§ 3.6 The Method of Sealed Bids

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§ 3.6 The Method of Sealed Bids

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.

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

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.

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

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 (100 - 31) = $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 $30.75 - he must pay (48 - 30.75) = $17.25 Ralph: Ralph gets a comic he values at $56 and his fair share is $20. He must pay (56 - 20) = $36.

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.` 69 - 31.25 + 17.25 + 36 = $91

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 = $22.75 as a bonus. The final results are as follows: Martin gets Radioactive Boy #1 and pays (69 - 22.75) = $46.25 Bart receives 31.25 + 22.75 = $54. Millhouse gets Itchy & Scratchy and (22.75 - 17.25) = $5.50 Ralph gets Fallout Boy and must pay (36 - 22.75) = $13.25

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.

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