Question

number of voters 4 8 10 6
1st choice c a b d
2nd choice d c d a
3rd choice b b a c
4th choice a d c b
find number of points candidate a receives under pairwise comparison (copelands method)
points =
find the winner of this election under pairwise comparison (copelands method)
winner =

Explanation:

Part 1: Points for Candidate A under Copeland's Method

Copeland's Method involves pairwise comparisons between each pair of candidates. For each pair (X, Y), we count how many voters prefer X over Y. A candidate gets 1 point for a win, 0.5 for a tie, and 0 for a loss in each pairwise comparison. First, find the total number of voters: \(4 + 8 + 10 + 6 = 28\).

Candidates are A, B, C, D. We need to compare A with B, A with C, A with D.

Step 1: A vs B

• Voters who prefer A over B: Look at ballots where A is ranked higher than B.
• Group 1 (4 voters): 1st=C, 2nd=D, 3rd=B, 4th=A → A is 4th, B is 3rd: B > A.
• Group 2 (8 voters): 1st=A, 2nd=C, 3rd=B, 4th=D → A is 1st, B is 3rd: A > B (8 voters).
• Group 3 (10 voters): 1st=B, 2nd=D, 3rd=A, 4th=C → B is 1st, A is 3rd: B > A.
• Group 4 (6 voters): 1st=D, 2nd=A, 3rd=C, 4th=B → A is 2nd, B is 4th: A > B (6 voters).
• Total for A > B: \(8 + 6 = 14\); B > A: \(4 + 10 = 14\). So it's a tie. A gets 0.5 points.

Step 2: A vs C

• Voters who prefer A over C:
• Group 1 (4 voters): 1st=C, 2nd=D, 3rd=B, 4th=A → C > A.
• Group 2 (8 voters): 1st=A, 2nd=C, 3rd=B, 4th=D → A > C (8 voters).
• Group 3 (10 voters): 1st=B, 2nd=D, 3rd=A, 4th=C → A > C (10 voters).
• Group 4 (6 voters): 1st=D, 2nd=A, 3rd=C, 4th=B → A > C (6 voters).
• Total for A > C: \(8 + 10 + 6 = 24\); C > A: \(4\). So A wins. A gets 1 point.

Step 3: A vs D

• Voters who prefer A over D:
• Group 1 (4 voters): 1st=C, 2nd=D, 3rd=B, 4th=A → D > A.
• Group 2 (8 voters): 1st=A, 2nd=C, 3rd=B, 4th=D → A > D (8 voters).
• Group 3 (10 voters): 1st=B, 2nd=D, 3rd=A, 4th=C → D > A.
• Group 4 (6 voters): 1st=D, 2nd=A, 3rd=C, 4th=B → A > D (6 voters).
• Total for A > D: \(8 + 6 = 14\); D > A: \(4 + 10 = 14\). Tie. A gets 0.5 points.

Step 4: Total Points for A

Sum the points: \(0.5 + 1 + 0.5 = 2\).

We need to calculate points for all candidates (B, C, D) similarly.

Candidate B:

• B vs A: Tie (0.5, as calculated earlier)
• B vs C:
• Voters prefer B over C:
• Group1: 1st=C, 2nd=D, 3rd=B, 4th=A → B > C (4)
• Group2: 1st=A, 2nd=C, 3rd=B, 4th=D → B > C (8)
• Group3: 1st=B, 2nd=D, 3rd=A, 4th=C → B > C (10)
• Group4: 1st=D, 2nd=A, 3rd=C, 4th=B → C > B (6)
• B > C: \(4 + 8 + 10 = 22\); C > B: \(6\). B wins (1 point)
• B vs D:
• Voters prefer B over D:
• Group1: 1st=C, 2nd=D, 3rd=B, 4th=A → B > D (4)
• Group2: 1st=A, 2nd=C, 3rd=B, 4th=D → B > D (8)
• Group3: 1st=B, 2nd=D, 3rd=A, 4th=C → B > D (10)
• Group4: 1st=D, 2nd=A, 3rd=C, 4th=B → D > B (6)
• B > D: \(4 + 8 + 10 = 22\); D > B: \(6\). B wins (1 point)
• Total for B: \(0.5 + 1 + 1 = 2.5\)

Candidate C:

• C vs A: Loss (0, since A won vs C)
• C vs B: Loss (0, since B won vs C)
• C vs D:
• Voters prefer C over D:
• Group1: 1st=C, 2nd=D, 3rd=B, 4th=A → C > D (4)
• Group2: 1st=A, 2nd=C, 3rd=B, 4th=D → C > D (8)
• Group3: 1st=B, 2nd=D, 3rd=A, 4th=C → D > C (10)
• Group4: 1st=D, 2nd=A, 3rd=C, 4th=B → C > D (6)
• C > D: \(4 + 8 + 6 = 18\); D > C: \(10\). C wins (1 point)
• Total for C: \(0 + 0 + 1 = 1\)

Candidate D:

• D vs A: Tie (0.5, as calculated earlier)
• D vs B: Loss (0, since B won vs D)
• D vs C: Loss (0, since C won vs D)
• Total for D: \(0.5 + 0 + 0 = 0.5\)

Now, compare points: A=2, B=2.5, C=1, D=0.5. So B has the highest points.

Answer:

2

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Part 2: Winner under Copeland's Method