1. **State the problem:** We have an election with three candidates A, B, and C, and voter preferences given as: - (ACB) = 2 voters - (BAC) = 5 voters - (CBA) = 4 voters - (CAB) = 2 voters We need to answer four questions: - Is there a majority winner? If not, who is the plurality winner? - Who wins using the Borda count method? - Who wins if we eliminate the candidate with the most last-place votes and then have a runoff? - Could the two voters with preference (CAB) change the runoff outcome by voting insincerely as (CBA)? --- 2. **Majority and plurality winner:** - Total voters = 2 + 5 + 4 + 2 = 13 - Majority winner needs more than half, i.e., > 6.5 votes. Count first-place votes: - A first-place: (ACB) 2 + (BAC) 5 = 7 - B first-place: (CBA) 4 - C first-place: (CAB) 2 Since A has 7 first-place votes, which is more than half, **A is the majority winner**. --- 3. **Borda count method:** Assign points: 1st place = 2 points, 2nd place = 1 point, 3rd place = 0 points. Calculate total points for each candidate: - For (ACB) 2 voters: A=2*2=4, C=2*1=2, B=2*0=0 - For (BAC) 5 voters: B=5*2=10, A=5*1=5, C=5*0=0 - For (CBA) 4 voters: C=4*2=8, B=4*1=4, A=4*0=0 - For (CAB) 2 voters: C=2*2=4, A=2*1=2, B=2*0=0 Sum points: - A: 4 + 5 + 0 + 2 = 11 - B: 0 + 10 + 4 + 0 = 14 - C: 2 + 0 + 8 + 4 = 14 B and C tie with 14 points each. --- 4. **Elimination of candidate with most last-place votes and runoff:** Count last-place votes: - A last-place: (CBA) 4 voters - B last-place: (ACB) 2 voters + (CAB) 2 voters = 4 voters - C last-place: (BAC) 5 voters Candidate C has the most last-place votes (5), so eliminate C. Runoff between A and B: Voters' preferences between A and B: - (ACB) 2 voters: prefer A over B - (BAC) 5 voters: prefer B over A - (CBA) 4 voters: prefer B over A (since C eliminated, next preference is B) - (CAB) 2 voters: prefer A over B Count votes: - A: 2 + 2 = 4 - B: 5 + 4 = 9 B wins the runoff. --- 5. **Could the two (CAB) voters change the runoff outcome by voting (CBA)?** If (CAB) voters vote (CBA) insincerely, their preference between A and B changes from A > B to B > A. New runoff votes: - A: (ACB) 2 voters + (CAB) 0 voters = 2 - B: (BAC) 5 + (CBA) 4 + (CAB changed) 2 = 11 B still wins, so the two voters cannot change the outcome. --- **Final answers:** - Majority winner? Yes, A - Borda count winner? B and C tie - Runoff winner after elimination? B - Can (CAB) voters change runoff outcome? No