Question

exercises - voting theory score: 0/20 answered: 0/10 question 8 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 =

Explanation:

Step1: Total voters

First, calculate total number of voters: \(4 + 8 + 10 + 6 = 28\).

Step2: Compare A vs B

Count voters preferring A over B: Voters with A above B: group 2 (8), group 3 (10? No, group 3 1st is B, 3rd is A. Wait, group 3: 1st B, 3rd A. Group 4: 1st D, 2nd A, 4th B. So A over B: group 2 (8: A 1st, B 3rd), group 4 (6: A 2nd, B 4th), group 1? Group 1: 1st C, 4th A, 3rd B. So A is below B in group 1. So A over B: \(8 + 6 = 14\)? Wait no, group 3: 1st B, 3rd A: so B is above A, so A is below B here. Group 1: 1st C, 3rd B, 4th A: B above A. So A over B: group 2 (8: A 1st, B 3rd), group 4 (6: A 2nd, B 4th). So \(8 + 6 = 14\). B over A: group 1 (4) + group 3 (10) = 14. So tie? Wait no, total voters 28. 14 vs 14? Wait maybe I messed up. Wait group 1: 1st C, 2nd D, 3rd B, 4th A. So A is 4th, B is 3rd: B above A. Group 2: 1st A, 2nd C, 3rd B, 4th D: A above B. Group 3: 1st B, 2nd D, 3rd A, 4th C: B above A (1st vs 3rd). Group 4: 1st D, 2nd A, 3rd C, 4th B: A above B (2nd vs 4th). So A over B: group 2 (8) + group 4 (6) = 14. B over A: group 1 (4) + group 3 (10) = 14. So A vs B: tie, so A gets 0.5 points? Wait no, Copeland's method: for each pair, candidate gets 1 point for win, 0.5 for tie, 0 for loss. Wait, no: Copeland's method is number of wins minus number of losses? Wait no, standard Copeland: for each pair (A vs B, A vs C, A vs D), count how many times A is preferred over opponent, opponent over A. For each pair, A gets 1 point if more voters prefer A, 0.5 if tie, 0 if less.

Wait let's list all pairs for A: A vs B, A vs C, A vs D.

Step3: A vs C

Count voters preferring A over C: group 2 (8: A 1st, C 2nd), group 3 (10: A 3rd, C 4th? Wait group 3: 1st B, 2nd D, 3rd A, 4th C. So A is 3rd, C is 4th: A above C. Group 4: 1st D, 2nd A, 3rd C, 4th B: A above C (2nd vs 3rd). Group 1: 1st C, 4th A: C above A. So A over C: group 2 (8) + group 3 (10) + group 4 (6) = 24? Wait group 3: A is 3rd, C is 4th: yes, A above C. Group 4: A 2nd, C 3rd: A above C. Group 2: A 1st, C 2nd: A above C. Group 1: C 1st, A 4th: C above A (4 voters). So A over C: 8 + 10 + 6 = 24. C over A: 4. So A wins vs C: 1 point.

Step4: A vs D

Count voters preferring A over D: group 2 (8: A 1st, D 4th), group 4 (6: A 2nd, D 1st? Wait group 4: 1st D, 2nd A: D above A. Group 1: 1st C, 2nd D, 4th A: D above A (4 voters). Group 3: 1st B, 2nd D, 3rd A: D above A (10 voters). Wait no: A over D: when is A above D? Group 2: A 1st, D 4th: A above D (8). Any others? Group 4: D 1st, A 2nd: D above A. Group 1: D 2nd, A 4th: D above A. Group 3: D 2nd, A 3rd: D above A. So A over D: 8. D over A: 4 + 10 + 6 = 20. So A loses vs D: 0 points.

Wait wait, I think I messed up A vs D. Let's re-express each voter group's preference for A vs D:

Group 1 (4 voters): 1st C, 2nd D, 3rd B, 4th A. So D is 2nd, A is 4th: D > A.

Group 2 (8 voters): 1st A, 2nd C, 3rd B, 4th D. So A > D.

Group 3 (10 voters): 1st B, 2nd D, 3rd A, 4th C. So D > A (D 2nd, A 3rd).

Group 4 (6 voters): 1st D, 2nd A, 3rd C, 4th B. So D > A (D 1st, A 2nd).

So A over D: only group 2: 8 voters. D over A: 4 + 10 + 6 = 20 voters. So A loses to D: 0 points.

Step5: A vs B (recheck)

Group 1 (4): 1st C, 3rd B, 4th A: B > A (B 3rd, A 4th).

Group 2 (8): 1st A, 3rd B: A > B (A 1st, B 3rd).

Group 3 (10): 1st B, 3rd A: B > A (B 1st, A 3rd).

Group 4 (6): 1st D, 2nd A, 4th B: A > B (A 2nd, B 4th).

So A over B: group 2 (8) + group 4 (6) = 14. B over A: group 1 (4) + group 3 (10) = 14. So tie: A gets 0.5 points.

Wait, but total voters 28. 14 vs 14: tie. So A vs B: 0.5 points.

A vs C: A wins (24 vs 4), so 1 point.

A vs D: A loses (8 vs 20), so 0 points.

Wait, but wait: how many candidates are there? C, A, B, D: 4 candidates. So each candidate has 3 pairs (vs B, vs C, vs D for A). Wait no: 4 candidates, so number of pairs for A is 3: A-B, A-C, A-D.

Wait let's recalculate A vs C:

Group 1: C 1st, A 4th: C > A (4)

Group 2: A 1st, C 2nd: A > C (8)

Group 3: A 3rd, C 4th: A > C (10)

Group 4: A 2nd, C 3rd: A > C (6)

So A > C: 8 + 10 + 6 = 24. C > A: 4. So A wins: 1 point. Correct.

A vs D:

Group 1: D 2nd, A 4th: D > A (4)

Group 2: A 1st, D 4th: A > D (8)

Group 3: D 2nd, A 3rd: D > A (10)

Group 4: D 1st, A 2nd: D > A (6)

So A > D: 8. D > A: 4 + 10 + 6 = 20. So A loses: 0 points. Correct.

A vs B:

Group 1: B 3rd, A 4th: B > A (4)

Group 2: A 1st, B 3rd: A > B (8)

Group 3: B 1st, A 3rd: B > A (10)

Group 4: A 2nd, B 4th: A > B (6)

Wait wait! Group 4: A 2nd, B 4th: so A > B (6 voters). So A > B: 8 + 6 = 14. B > A: 4 + 10 = 14. So tie: 0.5 points. Correct.

Now, sum the points for A: 0.5 (vs B) + 1 (vs C) + 0 (vs D) = 1.5? Wait no, wait: Copeland's method is number of wins minus number of losses? Wait no, standard Copeland's method: for each pair, candidate gets 1 point for a win, 0.5 for a tie, 0 for a loss. So total points for A:

- A vs B: tie (0.5)
- A vs C: win (1)
- A vs D: loss (0)

Total: 0.5 + 1 + 0 = 1.5? Wait but let's check again. Wait maybe I made a mistake in A vs B. Wait group 4: 6 voters, A is 2nd, B is 4th: so A is preferred over B here (6). Group 2: 8 voters, A 1st, B 3rd: A preferred (8). So A over B: 8 + 6 = 14. B over A: group 1 (4: B 3rd, A 4th) + group 3 (10: B 1st, A 3rd) = 14. So 14 vs 14: tie. So 0.5 points.

A vs C: A over C: 8 (group2) + 10 (group3) + 6 (group4) = 24. C over A: 4 (group1). 24 > 4: win, 1 point.

A vs D: A over D: 8 (group2). D over A: 4 (group1) + 10 (group3) + 6 (group4) = 20. 8 < 20: loss, 0 points.

So total points: 0.5 + 1 + 0 = 1.5. Wait but that seems odd. Wait maybe I messed up the pairs. Wait the candidates are A, B, C, D. So A has to compare with B, C, D. Let's confirm each pair:

1. A vs B:
- Voters preferring A: group2 (8) + group4 (6) = 14
- Voters preferring B: group1 (4) + group3 (10) = 14
- Tie: 0.5 points for A.

2. A vs C:
- Voters preferring A: group2 (8) + group3 (10) + group4 (6) = 24
- Voters preferring C: group1 (4)
- A wins: 1 point.

3. A vs D:
- Voters preferring A: group2 (8)
- Voters preferring D: group1 (4) + group3 (10) + group4 (6) = 20
- A loses: 0 points.

Total points: 0.5 + 1 + 0 = 1.5. Wait, but maybe the problem counts points as 1 for win, 0 for loss, and 0.5 for tie, so that's correct.

Wait but let's check the total number of voters again: 4 + 8 + 10 + 6 = 28. Correct.

Another way: for each pair, the number of voters preferring X over Y is the sum of voters where X is ranked higher than Y.

So for A vs B:

- A > B: group2 (A1, B3) → 8; group4 (A2, B4) → 6. Total 14.
- B > A: group1 (B3, A4) → 4; group3 (B1, A3) → 10. Total 14.
- Tie: 0.5 points.

A vs C:

- A > C: group2 (A1, C2) → 8; group3 (A3, C4) → 10; group4 (A2, C3) → 6. Total 24.
- C > A: group1 (C1, A4) → 4.
- A wins: 1 point.

A vs D:

- A > D: group2 (A1, D4) → 8.
- D > A: group1 (D2, A4) → 4; group3 (D2, A3) → 10; group4 (D1, A2) → 6. Total 20.
- A loses: 0 points.

Summing these: 0.5 + 1 + 0 = 1.5. So the number of points Candidate A receives is 1.5.

Wait, but maybe the problem expects integer points? Wait no, Copeland's method allows 0.5 points for ties. So the answer should be 1.5.

Answer:

\(1.5\)