Question

Question 7: [10pt total] Consider the following voter profile: Votes 13 9 8 10 Preferences А с B D B с D A с B D А D B с Calculate the following: 7)a) [3pt) Social ranking under the Plurality system. Plurality Social Ranking: 7)b) (3pt) Social ranking under the Instant Runoff system. Instant Runoff Social Ranking: 7)c) [4pt Condorcet winner (if one exists). Condorcet Winner:

Answer 1

7(A) Plurality Method Social ranking : -

VOTES	13	9	8	10
1ST COICE	A	B	C	D
2ND CHOICE	C	C	B	B
3RD CHOICE	B	D	D	C
4TH CHOICE	D	A	A	A

For Plurality Method we, see calculate the first choice votes for each candidate and the person who got the maximum first choice vote is declared winner.

Votes obtained by candidate

A got 13

B got 9

C got 8

D got 10

So, there social ranking is as follows

A(13) ---> D(10) ----> B(9) ----> C ( 8)

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(B) Social ranking under Instant runoff Method ; -

In this Method, We calculate the first choice for each candidate and then comapare whether any of them get the majority vote, if majority is there the person is declared winner and the remaining ranks are obtained. If there is no majority then we eliminate the person with the least votes and his votes are given to the candidate who is second choice after eliminating him in his column.

Successively again the votes are calculated and then least vote receiver is eliminated till the winner is not obtained.

VOTES	13	9	8	10
1ST COICE	A	B	C	D
2ND CHOICE	C	C	B	B
3RD CHOICE	B	D	D	C
4TH CHOICE	D	A	A	A

TOTAL VOTES = 40

MAJOIRT VOTE =21

A = 13

B = 9

C =8

D = 10

SINCE, NO ONE GETS 21, SO, ELIMINTING THE LEAST VOTES RECEIVER

C IS ELIMINATED AND VOTES OF C IS GIVEN TO B

VOTES	13	9	8	10
1ST COICE	A	B		D
2ND CHOICE			B	B
3RD CHOICE	B	D	D
4TH CHOICE	D	A	A	A

NOW NO. OF VOTES

A = 13

B = 9+8 = 17

D = 10

LEAST VOTES RECEIVER IS D, SO ELIMINATED AND VOTES ARE TRANSFERRED TO B

VOTES	13	9	8	10
1ST COICE	A	B
2ND CHOICE			B	B
3RD CHOICE	B
4TH CHOICE		A	A	A

A = 13

B = 17 + 10 = 27

SINCE A HAS LEAST VOTE, SO ELIMINATED.

B IS DECLARED AS WINNER

SOCIAL RANKINGS ARE GIVEN AS

B---> A---->D ---> C

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(c) Condorcet Winner

Condorcet Winner is determined if there is a choice that is preferred in every one to one comparison with the other choices, that should be the winner, This winner is called as condorcert winner.

Since we know that by plurality method we have A as a winner

VOTES	13	9	8	10
1ST COICE	A	B	C	D
2ND CHOICE	C	C	B	B
3RD CHOICE	B	D	D	C
4TH CHOICE	D	A	A	A

We take one on one comparison for the following

A vs B

A vs C

A vs D

B vs C

B vs D

C vs D

VOTES	13	9	8	10
1ST COICE	A	B
2ND CHOICE			B	B
3RD CHOICE	B
4TH CHOICE		A	A	A

(i) A vs B

A gets 13 votes

B gets 9+8+10 = 27 votes

A winner

(ii) A vs C

VOTES	13	9	8	10
1ST COICE	A		C
2ND CHOICE	C	C
3RD CHOICE				C
4TH CHOIC		A	A	A

A gets 13 votes

C gets 27 votes

C winner

(iii) A vs D

VOTES	13	9	8	10
1ST COICE	A			D
2ND CHOICE
3RD CHOICE		D	D
4TH CHOICE	D	A	A	A

A gets 13 votes

D gets 27 votes

D winner

(iv) B vs C

VOTES	13	9	8	10
1ST COICE		B	C
2ND CHOICE	C	C	B	B
3RD CHOICE	B			C
4TH CHOICE

B gets 9+10 = 19

C gets 13+8 = 21

C winner

(v) B vs D

VOTES	13	9	8	10
1ST COICE		B		D
2ND CHOICE			B	B
3RD CHOICE	B	D	D
4TH CHOICE	D

B gets 30

D gets 10

B winner

(vi) C vs D

VOTES	13	9	8	10
1ST COICE			C	D
2ND CHOICE	C	C
3RD CHOICE		D	D	C
4TH CHOICE	D

C gets 30

D gets 10

C winner

C is preffered maximum times (3 TIMES)

Therefore Our Concocert Winner is "C"

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