====================================================================== SOURCE FILE: ex02_tenth_ballot.yaml TABULATED FILE: ex02_tenth_ballot_tabulated.txt ====================================================================== election_title: "Exercise 2 — The tenth ballot: all ten ballots" scenario_description: |- The same nine ballots plus the forgotten tenth — a voter whose favorite is Alex (5), with Chris at 2 and everyone else at 0. The ballot ADDS more to Alex's total than to anyone else's, yet it flips the win away from Alex: scoring round Alex 32, Chris 26, Bella 25, Eli 22, Dana 20 — the extra 2 points lift Chris past Bella into the runoff, and Chris beats Alex head-to-head 6-4. STAR fails the PARTICIPATION criterion: casting an honest ballot can occasionally hurt your favorite (here, by changing WHO Alex has to face). Part of the exercise ex02_tenth_ballot.md. Ballots adapted from a RangeVoting.org example, posed as an exercise in Brendan W. Sullivan, "An Introduction to the Math of Voting Methods" (2022), ch. 5. voting_method: STAR num_winners: 1 ballots: |- Alex,Bella,Chris,Dana,Eli 3,2,4,5,0 # 4 voters: Dana first, Chris close second 3,2,4,5,0 3,2,4,5,0 3,2,4,5,0 3,5,0,0,4 # 3 voters: Bella first, Eli second 3,5,0,0,4 3,5,0,0,4 3,1,4,0,5 # 2 voters: Eli first, Chris second 3,1,4,0,5 5,0,2,0,0 # the forgotten TENTH ballot: Alex is this voter's favorite expected_winners: - Chris # file: ex02_tenth_ballot.yaml ====================================================================== TABULATION RESULTS ====================================================================== === Exercise 2 — The tenth ballot: all ten ballots === The same nine ballots plus the forgotten tenth — a voter whose favorite is Alex (5), with Chris at 2 and everyone else at 0. The ballot ADDS more to Alex's total than to anyone else's, yet it flips the win away from Alex: scoring round Alex 32, Chris 26, Bella 25, Eli 22, Dana 20 — the extra 2 points lift Chris past Bella into the runoff, and Chris beats Alex head-to-head 6-4. STAR fails the PARTICIPATION criterion: casting an honest ballot can occasionally hurt your favorite (here, by changing WHO Alex has to face). Part of the exercise ex02_tenth_ballot.md. Ballots adapted from a RangeVoting.org example, posed as an exercise in Brendan W. Sullivan, "An Introduction to the Math of Voting Methods" (2022), ch. 5. --- Runoff (Preference) Matrix --- Head-to-head / pairwise comparison Legend: For - Equal Support - Against * indicates Top 2 Finalist | * Alex | Bella | * Chris | Dana | Eli | ----------------------------------------------------------------------------- * Alex > | --- |7 - 0 - 3 |4 - 0 - 6 |6 - 0 - 4 |5 - 0 - 5 | Bella > | 3 - 0 - 7 | --- |3 - 0 - 7 |5 - 1 - 4 |7 - 1 - 2 | * Chris > | 6 - 0 - 4 |7 - 0 - 3 | --- |3 - 3 - 4 |5 - 0 - 5 | Dana > | 4 - 0 - 6 |4 - 1 - 5 |4 - 3 - 3 | --- |4 - 1 - 5 | Eli > | 5 - 0 - 5 |2 - 1 - 7 |5 - 0 - 5 |5 - 1 - 4 | --- | [Condorcet Winner] No Condorcet winner (majority cycle: Alex > Bella > Dana > Chris > Alex) [Divergence from STAR] STAR = Chris Choose-One (Plurality) = Dana (differs from STAR) RCV-IRV = Bella (differs from STAR) Approval = Alex (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/ex02_tenth_ballot_RCV-IRV_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 10 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 1 × 5, 0, 2, 0, 0 [Score Distribution] (how many ballots gave each star rating) Score Candidate 5 4 3 2 1 0 | Total Avg Alex 1 0 9 0 0 0 | 32 3.2 Bella 3 0 0 4 2 1 | 25 2.5 Chris 0 6 0 1 0 3 | 26 2.6 Dana 4 0 0 0 0 6 | 20 2.0 Eli 2 3 0 0 0 5 | 22 2.2 [STAR Voting: Scoring Round] The two highest-scoring candidates advance to the next round. Alex -- 32 -- First place Chris -- 26 -- Second place Bella -- 25 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 -- 4 Equal Support -- 0 Chris wins. Runoff math: 10 ballots cast − 0 Equal Support (no preference between the two finalists) ── 10 voters with a preference (majority = 6) Chris 6 (60%) · Alex 4 (40%) [STAR Voting: Winner — STAR Voting Method (single winner)] Chris