====================================================================== SOURCE FILE: ex02_bella_exits.yaml TABULATED FILE: ex02_bella_exits_tabulated.txt ====================================================================== election_title: "Exercise 2 — The tenth ballot: Bella withdraws" scenario_description: |- The original nine ballots with Bella's column removed — she withdrew before the count, and no scores change for anyone else. Scoring round: Alex 27, Chris 24, Eli 22, Dana 20 — with Bella gone, Chris takes her runoff slot and beats Alex head-to-head 6-3. A candidate who was NOT going to win still changed who does (a runoff-slot spoiler, the independence-of-irrelevant- alternatives family): Bella's presence was what kept Chris out of the runoff. 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,Chris,Dana,Eli 3,4,5,0 # 4 voters: Dana first, Chris close second 3,4,5,0 3,4,5,0 3,4,5,0 3,0,0,4 # 3 voters: (Bella was their favorite) Eli second 3,0,0,4 3,0,0,4 3,4,0,5 # 2 voters: Eli first, Chris second 3,4,0,5 expected_winners: - Chris # file: ex02_bella_exits.yaml ====================================================================== TABULATION RESULTS ====================================================================== === Exercise 2 — The tenth ballot: Bella withdraws === The original nine ballots with Bella's column removed — she withdrew before the count, and no scores change for anyone else. Scoring round: Alex 27, Chris 24, Eli 22, Dana 20 — with Bella gone, Chris takes her runoff slot and beats Alex head-to-head 6-3. A candidate who was NOT going to win still changed who does (a runoff-slot spoiler, the independence-of-irrelevant- alternatives family): Bella's presence was what kept Chris out of the runoff. 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 | * Chris | Dana | Eli | ----------------------------------------------------------------- * Alex > | --- |3 - 0 - 6 |5 - 0 - 4 |4 - 0 - 5 | * Chris > | 6 - 0 - 3 | --- |2 - 3 - 4 |4 - 0 - 5 | Dana > | 4 - 0 - 5 |4 - 3 - 2 | --- |4 - 0 - 5 | Eli > | 5 - 0 - 4 |5 - 0 - 4 |5 - 0 - 4 | --- | [Condorcet Winner] Condorcet Winner: Eli — STAR elected Chris instead (Eli was eliminated in the scoring round) [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 [Score Distribution] (how many ballots gave each star rating) Score Candidate 5 4 3 2 1 0 | Total Avg Alex 0 0 9 0 0 0 | 27 3.0 Chris 0 6 0 0 0 3 | 24 2.7 Dana 4 0 0 0 0 5 | 20 2.2 Eli 2 3 0 0 0 4 | 22 2.4 [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