Math 115 – 03A: Midterm Exam (Chapters 1 – 4)Name:
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 / 21st / 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.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.(a)How many coalitions are there for a weighted voting system consisting of 8 players?
(b)How many sequential coalitions are there for a weighted system consisting of 6 players?
6.In each of the weighted voting systems shown below, determine which players are dictators, have veto power, or are dummies. (5 points each)
(a)[6: 4, 2, 1]
(b)[6: 7, 3, 1]
(c)[6: 5, 3, 1]
7.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?
8.Find the Banzhaf power distribution for the system [25: 16, 9, 9, 6]. Is it fair? Explain. (10 points)
9.Find the Shapley-Shubik power distribution of the system [8: 6, 3, 2]. Is it air? 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 ofchocolate, banana, and strawberry.
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)
s1s2s3s4
Divider25%25%25%25%
Chooser 140%20%20%20%
Chooser 225%30%30%15%
Chooser 330%20%25%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. (12 points)
ABCD
House200000185000195000190000
Boat48000500004900052000
Car26000280003000028000
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)Who starts round 4?
(f)Which players will finish the “game” by playing Divider-Chooser on their own?
15.State the “Quota Rule” for an apportionment problem. (6 points)
Questions 16 – 19 deal with the following: The Interplanetary Federation of Fraternia 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.
16.Complete the table below:
StatePopulationStd. QuotaLow. QuotaUp. QuotaRounded Quota
A1137000
B807000
C3862000
D1498000
E1042000
F1654000
Total:
17.How many seats in the Congress should each planet receive under Hamilton’s method? Is the apportionment fair? Explain.
18.How many seats in the Congress should each planet receive under Jefferson’s method? Is the apportionment fair? Explain.
19.How many seats in the Congress should each planet receive under Webster’s method? Is the apportionment fair? Explain.