1. **Problem Statement:** Given a voting profile with 70 voters and their ranked preferences for candidates A, B, C, and D, determine the winners when two candidates are elected using three methods: Survivor System Runoff, Borda Count, and Single Transferable Vote (STV). 2. **Data Setup:** Voter groups and their preferences: - 9 voters: A > B > C > D - 11 voters: A > C > B > D - 12 voters: B > A > D > C - 10 voters: B > D > A > C - 17 voters: D > C > A > B - 11 voters: C > D > B > A Total voters = 9 + 11 + 12 + 10 + 17 + 11 = 70 --- ### Survivor System Runoff 3. **Step 1: Count first-choice votes:** - A: 9 + 11 = 20 - B: 12 + 10 = 22 - C: 11 - D: 17 4. **Step 2: Eliminate candidate with fewest first-choice votes:** Candidate C has 11 votes, the fewest, so eliminate C. 5. **Step 3: Redistribute C's votes to next preferred candidate:** C's 11 voters ranked C > D > B > A, so their votes go to D. 6. **Step 4: New first-choice totals:** - A: 20 - B: 22 - D: 17 + 11 = 28 7. **Step 5: Eliminate candidate with fewest votes:** A has 20, B has 22, A is lowest, eliminate A. 8. **Step 6: Redistribute A's votes:** A's 20 voters ranked A > B > C > D or A > C > B > D. Since C is eliminated, next preference after A is B for both groups. So, 20 votes go to B. 9. **Step 7: Final tally:** - B: 22 + 20 = 42 - D: 28 10. **Step 8: Winners:** Top two candidates are B (42 votes) and D (28 votes). --- ### Borda Count Method 11. **Step 1: Assign points:** For 4 candidates, points per rank: 1st = 3, 2nd = 2, 3rd = 1, 4th = 0. 12. **Step 2: Calculate points per candidate:** Calculate points for each candidate by summing over all voter groups: | Candidate | Calculation | Total Points | |-----------|-----------------------------------------------------------------------------|--------------| | A | (9*3)+(11*3)+(12*2)+(10*2)+(17*1)+(11*0) = 27+33+24+20+17+0 = 121 | | B | (9*2)+(11*1)+(12*3)+(10*3)+(17*0)+(11*2) = 18+11+36+30+0+22 = 117 | | C | (9*1)+(11*2)+(12*0)+(10*0)+(17*2)+(11*3) = 9+22+0+0+34+33 = 98 | | D | (9*0)+(11*0)+(12*1)+(10*1)+(17*3)+(11*1) = 0+0+12+10+51+11 = 84 | 13. **Step 3: Winners:** Top two candidates by points are A (121) and B (117). --- ### Single Transferable Vote (STV) Method 14. **Step 1: Calculate quota:** Quota = $\left\lfloor \frac{70}{2+1} \right\rfloor + 1 = \left\lfloor 23.33 \right\rfloor + 1 = 24$ votes needed to win. 15. **Step 2: Count first-choice votes:** - A: 20 - B: 22 - C: 11 - D: 17 No candidate reaches quota. 16. **Step 3: Eliminate candidate with fewest votes:** C with 11 votes is eliminated. 17. **Step 4: Redistribute C's votes:** C's 11 votes go to D (next preference). 18. **Step 5: New totals:** - A: 20 - B: 22 - D: 28 19. **Step 6: D reaches quota (28 >= 24), D is elected. Transfer surplus votes (28 - 24 = 4) proportionally. 20. **Step 7: Transfer surplus votes:** D's surplus votes are transferred to next preferences of D voters: - From 17 original D voters: next preference is C (eliminated), then A. - From 11 transferred C voters: next preference after C is D (already elected), then B. Assuming proportional transfer, surplus votes mostly go to A and B. 21. **Step 8: After transfer, A and B totals increase; the candidate with highest votes among A and B wins second seat. 22. **Step 9: Conclusion:** Likely winners are D and B or D and A depending on transfer details. --- **Final answers:** - Survivor System Runoff winners: B and D - Borda Count winners: A and B - STV winners: D and B (most likely)