1.6 Rankings

• Sometimes, we are not just interested in the first place winner of the election, but also want to know who comes in second, third …

• There are 2 methods which we can use to rank candidates:– Extended Method– Recursive Method

Extended Plurality Method

• Since A gets 10 votes, B gets 11 votes, C gets 4 votes, and D gets 2 votes so the order will be.

1st place winner: B

2nd place winner: A

3rd place winner: C

4th place winner: D

10 6 5 4 2

1st A B B C D

2nd C D C A C

3rd B C A D B

4th D A D B A

Extended Borda Count

A1A4A8A:10D:144th choice

B2C8B16D:20C:283rd choice

D3D12C:24B:30B:422nd choice

C4B16D:32C:40A:561st choice

1481014Number of voters

1point

2points3points

4points

A gets 79 points, B gets 106 points, C gets 104 points, and D gets 81 points. Therefore the order will be:

1st place winner: B

2nd place winner: C

3rd place winner: D

4th place winner: A

Extended Pairwise Comparisons

Number of voters

14 10 8 4 1

1st choice A C D B C

2nd choice B B C D D

3rd choice C D B C B

4th choice D A A A A

C gets 3 points, B gets 2 points, D gets 1 point, and A gets 0 point, so the order will be:

1st place winner: C

2nd place winner: B

3rd place winner: D

4th place winner: A

Extended Plurality with Elimination Method

Number of voters

14 10 8 4 1

1st choice A C D B C

2nd choice B B C D D

3rd choice C D B C B

4th choice D A A A A

Number of voters

14 10 8 4 1

1st choice A D D D D

2nd choice D A A A A

Number of voters

14 10 8 4 1

1st choice A C D D C

2nd choice C D C C D

3rd choice D A A A A

4th choice

B gets eliminated first, C gets eliminated on the 2nd round and then A. Therefore the order will be:

1st place: D

2nd place: A

3rd place: C

4th place: B37 votes, need 19 for a majority vote

Recursive Plurality Method

10 6 5 4 2

1st C D C C D

2nd D C D D C

10 6 5 4 2

1st A B B C D

2nd C D C A C

3rd B C A D B

4th D A D B A

B is the 1st place winner.

Now remove B and apply the same method again

A is 2nd place winner

Now remove A

10 6 5 4 2

1st A D C C D

2nd C C A A C

3rd D A D D A

C is the 3rd place winner

And D is the 4th place winner

Recursive Borda Count

A1A4A8A:10D:144th choice

B2C8B16D:20C:283rd choice

D3D12C:24B:30B:422nd choice

C4B16D:32C:40A:561st choice

1481014Number of voters

1point

2points

3points

4points

A gets 79 points, B gets 106 points, C gets 104 points, and D gets 81 points. Therefore the order will be:

1st place winner: B

Now remove B and create a new table to find the 2nd place winner

Work together on the board

Recursive Plurality with Elimination Method

Number of voters

14 10 8 4 1

1st choice A C D B C

2nd choice B B C D D

3rd choice C D B C B

4th choice D A A A A

Number of voters

14 10 8 4 1

1st choice A D D D D

2nd choice D A A A A

Number of voters

14 10 8 4 1

1st choice A C D D C

2nd choice C D C C D

3rd choice D A A A A

4th choice

D is the 1st place winner. Now remove D and do the whole process all over again.

Work together on the board