Math 103:18 Post-class assignment 9 due Apr 22, 2021
1. Eve, Fred, Gail, and Hank are four siblings who inherit a large piece of land. The land is not uniform, and has different features in different locations. A Divider is selected at random, and divides the land into 4 parts (“shares”), which we will call s1, s2, s3, and s4.
(a) Eve, Fred, Gail, and Hank each write their assessments of the value of each share, recorded in the following table. For each player, find his/her minimum fair share amount in dollars, find his/her bid list (list of which shares will be fair shares for him/her), and record these in the table:
value of s1
value of s2
value of s3
value of s4
Minimum fair share amount in dollars
Bid list
Eve
$240,000
$120,000
$100,000
$140,000
Fred
$350,000
$250,000
$150,000
$250,000
Gail
$400,000
$100,000
$100,000
$120,000
Hank
$250,000
$250,000
$250,000
$250,000
(b) Who must have been the Divider, and how can you tell?
(c) There is a "standoff" here between Eve and Gail. Suppose that before the standoff is resolved, Fred is given s4 and Hank is given s3. Describe in words, as specifically as possible in light of the information available, what is done at this stage to resolve the standoff.
(d) There is not enough information given to find the exact dollar value of the share which Gail will ultimately receive, after the standoff is resolved. Based on the available information, Gail will receive a share worth at least how much? Is this a fair share, in relation to the original problem of dividing the land among the four siblings?
2. Kelly and Lauren use the Divider-Chooser method to divide use of a retail space over the course of a year. As before
Each runs a different type of business. Their individual preferences are as follows:
Jan
Fe
Ma
Ap
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
total
Kelly
2
1
2
1
3
1
1
1
1
1
1
1
16
Lauren
1
2
2
2
2
3
1
1
1
1
1
1
18
(a) If Lauren is the Divider, at what point in the year does she make the division.?
(b) If we call the two shares into which Lauren has divided the year s1 and s2, how much are each of s1 and s2 worth to Kelly, as a fraction (or percentage) of what the whole year is worth?
(c) Describe Kelly's share (i.e. which share does she select) and Lauren's share (which share is she left with) at the end of the Divider-Chooser method.
Now Jason appears, with documents proving that he too has the rights to a fair share of the retail space. Rather than start over, they agree to use the Lone Chooser method, and build on what Kelly and Lauren have already done. This means that Jason is the Chooser, and Lauren is the first Divider (and she has already done the work of the first division). Here are all their preferences:
Jan
Fe
Ma
Ap
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
total
Jason
1
1
1
1
1
1
1
1
1
1
1
1
12
Kelly
2
1
2
1
3
1
1
1
1
1
1
1
16
Lauren
1
2
2
2
2
3
1
1
1
1
1
1
18
Subdivision:
(d) Describe the second division which Kelly makes, i.e. describe (in terms of specific months or parts of months) the 3 subshares k1, k2, k3 she creates from her share, and indicate how much each subshare is worth to her. You may describe the value to Kelly in terms of “Kelly points”, or as a fraction of the value of the whole year.
(e) Describe the second division which Lauren makes, i.e. describe (in terms of specific months or parts of months) the 3 subshares l1, l2, l3 she creates from her share, and indicate how much each subshare is worth to her. You may describe the value to Lauren in terms of “Lauren points”, or as a fraction of the value of the whole year.
Selection: Now Jason, the Chooser, returns.
(f) In the following table, write down how much each subshare of Kelly's is worth to Jason, and how much each subshare of Lauren's is worth to Jason. You may describe the value to Jason in terms of “Jason points”, or as a fraction of the value of the whole year.
Subshare
k1
k2
k3
l1
l2
l3
value to Jason
(g) Which subshares does Jason select?
(h) By filling out the following table, describe the final fair division of the year (indicating which months each player receives), and find the value of each player’s final share as a fraction (or percentage) of the value of the entire year:
Playe
Subshares received
Description
(which months or parts of months)
Fraction of value of the entire yea
Jason
Kelly
Lauren
3. Recall the scenario of problem 1 above, in which Eve, Fred, Gail, and Hank are four siblings who inherit a large piece of land, and use the Lone Divider method to produce a fair division. Suppose that Fred takes a second look at the numbers he originally wrote down, and decides to reevaluate the shares, so that the following is the full set of information before the division is ca
ied out:
Value of s1
Value of s2
Value of s3
Value of s4
Eve
$240,000
$120,000
$100,000
$140,000
Fred
$550,000
$150,000
$150,000
$150,000
Gail
$400,000
$100,000
$100,000
$120,000
Hank
$250,000
$250,000
$250,000
$250,000
(a) There is a standoff here among Eve, Fred, and Gail. Suppose that before the standoff is resolved, Hank is given s4. Describe in words, as specifically as possible in light of the information available, what is done at this stage to resolve the standoff. [Hint: there’s very little you can say specifically!]
(b) There is not enough information given to find the exact dollar value of the share which Gail will ultimately receive, after the standoff is resolved. Based on the available information, Gail will receive a share worth at least how much? Is this a fair share for her, in relation to the original problem of dividing the land among the four siblings?
Math 103:18 Post-class assignment 9 due Apr 22, 2021
1. Eve, Fred, Gail, and Hank are four siblings who inherit a large piece of land. The land is not uniform, and has different features in different locations. A Divider is selected at random, and divides the land into 4 parts (“shares”), which we will call s1, s2, s3, and s4.
(a) Eve, Fred, Gail, and Hank each write their assessments of the value of each share, recorded in the following table. For each player, find his/her minimum fair share amount in dollars, find his/her bid list (list of which shares will be fair shares for him/her), and record these in the table:
value of s1
value of s2
value of s3
value of s4
Minimum fair share amount in dollars
Bid list
Eve
$240,000
$120,000
$100,000
$140,000
Fred
$350,000
$250,000
$150,000
$250,000
Gail
$400,000
$100,000
$100,000
$120,000
Hank
$250,000
$250,000
$250,000
$250,000
(b) Who must have been the Divider, and how can you tell?
(c) There is a "standoff" here between Eve and Gail. Suppose that before the standoff is resolved, Fred is given s4 and Hank is given s3. Describe in words, as specifically as possible in light of the information available, what is done at this stage to resolve the standoff.
(d) There is not enough information given to find the exact dollar value of the share which Gail will ultimately receive, after the standoff is resolved. Based on the available information, Gail will receive a share worth at least how much? Is this a fair share, in relation to the original problem of dividing the land among the four siblings?
2. Kelly and Lauren use the Divider-Chooser method to divide use of a retail space over the course of a year. As before
Each runs a different type of business. Their individual preferences are as follows:
Jan
Fe
Ma
Ap
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
total
Kelly
2
1
2
1
3
1
1
1
1
1
1
1
16
Lauren
1
2
2
2
2
3
1
1
1
1
1
1
18
(a) If Lauren is the Divider, at what point in the year does she make the division.?
(b) If we call the two shares into which Lauren has divided the year s1 and s2, how much are each of s1 and s2 worth to Kelly, as a fraction (or percentage) of what the whole year is worth?
(c) Describe Kelly's share (i.e. which share does she select) and Lauren's share (which share is she left with) at the end of the Divider-Chooser method.
Now Jason appears, with documents proving that he too has the rights to a fair share of the retail space. Rather than start over, they agree to use the Lone Chooser method, and build on what Kelly and Lauren have already done. This means that Jason is the Chooser, and Lauren is the first Divider (and she has already done the work of the first division). Here are all their preferences:
Jan
Fe
Ma
Ap
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
total
Jason
1
1
1
1
1
1
1
1
1
1
1
1
12
Kelly
2
1
2
1
3
1
1
1
1
1
1
1
16
Lauren
1
2
2
2
2
3
1
1
1
1
1
1
18
Subdivision:
(d) Describe the second division which Kelly makes, i.e. describe (in terms of specific months or parts of months) the 3 subshares k1, k2, k3 she creates from her share, and indicate how much each subshare is worth to her. You may describe the value to Kelly in terms of “Kelly points”, or as a fraction of the value of the whole year.
(e) Describe the second division which Lauren makes, i.e. describe (in terms of specific months or parts of months) the 3 subshares l1, l2, l3 she creates from her share, and indicate how much each subshare is