Answers to the Even Exercises

Chapter 9

10a) A (with 4 first place votes)10b) B (with 15 points)10c) C (first D is eliminated, then B)10d) D

 

12a) A12b) B12c) E12d) E

 

14a) A14b) D14c) B14d) B

 

20a) Suppose everyone prefers A to B. Then A's Borda count will be greater than B's. Hence, B doesn't win.
20b) 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.
22) 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.
24) 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.
26a) 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.
26b) 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).

Chapter 11

8c) (1/2, 1/6, 1/6, 1/6)
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) Management wins 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.
22) You would rather be the chooser. That way, if you find one piece more desirable than the other, you can choose it.

Chapter 14

6a) Abe 14, Beth 18, and Charles 46b) Abe 15, Beth 19, and Charles 36c) Tell him he was fallen victim to the Alabama paradox.

8) Three geometry and two algebra sections are scheduled; calculus is canceled.
12)

State
Population
Apportionments
A
5,576,330
25
26
26
B
1,387,342
6
6
6
C
3,334,241
15
15
16
D
7,512,860
34
34
35
E
310,968
2
2
1
Totals
18,121,741
82
83
84

The Alabama paradox occurs when the apportionment for state E decreases from 2 to 1 as the house size increases from 83 to 84.
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.
24) With the Hamilton and Webster methods, the Liberals get 49 votes and the Tories get 50.
26) 40%
28a) North Carolina
28b) 45.88%
30a) California: 640,204; Utah: 745,571
30b) Absolute difference: 105,367; relative difference: 16.46%
30c) Absolute difference: 93,336.5; relative difference: 16.69%
30d) The absolute difference in district populations would be less if California had 52 seats and Utah had 4. With that revised apportionment, the relative difference would be greater. Thus, the Hill-Huntington method did not minimize absolute differences in district population. It minimized relative differences.
32) Algebra 3, geometry 1, and calculus 1.

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.