Skip to content

Exercise 7 — The vanishing votes that never vanished

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

A park board votes on which tree to plant. Nine ballot papers come back — two score the top pair equal, one bullet-votes Cedar, one is all zeros, one is completely blank. The runoff summary then announces: "Voters with a preference: 3 of 9." A board member stands up: "Six votes just vanished!" Your job: predict every number on that summary line before you see it — and then answer the board member.

You practice: reading STAR's two-line runoff summary — the decided-voters denominator, Equal Support, and the difference between a zero, an equal score, and a blank (see Two Denominators, One Winner).

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

Three trees, nine ballot papers, one row per voter (- = left blank):

# Aspen Birch Cedar the voter's story
1 5 5 0 loves both front-runners equally
2 5 5 3 same, with a nod to Cedar
3 3 3 5 Cedar first, equal on the other two
4 5 2 0 clear Aspen preference
5 4 1 0 clear Aspen preference
6 2 4 0 clear Birch preference
7 0 0 5 bullet vote for Cedar
8 0 0 0 turned in all zeros
9 – – – completely blank

Your task

  • (a) Scoring round: compute all three totals (what do - and 0 contribute?). Who are the finalists?
  • (b) For each of the nine ballots, say what it does in the runoff: counts for Aspen, counts for Birch, or Equal Support.
  • (c) Now write the summary line yourself: "Voters with a preference: __ of __ (__ Equal Support)" — and the percentages.
  • (d) Ballot 7 scored both finalists zero and ballot 9 is blank. Do they land in the same buckets everywhere? Where does the count tell them apart?
  • (e) Answer the board member, in two sentences, using only numbers from the report.

Solutions

(a) Scoring round — Aspen 24, Birch 20, Cedar 13 Both `-` and `0` contribute zero stars to the totals (the engine prints exactly that note). Aspen 5+5+3+5+4+2 = **24**, Birch 5+5+3+2+1+4 = **20**, Cedar 3+5+5 = **13**. **Aspen and Birch advance** — ballot 7's bullet 5 counted at full weight toward Cedar's 13; it just wasn't enough to make Cedar a finalist.
(b) Ballot by ballot Aspen-vs-Birch verdicts: ballots **4, 5 → Aspen**; ballot **6 → Birch**; ballots **1, 2, 3, 7, 8, 9 → Equal Support** (1 and 2 at 5–5, 3 at 3–3, 7 and 8 at 0–0, and the blank 9 scores no one). Note the range inside "Equal Support": loving both equally, rating both mid, supporting *neither*, and abstaining entirely all land in the same runoff bucket — because between *these two finalists* they express exactly the same thing: no preference.
(c) The summary line 
Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   Aspen         -- 2 -- First place
   Birch         -- 1
   Equal Support -- 6
 Aspen wins.
   Voters with a preference: 3 of 9 (6 Equal Support).
   Aspen 2 (67%) vs Birch 1 (33%); majority = 2.
**3 of 9, with 6 Equal Support** — and the line self-reconciles: 3 + 6 = 9. If you wrote "3 of 8 (5 Equal Support)", you tracked the *cast* ballots — a perfectly good instinct, and exactly the trap: on this line the engine counts every **tabulated** ballot, blank included, and reports the cast-vs-abstained split separately (the tabulation header prints *"Note: 1 of 9 ballots is marked as an abstention"*). Percentages use the decided voters only: Aspen 2 of 3 = 67%.
(d) Zero vs blank — same bucket here, different everywhere else In *this* runoff they coincide (both Equal Support), but the count still tells them apart. Ballot 7 is a **cast vote**: its Cedar 5 counted in the scoring round, and had Cedar been a finalist, ballot 7 would have had a preference to give. Ballot 9 is a **true abstention**: it contributed nothing anywhere and is flagged by the abstention note — it exists so turnout-style questions ([quorum](../../07_Concepts/topics/quorum.md)) can be answered honestly. Zero is an opinion; blank is the absence of one. (Canonical tiny demo: [the abstentions example](../02_Examples/cases/cases_pages/abstentions.md).)
(e) Answering "six votes vanished!" "All nine papers were counted — every score they contain is in the totals that chose the finalists (Cedar's 13 includes the bullet vote). Six of them scored Aspen and Birch *equally*, so in an Aspen-vs-Birch runoff they have no preference to express — the report says so on its face: 3 of 9 with a preference, 6 Equal Support, 3 + 6 = 9." Nothing vanished; the deeper treatment of why an equal-scores ballot is a *counted voice*, not a discount, is [Aren't equal-score votes discounted?](../01_Learn/reference/are_equal_score_votes_discounted.md)

Reading this fairly

No paradox here at all — this drill exists because the reporting is where real elections win or lose trust, and "votes vanished" is the most common misreading of any runoff summary. The two-denominator design (count everything in the scoring round; state the decided-voter split with Equal Support named inline) is precisely the answer to it.

Run it yourself

python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 01_STAR/05_Practice/cases/ex07_vanishing_votes.yaml

Source: ex07_vanishing_votes.yaml. Full audit report: mirror.

Where this comes from. Original to this repo (ballots and cast); the concept homes are Two Denominators, One Winner and the equal-score votes FAQ.

Back to the exercises set · curriculum home: Voting 301

file: ex07_vanishing_votes.md