Exercise 2 — The tenth ballot¶
Method: STAR (single winner) · 1 seat · Expected winner: Chris · full count →
A small club elects its coordinator with STAR Voting. Nine ballots are counted and a winner is announced — and then a tenth ballot turns up, cast on time and mislaid in the envelope pile. The tenth voter's favorite is the announced winner. Recount with their ballot included: can anything change?
You practice: the scoring round · the runoff pairing · the participation criterion (the no-show paradox) · the runoff-slot spoiler.
Work each part on paper before opening its solution. The YAML files at the bottom are runnable; their expected_winners answer keys are regression-tested, and each _tabulated mirror is the full audit report.
The ballots¶
Nine voters, five candidates, scores 0–5, three ballot styles (rows are candidates, columns are voter blocs):
| ×4 voters | ×3 voters | ×2 voters | |
|---|---|---|---|
| Alex | 3 | 3 | 3 |
| Bella | 2 | 5 | 1 |
| Chris | 4 | 0 | 4 |
| Dana | 5 | 0 | 0 |
| Eli | 0 | 4 | 5 |
The tenth ballot, found after the count: Alex 5 · Chris 2 · Bella, Dana, Eli 0.
Your task¶
- (a) Tabulate the nine counted ballots (STAR: scoring round, then the automatic runoff). Who wins? Would plain Score voting agree?
- (b) The tenth voter's favorite is Alex, scored a full 5. Write down your prediction for the 10-ballot recount.
- (c) Recount with all ten ballots. What happened — at which step, and by how many points? What does the tenth voter think of the outcome?
- (d) Back to the original nine ballots. Suppose instead that Bella had withdrawn before the count (ballots unchanged, her column simply removed). Predict, then tabulate.
- (e) Parts (c) and (d) flipped the winner the same way. Name the shared mechanism — and which methods are structurally immune.
Solutions¶
(a) The nine counted ballots
Score totals: Alex 9×3 = **27**, Bella 4×2+3×5+2×1 = **25**, Chris 4×4+2×4 = **24**, Eli 3×4+2×5 = **22**, Dana 4×5 = **20**. Alex and Bella advance; six voters score Alex above Bella (the ×4 and ×2 blocs), three the reverse.[Divergence from STAR]
STAR = Alex
Choose-One (Plurality) = Dana (differs from STAR)
RCV-IRV = Bella (differs from STAR)
RCV-RR = Eli (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.
Full round-by-round reports (generated for review):
RCV-IRV rounds: cases_tabulated/ex02_nine_ballots_RCV-IRV_tabulated.txt
RCV-RR round-robin: cases_tabulated/ex02_nine_ballots_RCV-RR_tabulated.txt
--- STAR Voting Method (single winner) ---
[STAR Voting]
Tabulating 9 ballots.
Count × Alex,Bella,Chris,Dana,Eli
4 × 3, 2, 4, 5, 0
3 × 3, 5, 0, 0, 4
2 × 3, 1, 4, 0, 5
[STAR Voting: Scoring Round]
The two highest-scoring candidates advance to the next round.
Alex -- 27 -- First place
Bella -- 25 -- Second place
Chris -- 24
Eli -- 22
Dana -- 20
Alex and Bella advance.
[STAR Voting: Automatic Runoff Round]
The candidate preferred in the most head-to-head matchups wins.
Alex -- 6 -- First place
Bella -- 3
Equal Support -- 0
Alex wins.
Runoff math:
9 ballots cast
− 0 Equal Support (no preference between the two finalists)
─
9 voters with a preference (majority = 5)
Alex 6 (67%) · Bella 3 (33%)
[STAR Voting: Winner — STAR Voting Method (single winner)]
Alex
(b) The prediction
The trap reads: "the ballot gives Alex the maximum and his rivals almost nothing — it can only help Alex." Write your answer down, then open (c).(c) The recount — helping your favorite hurts him
New totals: Alex 27+5 = **32**, Chris 24+2 = **26**, Bella 25+0 = **25**, Eli 22, Dana 20. Alex's lead *grows* — but the tenth ballot's 2 points for Chris nudge him past Bella (26 vs 25) into the runoff. And the Alex-vs-Chris matchup was never Alex's: the ×4 and ×2 blocs all score Chris 4 > Alex 3.Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Chris -- 6 -- First place
Alex -- 4
Equal Support -- 0
Chris wins.
Voters with a preference: 10 of 10 (no Equal Support).
Chris 6 (60%) vs Alex 4 (40%); majority = 6.
(d) Bella withdraws — the shield effect
Remove Bella's column from the original nine ballots and nothing else moves: Alex **27**, Chris **24**, Eli 22, Dana 20. Chris inherits the second runoff slot — and beats Alex head-to-head 6–3.[Divergence from STAR]
STAR = Chris
Choose-One (Plurality) = Eli (differs from STAR)
RCV-IRV = Eli (differs from STAR)
Approval = Alex (differs from STAR)
RCV-RR (Condorcet) = Eli (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) sides with RCV-IRV, so STAR is the outlier
here — STAR need not elect the Condorcet candidate.
Full round-by-round reports (generated for review):
RCV-IRV rounds: cases_tabulated/ex02_bella_exits_RCV-IRV_tabulated.txt
RCV-RR round-robin: cases_tabulated/ex02_bella_exits_RCV-RR_tabulated.txt
[Runoff Reversal]
- Score Round Winner(s) = (Alex)
- Runoff Round Winner = (Chris)
Candidate Alex earned the highest total score, but
Candidate Chris won the automatic runoff — not a malfunction,
STAR working as designed: the runoff elects the finalist preferred
by the majority (of voters with a preference).
--- STAR Voting Method (single winner) ---
[STAR Voting]
Tabulating 9 ballots.
Count × Alex,Chris,Dana,Eli
4 × 3, 4, 5, 0
3 × 3, 0, 0, 4
2 × 3, 4, 0, 5
[STAR Voting: Scoring Round]
The two highest-scoring candidates advance to the next round.
Alex -- 27 -- First place
Chris -- 24 -- Second place
Eli -- 22
Dana -- 20
Alex and Chris advance.
[STAR Voting: Automatic Runoff Round]
The candidate preferred in the most head-to-head matchups wins.
Chris -- 6 -- First place
Alex -- 3
Equal Support -- 0
Chris wins.
Runoff math:
9 ballots cast
− 0 Equal Support (no preference between the two finalists)
─
9 voters with a preference (majority = 5)
Chris 6 (67%) · Alex 3 (33%)
[STAR Voting: Winner — STAR Voting Method (single winner)]
Chris
(e) The shared mechanism — and who is immune
Both flips are **runoff-slot events**: nothing changed Alex's standing in the scoring round (he leads it in all three variants), but a one-or-two-point perturbation — one extra ballot, one withdrawn also-ran — swapped **who he must face**, and Chris beats him head-to-head in every variant. The runoff is the same feature that gives STAR its majoritarian bite (see [Runoff Reversal](../02_Examples/runoff_overturns_leader/README.md)); these are the criteria it costs. Structurally immune to participation failures: the pure point-summing methods — **Score** and **Approval** (and Choose-One) — where an extra supportive ballot can only add to your favorite's total. Score elects Alex in all three variants of this exercise. Not immune: STAR, RCV-IRV, top-two runoffs, and Condorcet methods. The honest scorecard, including how *rarely* STAR's version bites compared to IRV's elimination machinery, is the [Participation topic hub](../../07_Concepts/topics/participation/README.md)'s job — and note the repo's live multi-method pair ([participation_no_show](../../method_comparisons/participation_no_show/README.md)), where the *same* sincere ballots reward the voters under STAR and punish them under RCV-IRV. This exercise is the deliberately-constructed counterpoint: the rare STAR-side failure, so the claim "STAR fails participation too, far more rarely" comes with a runnable example instead of a hand-wave.Five methods, four winners¶
A closing curiosity from the engine's [Divergence from STAR] block on the base nine ballots: this little electorate splits the methods four ways — Choose-One elects Dana, RCV-IRV elects Bella, Ranked Robin elects Eli, and STAR (with Score agreeing) elects Alex. Engineered ballots disagree loudly; which notion of "best winner" should prevail is three winner notions' question.
Reading this fairly¶
A constructed classroom electorate with one-point margins — it demonstrates the mechanism honestly and says nothing about frequency. Every method fails something; what matters is which failures, how often, under sincere votes. Apply the four-part test before turning any single construction into a verdict, and see STAR's honest limits for the balanced scorecard.
Run it yourself¶
python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 01_STAR/05_Practice/cases/ex02_nine_ballots.yaml
python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 01_STAR/05_Practice/cases/ex02_tenth_ballot.yaml
python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 01_STAR/05_Practice/cases/ex02_bella_exits.yaml
Sources: ex02_nine_ballots.yaml · ex02_tenth_ballot.yaml · ex02_bella_exits.yaml. Full audit reports: nine ballots · ten ballots · Bella withdraws.
Where these ballots come from. Adapted (candidates renamed, prose our own) from a worked example published on RangeVoting.org — a Score-voting advocacy site, so its framing favors Score; the arithmetic is method-neutral — and posed as an add-a-voter exercise in Brendan W. Sullivan, An Introduction to the Math of Voting Methods (619 Wreath Publishing, 2022), ch. 5, a method-neutral general-education textbook whose practice-problems-with-solutions format this exercises set borrows.
Back to the exercises set · curriculum home: Voting 301