Answers to the Even Exercises
Chapter
9
| 8a) A (with 4 first place votes) | 8b) B (with 15 points) | 8c) C (first D is eliminated, then B) | 8d) D |
| 10a) A | 10b) B | 10c) E | 10d) E |
| 12a) A | 12b) D | 12c) B | 12d) B |
18a)
Suppose everyone prefers A to B. Then A's Borda count will be greater than B's.
Hence, B doesn't win.
18b) Suppose alternative A is the winner under a Borda
count. Suppose the only change is that one voter puts A above B. This will increase
A's Borda count and decrease B's. All other alternatives have unchanged counts.
So A still wins.
20) Suppose everyone prefers A to B. Since B cannot have
any first place votes, B is eliminated in the first stage. Hence, B doesn't win.
22) C beats both A and B head-to-head so C is the Condorcet winner. The plurality
with runoff method results in B winning the election. Since the Condorcet winner
did not win the election, this shows that the plurality with runoff method fails
to satisfy the CWC.
24a) If the first election is contested using plurality
voting, then A wins with 2 first place votes. If the second election is contested
the same way, then A and B tie with 2 first place votes. Thus, B has gone from
non-winner status to winner status even though no voter reversed the order in
which he or she had B and A ranked.
24b) If the Hare system is used instead
of plurality, we still have A winning the first election and A and B tying for
the win in the second election. Thus, the reasoning here is the same as in part
(a).
| 32a) D | 32b) A, B, D, F | 32c) B, D, F | 32d) A, B, D, F |
Chapter 11
6) The
last juror in the permutation is the pivotal voter.
12) {A, B}, {A, B, C},
{A, B, D}, {A, C}, {A, C, D}, {A, D}, {A, B, C, D}, {B, C, D}
12a) In {A,
B}, both are critical. In {A, B, C}, A is critical. In {A, B, D}, A is critical.
In {A, C}, both are critical. In {A, C, D}, A is critical. In {A, D}, both are
critical. In {A, B, C, D}, none are critical. In {B, C, D}, all are critical.
12b) (12, 4, 4, 4)
| 14a) (4, 0) | 14b) (4, 4, 4) | 14c) (6, 2, 2) | 14d) (8, 8, 8, 0) | 14e) (12, 4, 4, 4) |
Chapter 13
2)
Calvin gets the cannon, the doubloon, the sword, the cannon ball, the wooden leg,
the flag, the crow's nest, and 3/7 of the unopened chest. Hobbes gets the anchor,
the figurehead, and 4/7 of the unopened chest.
4) Managementwins the salary
increase issue and 1/2 of the base salary issue. Labor wins benefits, vacation
time and 1/2 of the base salary issue.
8) Mary receives the car and pays $15,081.25
cash. John receives $15,081.25 cash.
10) Mary receives the car and $13,668.75
cash. John receives the house and pays $13,668.75 cash.
12) E receives the Duesenberg and Cord and pays $8500. F receives the Bentley
and Pierce-Arrow and pays $7500. G receives the Ferrari and $16,000 cash.
Chapter 14
| 6a) Abe 14, Beth 18, and Charles 4 | 6b) Abe 15, Beth 19, and Charles 3 | 6c) Tell him he was fallen victim to the Alabama paradox. |
8) Three geometry and two
algebra sections are scheduled; calculus is canceled.
10a) A gets 42 members,
B gets 27, C gets 30, and D gets 1.
10b) A gets 43 members, B gets 26, C gets
29, and D gets 2.
10c) States B and C had increased populations and decreased
apportionments. Although the population of state D decreased slightly, its apportionment
increased.
12) Last year's census gives Ash 40 seats, Beech 16, Chestnut 18,
Date 25, and The desert 1. This year's census gives Ash 41 seats, Beech 15, Chestnut
17, Date 25, and The desert 2. The population paradox occurs here since Chestnut
gained population and lost a seat while the desert lost population and gained
a seat.
16) Algebra 2, geometry 2, and calculus 1.
20) Hamilton's method
rounds the percentages to 92, 2, 2, 2, 1, and 1. The quota condition is satisfied.
Jefferson's method rounds the percentages to 95, 1, 1, 1, 1, and 1. The quota
condition is violated.
Webster's method rounds the percentages to 90, 2, 2,
2, 2, and 2. The quota condition is violated.
22) Algebra 3, geometry 1,
and calculus 1.
32) 40%
34a) North Carolina
34b) 45.88%
Chapter 15
6) The batter's optimal strategy is to guess fastball 3/4 of the
time and curve 1/4 of the time. The pitcher's optimal strategy is to throw the
fastball 1/2 of the time and the curve 1/2 of the time.
8) The batter should
always guess knuckleball and the pitcher should always throw a knuckleball.
10) The offense should should run 5/8 of the time and pass 3/8 of the time. The
defense should defend against the run 3/4 of the time and against the pass 1/4
of the time.