Contemporary MathematicsContemporary Mathematics at Nebraska

Example9.1

Sam and Omar have lived together for the last 3 years, during which time they shared the expense of purchasing several items for their home. Sam has accepted a job in another city, and now they find themselves needing to divide their shared assets.

Each records their value of each item, as shown below.

Use the sealed bids procedure to allocate the items between Sam and Omar.

Sam's total valuation of the items is $150+$200+$250+$50 = $650, making a fair share for Sam $650 2 = $325.

Omar's total valuation of the items is $100+$250+$150+$100 = $600, making a fair share for Omar $600 2 = $300.

Each item is now awarded to the highest bidder. Sam will receive the couch and video game system, providing $150+$250 = $400 of value to Sam. Since this exceeds his fair share, he has to pay the difference, $75, into a holding pile.

Omar will receive the TV and surround sound system, providing $250+$100=$350 in value. This is more than his fair share, so he has to pay the difference, $50, into the holding pile.

Thus, in the initial allocation, Sam receives the couch and video game system and pays $75 into the holding pile. Omar receives the TV and surround sound system and pays $50 into the holding pile. At this point, both players would feel they have received a fair share.

There is now $125 remaining in the holding pile. This is the surplus from the division. This is now split evenly, and both Sam and Omar are given back $62.50. Since Sam had paid in $75, the net effect is that he paid $12.50. Since Omar had originally paid in $50, the net effect is that he receives $12.50.

Thus, in the final allocation, Sam receives the couch and video game system and pays $12.50 to Omar. Omar receives the TV and surround sound system and receives $12.50. At this point, both players feel they have received more than a fair share.

Example9.2

Four small graphic design companies are merging operations to become one larger corporation. In this merger, there are a number of issues that need to be settled. Each company is asked to place a monetary value (in thousands of dollars) on each issue:

We can use the method of sealed bids to resolve these issues. We can allocate the issues as if they were items, using the usual sealed bids rules. Then, whichever company wins each issue can make the decision on that issue.

The items would then be allocated to the company that bid the most for each:

• Company name would be awarded to Graphix
• Company location would be awarded to DesignByMe
• CEO would be awarded to Super Designs
• Chair of the board would be awarded to Graphix

In the table below, we calculate the first and final settlements. Recall that all amounts are in thousands of dollars. Note that after the first settlement, the total surplus is \(2.50+3.25-5.50+7.25 = 7.50 \text{,}\) so each company gets \(7.50/4=1.875\) as their extra share.

In summary, the final settlement is:

• Super Designs wins the CEO, and pays $625 ($0.625 thousand)
• DesignByMe wins the company location and pays $1,375 ($1.375 thousand)
• LayoutPros wins no issues, but receives $7,375 in compensation
• Graphix wins the company name and chair of the board, and pays $5,375.

Exploration9.1

Jamal, Maggie, and Kendra are dividing an estate consisting of a house, a vacation home, and a small business. Their valuations (in thousands) are shown below. Determine the final settlement.

Example9.4

Suppose Chelsea and Mariah are sharing an apartment, and need to split the chores for the household. They list the chores, assigning a negative dollar value to each item; in other words, the amount they would pay for someone else to do the chore (a per week amount). We will assume, however, that they are committed to doing all the chores themselves and not hiring someone else.

Now we award each item to the player who bid the highestbut remember that the values are negative. So, for example, $8 is greater than $10, and thus Mariah is awarded the vacuuming. This makes sense, since her value of $8 indicates that she dislikes it less than Chelsea, who valued it at $10. Similarly, we give cleaning the bathroom and doing the dishes to Chelsea, while Mariah also gets dusting. Now we can find the first and final settlements, given in the table below.

The overall surplus is 71=6, so each player receives a surplus of 6 2=3, giving the final settlement values shown in the table above. Notice that the cash in the final settlement comes out even: Mariah pays Chelsea $4, with nothing left over. This is a good way to check we've done our math right.