§ 3.6 the method of sealed bids. example: example: martin, bart, millhouse and ralph stumble upon a...

§ 3.6 The Method of Sealed Bids§ 3.6 The Method of Sealed Bids

Example: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).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.


Step 1 (Bidding).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


Step 2 (Allocation).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.


Step 3 (Payments).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

$20


Step 3 (Payments). Cont`dStep 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.


Step 3 (Payments). Cont`dStep 3 (Payments). Cont`d

Now we need to consider how much money is in the `pot.`

69 - 31.25 + 17.25 + 36 = $91


Step 4 (Dividing the Surplus). Step 4 (Dividing the Surplus).

The surplus money is divided equally amongst all the participants.

In this case each kid gets91/4 = $22.75 as a bonus.

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

Notes: The Method of Notes: The Method of Sealed BidsSealed 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: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 $150 $165

Vanity $170 $200 $260

Tapestry

$400 $250 $500

§ 3.6 the method of sealed bids. example: example: martin, bart, millhouse and ralph stumble upon a...

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