1. § 3.6 The Method of Sealed Bids

2. Example: Martin, Bart, Millhouse and Ralph stumble upon a box containing three comics: Radioactive Man #1, Fallout Boy Annual #3 and Itchy & Scratchy Comics #5. The kids have decided to split up the loot using the Method of Sealed Bids.
Step 1 (Bidding).
Each player (in this case Martin, Bart, Millhouse and Ralph) makes a bid for each item, giving his honest assessment of the value of each.

3. Example: Martin, Bart, Millhouse and Ralph stumble upon a box containing three comics: Radioactive Man #1, Fallout Boy Annual #3 and Itchy & Scratchy Comics #5. The kids have decided to split up the loot using the Method of Sealed Bids.
Step 1 (Bidding).
Each player (in this case Martin, Bart, Millhouse and Ralph) makes a bid for each item, giving his honest assessment of the value of each.
The bids appear below:
Martin
Bart
Millhouse
Ralph
Radioactive Man
$100
$70
$67 $3
Fallout Boy $4
$15 $8
$56
Itchy & Scratchy
$20
$40
$48
$21

4. Example: Martin, Bart, Millhouse and Ralph stumble upon a box containing three comics: Radioactive Man #1, Fallout Boy Annual #3 and Itchy & Scratchy Comics #5. The kids have decided to split up the loot using the Method of Sealed Bids.
Step 2 (Allocation).
Each item goes to the highest bidder for that item.
In this case, Radioactive Man #1 goes to Martin, Fallout Boy Annual #3 goes to Ralph and Millhouse gets Itchy & Scratchy. Bart gets nothing.

5. Example: Martin, Bart, Millhouse and Ralph stumble upon a box containing three comics: Radioactive Man #1, Fallout Boy Annual #3 and Itchy & Scratchy Comics #5. The kids have decided to split up the loot using the Method of Sealed Bids.
Step 3 (Payments).
Depending on what item a player gets in Step 2, he/she will owe money or be owed money.
First we must calculate each player`s fair share (under his/her value system):
Martin
Bart
Millhouse
Ralph
Radioactive Man
$100
$70
$67 $3
Fallout Boy $4
$15 $8
$56
Itchy & Scratchy
$20
$40
$48
$21
Total
$124
$125
$123
$80
Fair Share
$31
$31.25
$30.75

6. Example: Martin, Bart, Millhouse and Ralph stumble upon a box containing three comics: Radioactive Man #1, Fallout Boy Annual #3 and Itchy & Scratchy Comics #5. The kids have decided to split up the loot using the Method of Sealed Bids.
Step 3 (Payments). Cont`d
Martin: Under his own value system Martin`s fair share is $31, but he got a comic a comic worth $100. This means Martin must pay ( ) = $69 into a `pot`.
Bart: Since Bart gets none of the comics he is instead awarded his fair share in cash - he gets $31.25
Millhouse: Millhouse gets a comic worth $48, but his fair share is $ he must pay ( ) = $17.25
Ralph: Ralph gets a comic he values at $56 and his fair share is $20. He must pay ( ) = $36.

7. Example: Martin, Bart, Millhouse and Ralph stumble upon a box containing three comics: Radioactive Man #1, Fallout Boy Annual #3 and Itchy & Scratchy Comics #5. The kids have decided to split up the loot using the Method of Sealed Bids.
Step 3 (Payments). Cont`d
Now we need to consider how much money is in the `pot.`
= $91

8. Example: Martin, Bart, Millhouse and Ralph stumble upon a box containing three comics: Radioactive Man #1, Fallout Boy Annual #3 and Itchy & Scratchy Comics #5. The kids have decided to split up the loot using the Method of Sealed Bids.
Step 4 (Dividing the Surplus).
The surplus money is divided equally amongst all the participants.
In this case each kid gets 91/4 = $ as a bonus.
The final results are as follows: Martin gets Radioactive Boy #1 and pays ( ) = $46.25 Bart receives = $54. Millhouse gets Itchy & Scratchy and ( ) = $5.50 Ralph gets Fallout Boy and must pay ( ) = $13.25

9. Notes: The Method of Sealed Bids
Each player must have enough money to play the game.
Each player must be willing to accept money in place of any item.

10. Example: (exercise 39, pg 124) Three Sisters (Ana, Belle and Chloe) wish to use the method of sealed bids to divide up 4 pieces of furniture they shared as children. Their bids on each of the items are given in the following table Describe the final outcome of this fair-division game.
Ana
Belle
Chloe
Dresser
$150
$300
$275
Desk
$180
$165
Vanity
$170
$200
$260
Tapestry
$400
$250
$500