185
Math 115–02A: Midterm Exam
Name:
Spring 2013
Put your work on separate paper and turn it in with this sheet. Please put your answers on this sheet.
The NFL is looking to choose an “assistant to the commissioner”. The candidates are (A) Aikman, (B) Bradshaw, (C) Czonka,
(D) Davis, and (E) Edwards. The preference schedule for the election is shown below, and it is used for questions 1 - 3. Each is
worth 10 points.
# of Voters
20
14
8
7
2
st
A
B
D
C
B
2nd
C
E
B
A
D
3rd
B
D
C
E
E
4th
E
C
E
B
A
5th
D
A
A
D
C
1
1. Who will be the “assistant to the commissioner” under the plurality with elimination method?
2. Who will be the “assistant to the commissioner” under the Borda Count method?
3. Which, if any, candidate is the Condorcet candidate? Support you answer. Comment on the
fairness of your answers from #1 and #2.
4. An election is held among six candidates (A, B, C, D, E, and F). The method of pairwise
comparisons (Copeland’s Method) is used to determine a winner. Candidates A, B, and C
each win 2 head-to-head comparisons, candidate D wins 4, and candidate E wins none. How
many does candidate F win? What can be said about candidate F, be as specific as possible?
Hint: Think about how many total comparisons there should be. (10 points)
5. Consider the weighted voting system [q: 5, 4, 3, 2, 2, 1].
(5 points each)
(a) What is the largest possible value of q for this system to be valid?
(b) What is the smallest possible value of q for this system to be valid?
(c) If no one is to have veto power, what is the largest possible value of q?
6. (a) How many coalitions (Banzhaf) are there for a weighted voting system consisting of 8
players? (5 points)
(b) How many sequential coalitions (Shapley-Shubik) are there for a weighted system
consisting of 6 players? (5 points)
7. Consider the weighted voting system [12 : 8, 6, 4, 2, 1] . For each coalition, determine which
player(s) are critical players. (5 points each)
(a) { P1 P2 P3 }
(b) {P2 P3 P4 P5 }
(c)
{P1 P2 P3 P4 }
8. Consider the weighted voting system [10 : 8, 4, 4, 2] .
determine which player is the pivotal player. (5 points each)
For each sequential coalition,
(a) P1 P3 P2 P4
(b) P4 P3 P1 P2
(c) P3 P2 P4 P1
9. Find the Banzhaf power distribution for the system [25: 16, 9, 9, 6]. Is it fair? Explain.
(10 points)
Questions 10 and 11 deal with the following: Dale likes strawberry and banana equally, but values chocolate twice
as much. Dale has a cake that cost $18 that is one-third each of chocolate, banana, and strawberry.
C
S
B
10. Determine the value (in $) of each section of the cake “in Dale’s eyes”.
(5 points)
11. Dale is offered cake made up of 20° chocolate, 45° banana and 25° strawberry. How much
does Dale believe this piece is worth? If Dale is sharing this cake with 2 other people, would
he consider this amount to be a “fair share”? (8 Points)
12. Four people are to divide a cake using the lone divider method. Given the table below,
describe all possible fair divisions. (12 Points)
Divider
Chooser 1
Chooser 2
Chooser 3
s1
25%
40%
25%
30%
s2
25%
20%
30%
20%
s3
25%
20%
30%
25%
s4
25%
20%
15%
25%
13. Four heirs are going to divide an estate using the method of sealed bids. The bids are shown
below in the table. Describe the outcome, including dealing with any surplus money.
(15 points)
House
Boat
Car
A
200000
48000
26000
B
185000
50000
28000
C
195000
49000
30000
D
190000
52000
28000
14. Six players are to divide a piece of land using the last diminisher method. In round 1 player
3 plays, but everyone else passes. In round 2 everyone passes. In round 3 each player plays.
In round 4, no one plays. Use this information to answer the following. (2 points each)
(a) Who gets a “fair share” at the end of round 1?
(b) Who starts round 2?
(c) Who gets a “fair share” at the end of round 2?
(d) Who gets a fair share in round 3?
(e) Which players will finish the “game” by playing Divider-Chooser on their own?
Questions 15 – 17 deal with the following: The Interplanetary Federation of Planets Consists of 6 planets: Alderaan,
Bespin, Coruscant, Dagobah, Endor, and Felucia. The federation is governed by a Congress consisting of 200 seats.
Each problem is worth 10 points.
15. Complete the table below:
State
A
Population
1137000
B
1498000
C
3862000
D
807000
E
1042000
F
Total:
1654000
Std. Quota
Low. Quota
Up. Quota
Rounded Quota
16. How many seats in the Congress should each planet receive under Hamilton’s method? Is
the apportionment fair? Explain.
17. How many seats in the Congress should each planet receive under Jefferson’s method? Is the
apportionment fair? Explain.