Chapter 3, Problem 1E

Henry, Tom, and Fred are breaking up their partnership and dividing among themselves the partnership’s real estate assets equally owned by the three of them. The assets are divided into three shares $\left(s_1,s_2\text{ and }{s}_3\right)$. Table 3-12 shows the values of the shares to each player expressed as a percent of the total value of the assets.

Table 3-12

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

a. Which of the shares are fair shares to Henry?

b. Which of the shares are fair shares to Tom?

c. Which of the shares are fair shares to Fred?

d. Find all possible fair divisions of the assets using $s_1,s_2\text{ and }{s}_3$ as shares.

e. Of the fair divisions found in (d), which one is the best?

To determine

(a)

To find:

Fair shares for Henry from the given table.

Answer to Problem 1E

Solution:

Fair shares for Henry are $s_2\text{ and }{s}_3$.

Explanation of Solution

Given:

The given table for value of shares to each player is shown in table 1.

Table 1

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

Fair share for each player should be $\ge \frac{100}{N}\%$, where $N$ is the total number of players.

Calculation:

The value of shares to each player is,

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

There are total 3 players in which assets will be divided so the fair share for each player would be $\ge \frac{100}{3}\%$ or $33\frac{1}{3}\%$.

Then according to table fair shares for Henry will be $s_2\text{ and }{s}_3$ because these are greater than fair share.

Conclusion:

Thus, fair shares for Henry are $s_2\text{ and }{s}_3$.

To determine

(b)

To find:

Fair shares for Tom from the given table.

Answer to Problem 1E

Solution:

Fair shares for Tom are $s_2\text{ and }{s}_3$.

Explanation of Solution

Given:

The given table for value of shares to each player is shown in table 1.

Table 1

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

Fair share for each player should be $\ge \frac{100}{N}\%$, where $N$ is the total number of players.

Calculation:

The value of shares to each player is,

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

There are total 3 players in which assets will be divided so the fair share for each player would be $\ge \frac{100}{3}\%$ or $33\frac{1}{3}\%$.

Then according to table fair shares for Tom will be $s_2\text{ and }{s}_3$ because these are greater than fair share.

Conclusion:

Thus, fair shares for Tom are $s_2\text{ and }{s}_3$.

To determine

(c)

To find:

Fair shares for Fred from the given table.

Answer to Problem 1E

Solution:

Fair shares for Fred are $s_1,s_2\text{ and }{s}_3$.

Explanation of Solution

Given:

The given table for value of shares to each player is shown in table 1.

Table 1

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

Fair share for each player should be $\ge \frac{100}{N}\%$, where $N$ is the total number of players.

Calculation:

The value of shares to each player is,

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

There are total 3 players in which assets will be divided so the fair share for each player would be $\ge \frac{100}{3}\%$ or $33\frac{1}{3}\%$.

Then according to table fair shares for Fred will be $s_1,s_2\text{ and }{s}_3$ because all are equal to the fair share.

Conclusion:

Thus, fair shares for Fred are $s_1,s_2\text{ and }{s}_3$.

To determine

(d)

To find:

All possible fair divisions of the assets using given table.

Answer to Problem 1E

Solution:

The fair division of assets is possible in two ways:

i. Henry gets $s_2$, Tom gets $s_3$, and Fred gets $s_1$.

ii. Henry gets $s_3$, Tom gets $s_2$, and Fred gets $s_1$.

Explanation of Solution

Given:

The given table for value of shares to each player is shown in table 1.

Table 1

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

Fair share for each player should be $\ge \frac{100}{N}\%$, where $N$ is the total number of players.

Calculation:

The value of shares to each player is,

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

There are total 3 players in which assets will be divided so the fair share for each player would be $\ge \frac{100}{3}\%$ or $33\frac{1}{3}\%$.

Henry and Tom both have fair shares $s_2\text{ and }{s}_3$ and Fred has all three shares are fair. So the fair division of assets is possible in two ways:

i. Henry gets $s_2$, Tom gets $s_3$, and Fred gets $s_1$.

ii. Henry gets $s_3$, Tom gets $s_2$, and Fred gets $s_1$.

Conclusion:

Thus, the fair division of assets is possible in two ways:

i. Henry gets $s_2$, Tom gets $s_3$, and Fred gets $s_1$.

ii. Henry gets $s_3$, Tom gets $s_2$, and Fred gets $s_1$.

To determine

(e)

To find:

The best fair division among the fair divisions found in part (4).

Answer to Problem 1E

Solution:

The best fair division of assets is: Henry gets $s_2$, Tom gets $s_3$ and Fred gets $s_1$.

Explanation of Solution

Given:

The given table for value of shares to each player is shown in table 1.

Table 1

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

Fair share for each player should be $\ge \frac{100}{N}\%$, where $N$ is the total number of players.

Calculation:

The value of shares to each player is,

	S1	S2	S3
$\text{Henry}$	$25\%$	$40\%$	$35\%$
$\text{Tom}$	$28\%$	$35\%$	$37\%$
$\text{Fred}$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$	$33\frac{1}{3}\%$

There are total 3 players in which assets will be divided so the fair share for each player would be $\ge \frac{100}{3}\%$ or $33\frac{1}{3}\%$.

Henry and Tom both have fair shares $s_2\text{ and }{s}_3$ and Fred has all three shares are fair. So the fair division of assets is possible in two ways:

i. Henry gets $s_2$, Tom gets $s_3$, and Fred gets $s_1$.

ii. Henry gets $s_3$, Tom gets $s_2$, and Fred gets $s_1$.

The best fair division is the one in which players are more happy. Henry would be more happy in choice (i) and Tom also would be more happy in choice (i). Fred is happy in equally in both choices.

So the best fair division of assets is: Henry gets $s_2$, Tom gets $s_3$, and Fred gets $s_1$.

Conclusion:

Thus, the best fair division of assets is: Henry gets $s_2$, Tom gets $s_3$, and Fred gets $s_1$.