5) We have to find the winner using the Copeland method.
In this method we compare each pair of candidates and see which one is preferred.
The candidate that wins is awarded one point.
If it is a tie, each one gets 1/2 points.
In this case, we have to consider the number of votes per column.
Then, we will have 4!/(2! * 2!) = 24/4 = 6 pairs.
We compare each pair:
-AB:
For the first column: A
For the second column: B
For the third column: A
For the fourth column: B
For the fifth column: B
For the sixth column: A
Then, for A we have 7 + 11 + 12 = 30.
For B we have 10 + 6 + 6 = 22.
Then A earns one point.
For the next pairs we repeat the process:
-AC:
We will directly count the votes for each candidate.
Votes for A: 11 + 6 + 12 = 29.
Votes for C: 7 + 10 + 6 = 23.
Then A earns another point.
-AD:
Votes for A: 7 + 12 = 19
Votes for D: 10 + 11 + 6 + 6 = 33
Then D earns one point.
-BC:
Votes for B: 10 + 6 + 6 + 12 = 34
Votes for C: 7 + 11 = 18
Then B earns one point.
-BD:
Votes for B: 10 + 6 + 12 = 28
Votes for D: 7 + 11 + 6 = 24
Then B earns another point.
-CD:
Votes for C: 7 + 10 +6 + 12 = 35
Votes for D: 10 + 6 = 16
Then C earns a point.
The total points are:
A: 2 points
B: 2 points
C: 1 point
D: 1 point
We have a tie between A and B.
We don't have a rule that is applied to break the tie, but we can look at the pair AB to see which one wins in that case. If we look A > B, so we can consider A to be the winner after the tie.
6) We have to apply the plurality method.
In this case we only look at the first choices.
Then, from the first column, C gets 7 votes.
From the second column, B gets 10 votes.
From the third column, D gets 12 votes.
From the fourth column, B gets another 6 votes.
From the fifth column, D gets another 6 votes.
From the sixth columns, A gets 12 votes.
Then, we can summarize as:
A = 12
B = 10 + 6 = 16
C = 7
D = 12 + 6 = 18
Then, D is the winner according to the plurality method.
Answer:
5) The Copeland's method gives a tie between A and B. We can break the tie comparing the pair (A,B), where A wins over B.
6) The plurality method gives D as the winner.