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Exercise 6 — Bullet voting backfires

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

A club books a speaker. Ari's four fans genuinely like Bree too — their honest ballot is Ari 5, Bree 3, Cash 0. But the night before the vote, one of them argues: "scoring Bree helps Ari's rival — zero her, and Ari walks in." All four bullet vote. Predict what they wake up to.

▶ Live on BetterVoting: honest vote · results ↗ · strategic vote · results ↗ (elections x4dkfd / 7f4f7q, Test IDs BV2193–94; each also carries RCV-IRV and Ranked Robin races).

You practice: strategic voting in STAR — specifically bullet voting (supporting only your favorite) — and why the automatic runoff makes it a gamble rather than a free lift.

Work each part on paper before opening its solution. Both YAMLs are runnable; their expected_winners keys are regression-tested, and the _tabulated mirrors are the full audit reports.

The ballots

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

Honest profile:

×4 Ari fans ×4 Cash fans ×1 Bree fan
Ari 5 0 0
Bree 3 2 5
Cash 0 5 1

The same nine ballots as marked paper — four identical Ari-fan ballots, four identical Cash-fan ballots, and the one Bree fan whose Cash 1 will decide the runoff:

The ballots as marked — the filled bubble is the score given, and the score is the number in its column:

# Ballot as marked Ari Bree Cash
1 A 0–5 STAR ballot — Ari fan 1 of 4 — honest: Bree really is a second choice: Ari 5, Bree 3, Cash 0. 5 3 0
2 A 0–5 STAR ballot — Ari fan 2 of 4 — same honest ballot: Ari 5, Bree 3, Cash 0. 5 3 0
3 A 0–5 STAR ballot — Ari fan 3 of 4 — same honest ballot: Ari 5, Bree 3, Cash 0. 5 3 0
4 A 0–5 STAR ballot — Ari fan 4 of 4 — same honest ballot: Ari 5, Bree 3, Cash 0. 5 3 0
5 A 0–5 STAR ballot — Cash fan 1 of 4 — Bree tolerable: Ari 0, Bree 2, Cash 5. 0 2 5
6 A 0–5 STAR ballot — Cash fan 2 of 4 — same ballot: Ari 0, Bree 2, Cash 5. 0 2 5
7 A 0–5 STAR ballot — Cash fan 3 of 4 — same ballot: Ari 0, Bree 2, Cash 5. 0 2 5
8 A 0–5 STAR ballot — Cash fan 4 of 4 — same ballot: Ari 0, Bree 2, Cash 5. 0 2 5
9 A 0–5 STAR ballot — The lone Bree fan — her Cash 1 outranks Ari's 0: Ari 0, Bree 5, Cash 1. 0 5 1

Strategic profile — identical except the Ari fans bullet vote: their column becomes Ari 5, Bree 0, Cash 0. Only the top four ballots change; every other mark on the page stays exactly where it was:

The ballots as marked — the filled bubble is the score given, and the score is the number in its column:

# Ballot as marked Ari Bree Cash
1 A 0–5 STAR ballot — Ari fan 1 of 4 — BULLET: Bree zeroed (honest was 5,3,0): Ari 5, Bree 0, Cash 0. 5 0 0
2 A 0–5 STAR ballot — Ari fan 2 of 4 — BULLET: Bree zeroed: Ari 5, Bree 0, Cash 0. 5 0 0
3 A 0–5 STAR ballot — Ari fan 3 of 4 — BULLET: Bree zeroed: Ari 5, Bree 0, Cash 0. 5 0 0
4 A 0–5 STAR ballot — Ari fan 4 of 4 — BULLET: Bree zeroed: Ari 5, Bree 0, Cash 0. 5 0 0
5 A 0–5 STAR ballot — Cash fan 1 of 4 — unchanged: Ari 0, Bree 2, Cash 5. 0 2 5
6 A 0–5 STAR ballot — Cash fan 2 of 4 — unchanged: Ari 0, Bree 2, Cash 5. 0 2 5
7 A 0–5 STAR ballot — Cash fan 3 of 4 — unchanged: Ari 0, Bree 2, Cash 5. 0 2 5
8 A 0–5 STAR ballot — Cash fan 4 of 4 — unchanged: Ari 0, Bree 2, Cash 5. 0 2 5
9 A 0–5 STAR ballot — The lone Bree fan — unchanged, and she decides the runoff: Ari 0, Bree 5, Cash 1. 0 5 1

Your task

  • (a) Tabulate the honest profile (STAR: scoring round, finalists, runoff). Who wins, and what do Ari's fans get?
  • (b) Before computing: write down who you think wins after the four fans switch to bullet ballots — and who you think Ari's fans believe will win.
  • (c) Tabulate the strategic profile. Walk through exactly where the plan works and where it detonates.
  • (d) The gamble had a hidden precondition. State the general rule: when is bullet voting in STAR safe, and when is it playing with fire?

Solutions

(a) Honest — Bree wins, 5–4; Ari's fans get their second choice 
[Divergence from STAR]
  STAR                   = Bree
  Choose-One (Plurality) = Ari   (differs from STAR)
  RCV-IRV                = Cash   (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/ex06_bullet_honest_RCV-IRV_tabulated.txt

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

[STAR Voting]
 Tabulating 9 ballots.
Count × Ari,Bree,Cash
    4 ×   5,   3,   0
    4 ×   0,   2,   5
    1 ×   0,   5,   1

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Bree          -- 25 -- First place
   Cash          -- 21 -- Second place
   Ari           -- 20
 Bree and Cash advance.

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

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Bree
Ari misses the runoff by one point (20 vs Cash's 21), and the broad compromise **Bree beats Cash 5–4** — the Ari fans' 3s are precisely what put her there. Their honest ballots bought their second choice. (A side note the engine's divergence block volunteers: on these same honest opinions, *RCV-IRV elects Cash* — Bree is squeezed out on first choices, [exercise 5](ex05_center_squeeze.md)'s lesson making a cameo. The honest STAR outcome these fans are about to gamble away is one other methods wouldn't even have found.)
(b) The prediction The fans' theory: zeroing Bree drops her below Ari, Ari reaches the runoff, and Ari's four maximums carry the day. Write down your own call — winner, finalists, margin — then open (c).
(c) Strategic — Cash wins; the plan works halfway, then detonates 
[Divergence from STAR]
  STAR                   = Cash
  Choose-One (Plurality) = Ari   (differs from STAR)
  Approval               = Ari   (differs from STAR)

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

[STAR Voting]
 Tabulating 9 ballots.
Count × Ari,Bree,Cash
    4 ×   5,   0,   0
    4 ×   0,   2,   5
    1 ×   0,   5,   1

[STAR Voting: Scoring Round]
 The two highest-scoring candidates advance to the next round.
   Cash          -- 21 -- First place
   Ari           -- 20 -- Second place
   Bree          -- 13
 Cash and Ari advance.

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

[STAR Voting: Winner — STAR Voting Method (single winner)]
 Cash
The first half works perfectly: Bree crashes 25 → 13, out of the runoff; Ari is a finalist. Then the trap: **the runoff Ari inherits is against Cash**, and zeroing Bree never created a single Ari-over-Cash preference — those are still 4 (Ari's fans) versus 5 (Cash's fans plus the Bree fan, who scored Cash 1 > Ari 0). **Cash wins 5–4.** The fans demoted their sure second choice and elected their zero. Under the honest count they got Bree; the "clever" ballots got them their nightmare.
(d) The general rule Bullet voting is *safe* only when your favorite would **win the runoff against whoever replaces the candidate you zeroed** — which is exactly what these fans never checked (Ari loses to Cash 4–5 no matter what). Zeroing a compromise you actually prefer to the field does two things at once: it may promote your favorite into the runoff, and it *simultaneously* removes your insurance if your favorite still loses there. That double edge is by design — the runoff is why STAR "discourages bullet voting" (glossary: [bullet voting / tactical minimization](../../07_Concepts/GLOSSARY.md)) — and it's the honest version of that claim: not *impossible* to game (nothing is — [Gibbard](../../07_Concepts/topics/gibbard_satterthwaite_theorem.md)), but a strategy that pays only with information voters rarely have, and costs dearly when the guess is wrong. The balanced treatment across methods is [strategic voting](../../07_Concepts/topics/strategic_voting.md).

Reading this fairly

Note what this exercise is and isn't: a sincere-vs-strategic pair, so per the four-part test it says so loudly — the failure here is the strategy's, not the method's, and the same "zero the compromise" gamble misfires in Approval and ranked methods too. One honest profile, one coordinated deviation, both runnable; margins of one point, engineered for clarity.

Run it yourself

python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 01_STAR/05_Practice/cases/ex06_bullet_honest.yaml
python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 01_STAR/05_Practice/cases/ex06_bullet_backfire.yaml

Sources: ex06_bullet_honest.yaml · ex06_bullet_backfire.yaml. Full audit reports: honest · strategic.

Where this comes from. Original to this repo (ballots and cast). Concept homes: strategic voting across the Equal Vote methods and the second-round FAQ (why the runoff exists).

Back to the exercises set · curriculum home: Voting 301

file: ex06_bullet_backfire.md