MGF 1107/ Classroom examples/ Chapter 13
1. Two legislative aides for a congressman are assigned to research 12 proposed bills. The congressman gives each aide 6 bills to work on. Juan complains that the division of the workload is unfair in that several of his bills are much more complex than those given to Patsy. How can they divide the work so that neither of them is envious of the other’s share of the load?
2. Use the lone-divider method to parcel a plot of land into 3 parcels s1, s2, and s3 if the chooser bids are:
a) C1: {s1, s2} b) C1: {s2}
C2: {s1} C2: {s2}
3. Four countries want to divide a piece of land using the lone-divider method. The divider D slices the land into 4 parcels s1, s2, s3 and s4. The value of each parcel (as a percentage of the total value of the land) in each chooser’s eye is given in the following table.
|
|
s1 |
s2 |
s3 |
s4 |
|
C1 |
10% |
50% |
30% |
10% |
|
C2 |
25% |
40% |
15% |
20% |
|
C3 |
15% |
30% |
20% |
35% |
a) What should each chooser’s bids be?
b) Describe a possible fair-division of the land.
4. Alicia and Bruce are getting divorced and must divide up the following items:
a. Retirement account. A six-figure retirement account has been built up from Bruce’s employment over several years. This is valuable to both spouses, but it is more so to Alicia because Bruce will have more opportunity than Alicia to reestablish such a fund before reaching retirement.
b. Home. This is a nice but not particularly extravagant house. Because Bruce must remain close to his business, it is worth more to him than to Alicia.
c. Summer cottage. This is a year-round house that is considerably less valuable than their home. But Alicia realizes that she could live there quite comfortably.
d. Investments. These are largely mutual funds that are of considerably less value than the retirement account.
e. Car. This is of more value to Bruce because Alicia
drives a company car.
f. Dog. Alicia inherited Snoopy when her sister died but Bruce has
also grown attached to him.
Bruce and Alicia are asked to distribute 100 points over the 5 items in a
way that reflects the relative worth of the items to them. This is done independently
and simultaneously with the results as follows:
|
Item |
Alicia |
Bruce |
|
Retirement account |
46 |
32 |
|
Home |
20 |
35 |
|
Summer Cottage |
15 |
10 |
|
Investments |
10 |
10 |
|
Car |
5 | 10 |
| Dog | 4 | 3 |
Use the adjusted winner procedure to determine a fair allocation of the assets.
5. In September of 1978, President Jimmy Carter invited President Anwar Sadat of Egypt and Prime Minister Menachem Begin of Israel to meet with him at Camp David to try to settle a 30-year-old dispute between the two countries. There were 6 principal areas of disagreement.
1. The Sinai Peninsula. This large tract of land, conquered by Israel during the six-day war in 1967, was the most important issue to both sides. For Israel, the Sinai provided a military buffer in case of a possible Egyptian attack. The land included captured oil fields that were of significant economic value. For Egypt, the Sinai was of such great importance that no agreement could be achieved that did not include Egyptian control over this territory.
2. Diplomatic recognition of Israel. Since its creation in 1948, Israel has not been recognized as a legitimate and sovereign nation by its Arab neighbors. This was important to Israel because it would help liberate it from its pariah status in the region. Egypt balked at normalizing relations because it would damage Egypt’s relations with other Arab nations.
3. The West Bank and the Gaza strip. This land was more geographically and historically important to Israel than even the Sinai. Begin considered them to be part of Israel and not occupied foreign land. For Egypt, these two territories had little economic or strategic worth.
4. Formal linkage of accords and Palestinian autonomy. Sadat pushed for Israeli recognition of the Palestinians’ right to self-determination, although he realized that a truly substantive agreement on this issue could not be negotiated by Egypt alone. Begin opposed any linkage between an Egyptian-Israeli agreement and the issue of Palestinian autonomy.
5. Israeli recognition of Palestinian rights. From the Egyptian perspective, some form of Israeli recognition of the rights of Palestinians was deemed necessary. This was difficult for Israel to agree to because of competing sovereignty claims between the Israelis and the Palestinians.
6. Jerusalem. The United Nations demanded in 1949 that the city be internationalized because of competing religious and political claims. It was split into eastern and western sections until Israel captured and unified the city in 1967. For Israel, Jerusalem was its capital and could not be relinquished. While Egypt faced pressure from other Arab nations on this issue, Sadat did not push strenuously, and seemed willing to leave the issue for the future.
Brams and Taylor assigned the following hypothetical point allocations based on the preceding analysis of each side’s interests.
|
|
Israel |
Egypt |
|
Sinai |
35 |
55 |
|
Diplomatic recognition |
10 |
5 |
|
West Bank/ Gaza Strip |
20 |
10 |
|
Linkage |
10 |
5 |
|
Palestinian rights |
5 |
20 |
|
Jerusalem |
20 |
5 |
Apply the adjusted winner procedure to the Camp David negotiations.
6. In her last will and testament, Grandma plays a little joke on her four grandchildren (Art, Betty, Carla, and Dave) by leaving just three valuable items – a house, a Rolls Royce, and a Picasso painting -- with the stipulation that the items must remain with the grandchildren (not sold to outsiders) and must be divided fairly in equal shares among them. Each of the grandchildren submits a sealed bid which represents his or her honest assessment of the dollar value of each item. The bids are shown below.
|
|
Art |
Betty |
Carla |
Dave |
|
House |
220,000 |
250,000 |
211,000 |
198,000 |
|
Rolls Royce |
40,000 |
30,000 |
47,000 |
52,000 |
|
Picasso |
280,000 |
240,000 |
234,000 |
190,000 |
Use the Knaster inheritance procedure to accomplish the fair division.
7. Three sisters (Ana, Betty, and Carmen) wish to use the Knaster inheritance procedure
to divide up 4 pieces of furniture they shared as children. Here are their bids:
|
|
Ana |
Betty |
Carmen |
|
Dresser |
$150 |
$300 |
$275 |
|
Desk |
$180 |
$150 |
$165 |
|
Vanity |
$170 |
$200 |
$260 |
|
Tapestry |
$400 |
$250 |
$500 |
What is the outcome of this fair division problem?
8. Suppose we have four items (W, X, Y, and Z) and four people
(Bob, Carol, Ted, and Alice). Assume that each of the people spreads 100 points
over the items (as in the Adjusted Winner Procedure) to indicate the relative
worth of each item to that person:
|
Bob
|
Carol
|
Ted
|
Alice
|
|
|
W
|
25
|
50
|
30
|
25
|
|
X
|
25
|
30
|
20
|
25
|
|
Y
|
25
|
10
|
20
|
30
|
|
Z
|
25
|
10
|
30
|
20
|
Suppose Bob is given X, Carol gets W, Ted gets Z, and Alice
gets Y. Is this allocation:
a) proportional?
b) envy-free?
c) equitable?
d) Pareto-optimal?
9. Suppose we have two items (X and Y) that must be divided
by Jose and Manny. Assume that Jose and Manny each spread 100 points over
the items (as in the Adjusted Winner Procedure) to indicate the relative worth
of each item to that person:
|
Jose
|
Manny
|
|
|
X
|
40
|
60
|
|
Y
|
60
|
40
|
Suppose Jose is given X and Manny is given Y. Is this allocation:
a) proportional?
b) envy-free?
c) equitable?
d) Pareto-optimal?