- 5 -- First place
   Apple         -- 4
   Equal Support -- 0
 Cherry wins.
   Runoff math:
     9  ballots cast
   − 0  Equal Support (no preference between the two finalists)
     ─
     9  voters with a preference  (majority = 5)
           Cherry 5 (56%)  ·  Apple 4 (44%)

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Cherry
**Cherry wins** — and unlike Score's bare total, the runoff line shows Cherry beating the Choose-One winner head-to-head, 5 voters to 4.
(f) The tally — and the verdict on Apple | Method | Winner | |---|---| | Choose-One | **Apple** | | RCV-IRV | **Banana** | | Approval | **Cherry** | | Score | **Cherry** | | STAR | **Cherry** | | Ranked Robin | **Cherry** | Three different winners from one electorate (the engine's `[Divergence from STAR]` block prints exactly this on screen). The strongest statement against Apple: **Apple is the [Condorcet loser](../../07_Concepts/GLOSSARY.md)** — Apple loses head-to-head to Cherry (4–5) *and* to Banana (4–5), yet Choose-One elects it. Cherry, meanwhile, is the Condorcet winner. Which verdict is "right" is a genuine question with three defensible answers — see [three winner notions](../01_Learn/properties_and_limits/STAR_three_winner_notions.md) — but "the candidate a majority beats head-to-head twice over" is a hard winner to defend.

Reading this fairly

Engineered, and openly so: the ballots are built so the verdicts fan out (a real electorate this small usually agrees with itself). The point is not "Choose-One always does this" — it is that the ballots alone don't determine the winner, so arguing about methods is arguing about which information gets counted. The four-part test applies before any of these verdicts becomes a talking point.

Run it yourself

python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 01_STAR/05_Practice/cases/ex03_five_verdicts.yaml

Source: ex03_five_verdicts.yaml. Full audit report: mirror.

Where this comes from. The one-electorate-many-methods format follows the worked comparisons in Brendan W. Sullivan, An Introduction to the Math of Voting Methods (619 Wreath Publishing, 2022), ch. 5 (his Indian/Pizza/Thai running example, where the Plurality loser sweeps the cardinal methods); the ballots and cast here are this repo's own, built on the STAR 0–5 scale.

Back to the exercises set · curriculum home: Voting 301

file: ex03_five_verdicts.md