Question

The Coombs method for plurality voting is very similar to the Plurality with Elimination method we practiced in class. The only difference is that the elimination round eliminates the candidate with the largest number of last-place votes (rather than the candidate with the least number of first place votes).

Use the Coombs method to determine the winner of a race between candidates A, B, C, and D, where voters have the following preferences between the candidates:

Rank of preferences:	# of voters with that preference:
A > B > C > D	14
C > B > D > A	10
D > C > B > A	8
B > D > C > A	4
C > D > B > A	1

Also, did this example of counting using the Coombs method violate the Independence of Irrelevant Alternatives fairness axiom? How can you tell?

Answer 1

Modifying the vote table:

We will show the preference position, 1^st, 2^nd, 3^rd and 4^th for each of the no of votes now.

Hence,

Number of votes	14	10	8	4	1
1st	A	C	D	B	C
2nd	B	B	C	D	D
3rd	C	D	B	C	B
4th	D	A	A	A	A

Here total no of votes = 14 + 10 + 8 + 4 + 1 = 37

For majority, the no of votes should be more than 50%, i.e., more than 37/2. Hence the no of votes should be minimum 19

Now calculating the no of votes for each 1^st position case

Number of votes	14	10	8	4	1
1st	A	C	D	B	C

A = 14

B = 4

C = 11

D = 8

Here, none of the candidates have minimum 19 votes.

Hence, we will apply Coombs Method

According to the coombs method, If none are in majority, we will eliminate the highest voted person in the last preference.

Here the last preference is the 4^th place.

Number of votes	14	10	8	4	1
4th	D	A	A	A	A

D = 14

A = 10+8+4+1 = 23

Hence A will be eliminated from the whole table and the rest candidates will now move up.

Eliminating A

Number of votes	14	10	8	4	1
1st		C	D	B	C
2nd	B	B	C	D	D
3rd	C	D	B	C	B
4th	D	A	A	A	A

Now taking each member up if any cell is empty due to elimination of A

Number of votes	14	10	8	4	1
1st	B	C	D	B	C
2nd	C	B	C	D	D
3rd	D	D	B	C	B

Now total no of votes for each candidate in 1^st position is:

B = 14 + 4 = 18

C = 10 + 1 = 11

D = 8

Now also no one attained the majority.

Hence we will go through the process again. We have to eliminate the maximum voted candidate from the 3^rd position now.

Number of votes	14	10	8	4	1
3rd	D	D	B	C	B

D = 14 + 10 = 24

B = 8 + 1 = 9

C = 4

Hence D will be eliminated.

Number of votes	14	10	8	4	1
1st	B	C		B	C
2nd	C	B	C
3rd			B	C	B

Hence taking one up to fill the positions of D

Number of votes	14	10	8	4	1
1st	B	C	C	B	C
2nd	C	B	B	C	B

Now taking the vote count:

B = 14 + 4 = 18

C = 10 + 8 + 1 = 19

Now we got the majority. As 19 is our majority and C attained it.

Hence the winner is C

No, it doesn’t violate the Independence of Irrelevant Alternatives fairness axiom.

Reason: The axiom says that, if something is a majority in the first place compared to the other one, then while taking the decision, the second person should never become a majority.

Here in the starting, the vote count for B was 4 and C was 11. Hence C was a majority among these two. While deciding the winner, C remained the majority and wont the voting. Hence it is not violating the axiom.

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