SOLUTIONS TO HW 1

4. (a) 55 students voted.

(b) 28 votes for majority.

(c) A and B both had the most votes at 16.

(d) C only had 11 votes.

(e) C had 4

(f) D had 19

10. (a)There would be a tie between A and B.

(b) B has fewer last place votes than A, so the exam would be on December 20, 9:00pm.

12. (a) CADB--C (great, student can go home on the 22 and some people will still be in town to give a ride), A, D (bad, student wouldn't be able to leave on the 22nd with this option), B (worst, student has a conflicting exam).

(b) If she votes ACDB she will break the tie between A and B for 1st place votes.

21. (a) 40% x 100 voters = 40 votes.

25% x 100 voters = 25 votes.

20% x 100 voters = 20 votes.

15% x 100 voters = 15 votes.

A=270 pts, B=295 pts, C=175 pts, D=245. B wins.

(b) 40N, 25N, 20N, and 15N votes. B wins with 295N pts.

(c)The winner will always be B no matter what N equals (how many voters there are), because N doesnt change the fact that B has the highest percentage of points.

27. There are 21 total votes. E has no 1st place votes, and is eliminated. B has the next fewest 1st place votes--eliminated. B's votes can't go to E, so they go to A. A now has 5+3+3=11 votes which is the majority. A wins.

35. Total votes=26

A&B--A (13), B(13); tie, each get 1/2 pt

A&C--A(21), C(5); A gets 1pt

A&D--A(14), D(12); A gets 1 pt

B&C--B(16), C(10); B gets 1pt

B&D--B(8), D(18); D gets 1pt

C&D--C(13), D(13);tie, each get 1/2 pt

A has the most pts at 2.5 pts and wins.

41. (a) ACDB. A wins with 12 1st place votes, then C with 9, then D with 5, and lastly B with 1.

(b) ACDB. A=74, C=72 D=60 B=55

(c) CADB. At first A=12 B=1 C=9 D=5. So B is eliminated and its 1 vote goes to A so A=13. Next D is eliminated and D's 5 votes go to C, so C=14 pts. C has more than A, so C wins.

(d)A&B--A (21), B(6); A gets 1 pt

A&C--A(13), C(14); C gets 1pt

A&D--A(13), D(14); D gets 1 pt

B&C--B(9), C(18); C gets 1pt

B&D--B(13), D(14); D gets 1pt

C&D--C(13), D(14); D gets 1 pt

DCAB. D=3pts, C=2pts, A=1pt, B=0pts

52. (a) A&B--A (12), B(15); B gets 1 pt

A&C--A(12), C(15); C gets 1pt

A&D--A(10), D(17); D gets 1 pt

B&C--B(8), C(19); C gets 1pt

B&D--B(11), D(16); D gets 1pt

C&D--C(15), D(12); C gets 1 pt

C is the Condorcet candidate as C is prefered when compared to each of the other 3 candidates.

(b) To start, A=10, B=6, C=5, D=6. C is eliminated first, 5 votes go to B. Then D is eliminated, 2 votes go to A and 4 go to B. B is the winner with 15 votes over A's 12.

(c)If D drops out, 2 more votes go to A and 4 go to C. (A=12, B=6, C=9). B is eliminated and 6 votes go to C, who now has 15 votes. C wins.

(d)The Condorcet Criterion is violated using plurality with elimination, as the condorcet candidate doesn't win. Between (b) and (c), the Independence of Irrelevant Alternatives Citerion is violated. B should have remained the winner dispite D dropping out.

64. (a) B wins. (A is eliminated, then D, then C)

(b)Still under debate.

(c)Ditto

71. (a) One example is:

4 3 2

B A A

C C B

A B C

(b)The Condorcet Loser might have, or be passed down, enough votes to keep form being eliminated.

(c) The condorcet loser would have mostly last place votes, and so his/her points would not add up to enough for a win with the Borda count.