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## Section9.1The Sealed Bids Method of Fair Division

The Sealed Bids method provides a method for discrete fair division, allowing for the division of items that cannot be split into smaller pieces, like a house or a car. Because of this, the method requires that all parties have a large amount of cash at their disposal to balance out the difference in item values.

The method begins by compiling a list of items to be divided. Then:

1. Each party involved lists, in secret, a dollar amount they value each item to be worth. This is their sealed bid.
2. The bids are collected. For each party, the value of all the items is totaled, and divided by the number of parties. This defines their fair share.
3. Each item is awarded to the highest bidder.
4. For each party, the value of all items received is totaled. If the value is more than that party's fair share, they pay the difference into a holding pile. If the value is less than that party's fair share, they receive the difference from the holding pile. This ends the initial allocation.
5. In most cases, there will be a surplus, or leftover money, in the holding pile. The surplus is divided evenly between all the players. This produces the final allocation.

While the assumptions we made for fair division methods specified that an arbitrator should not be necessary, it is common for an independent third party to collect the bids and announce the outcome. While not technically necessary, since the method can be executed without a third party involved, this protects the secrecy of the bids, which can sometimes help avoid resentment or bad feelings between the players.

###### 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.

 Sam Omar Couch $150$100 TV $200$250 Video game system $250$150 Surround sound system $50$100

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:

 Super Designs DesignByMe LayoutPros Graphix Company name $5$3 $3$6 Company location $8$9 $7$6 CEO $10$5 $6$7 Chair of the board $7$6 $6$8

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 settlemebt, 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.

 Jamal Maggie Kendra House $250$300 $280 Vacation home$170 $180$200 Small business $300$255 $270 Solution The fair shares are: • Jamal's total value is$250 + $170 +$300 = $720. His fair share is$240 thousand.
• Maggie's total value is $300 +$180 + $255 =$735. Her fair share is $245 thousand. • Kendra's total value is$280 + $200 +$270 = $750. Her fair share is$250 thousand.

In the first settlement,

• Jamal receives the business, and pays $300 -$240 = $60 thousand into holding. • Maggie receives the house, and pays$300 - $245 =$55 thousand into holding.
• Kendra receives the vacation home, and gets $250 -$200 = $50 thousand from holding. There is a surplus of$60 + $55 -$50 = $65 thousand in holding, so each person will receive$21,667 from surplus. In the final settlement,

• Jamal receives the business, and pays $38,333. • Maggie receives the house, and pays$33,333.