Survey of Math Chapter 13: Knaster Inheritance
The knaster inheritance procedure is a means of allocating items or issues to more than two parties in an equitable manner. The downside of this procedure is that it requires the parties to have wads of cash handy, which is used to buy out other members of the group. In Knaster inheritance, the parties assign a dollar value to each item they are going to divide.
Example Three children must make fair division of a painting and sculpture left to them by their mother. The value (in dollars) each child places on the objects is given below. These values should be assigned by the three children independently of each other.
| Item | Sasha | Ralph | Fergus |
|---|---|---|---|
| Painting | $4000 | $6300 | $6000 |
| Sculpture | $2300 | $1800 | $2400 |
Use the Knaster inheritance procedure to resolve this dispute.
Solution
We deal with each item separately.
Painting: The high bidder is awarded the painting. In this case, it is Ralph. Since there are three children, however, Ralph's fair share is only 1/3 of what he thinks the painting is worth, which is $6300/3 = $2100. Therefore, he puts into a temporary ``kitty'' (pot of money) 2/3 of what he bid on the painting, which would be $4200 in this case.
Each of the other children withdraws from the kitty 1/3 of the amount they bid on the painting, which would be their fair share of the painting:
Sasha: 1/3 x $4000 = $1333.33.
Fergus: 1/3 x $6000 = $2000.00.
Kitty: $4200 - $1333.33 - $2000.00 = $866.67.
At this point, every child feels that they have received 1/3 of the value of the painting, so the distribution of the painting is fair. However, the remaining money in the kitty is split 3 ways, and each child receives an additional $866.67/3 = $288.89, so everyone walks away feeling they got $288.89 more than their fair share!
At this point, the fair division has dealt with the painting:
Sasha: $1333.33 + $288.89 = $1622.22.
Ralph: painting - $4200.00 + $288.89 = painting - $3911.11.
Fergus: $2000.00 + $288.89 = $2288.89.
We can use the same procedure for the sculpture.
Sculpture: The high bidder is awarded the sculpture. In this case, it is Fergus. Since there are three children, however, Fergus's fair share is only 1/3 of what he thinks the painting is worth, which is $2400/3 = $800. Therefore, he puts into the kitty 2/3 of what he bid on the sculpture, which is $1600.
Each of the other children withdraws from the kitty 1/3 of the amount they bid on the sculpture, which would be their fair share of the sculpture:
Sasha: 1/3 x $2300 = $766.67.
Ralph: 1/3 x $1800 = $600.00.
Kitty: $1600 - $766.67 - $600.00 = $233.33.
The remaining money in the kitty is split 3 ways, and each child receives an additional $77.78.
At this point, the fair division has dealt with the sculpture:
Sasha: $766.67 + $77.78 = $844.45.
Ralph: $600.00 + $77.78 = $677.78.
Fergus: sculpture - $1600.00 + $77.78 = sculpture - $1522.22.
Final Distribution: The final result of the fair division process is the following distribution:
Sasha: $1622.22 + $844.45 = $2466.67
Ralph: painting - $3911.11 + $677.78 = painting - $3233.33.
Fergus: $2288.89 + sculpture - $1522.22 = sculpture + $766.67.
Ralph needs to have $3233.23 on hand to pay off Sasha and Fergus. This is a drawback of the Knaster inheritance procedure, since it is quite possible that Ralph doesn't have that much money.