MA 111 Agenda - Monday, 12.5.2011(Notes from Elizabeth Weaver.)12.5.1 The Method of Sealed Bids• The method of sealed bids can be used in fair division problems with any number of players where the set S is discrete.• The Method of Sealed Bids:1. Bidding: Each player makes a bid (in dollars) for each of the items in the set S and places it in a sealed envelopeto maintain the privacy condition.2. Allocation: Each item will go to the highest bidder for that item. (Ties can be broken with a coin flip.)3. First Settlement: First calculate each player’s fair share of the estate. If the items a player received in step 2total more than his fair share, he must pay the difference in cash. If the item s a player received in step 2 totalless than his fair share, he receives the difference in cash.4. Division of the Surplus: Calculate the difference between the amount the players paid and the amount theplayers received in step 3. This amount is the surplus, and it is to be divided among all of the players equally.5. Final Settlement: This is obtained by adding each player’s share of the surplus money to the first settlementobtained in ste p 3.• Example: In her last will and testament, Granny leaves her estate to Jed, Elly May, Jethro and Mr. Drysdale withthe stipulation that it must be divided equally and remain with them (not be sold to outsiders). Below is a listing ofeach player’s bids on the items.Jed Elly May Jethro Mr. DrysdaleCabin $220,000 $250,000 $211,000 $198,000Rolls Royce $40,000 $30,000 $47,000 $52,000Picasso $280,000 $240,000 $234,000 $190,000TotalFair Share(a) Who won which items?(b) How much should each person receive from (or pay to) the estate?(c) How much surplus is there? How much does each person receive of the surplus?(d) Describe which players get what items (and how much money they pay or receive) in the final settlement.• What are the disadvantages of the method of sealed bids?• Example: Chuck and Sarah are getting a divorce. The only joint property (of value) they own is their house. Theydecide to use the method of sealed bids. Chuck bids $140,000 on the house, Sarah bids $164,000. Sarah bid higherand therefore w ins the house. She pays her fair-share into the estate, while Chuck receives his in cash.(a) How much is each person’s fair-share?(b) How much is the surplus?(c) Describe the final settlement.• Example: Elmo, Grover, and Bert are dividing four pieces of furniture using the method of sealed bids. Their bidson each of the items are given in the following table.Elmo Grover BertDresser $150 $300 $275Desk $180 $150 $165Vanity $170 $200 $260Table $400 $250 $500(a) What is the value of each person’s fair share?(b) How much should each person receive from (or pay to) the estate?(c) How much surplus is there? How much does each person receive of the surplus?(d) Describe which players get what items (and how much money they pay or receive) in the final settlement.• Example: Jack, Janet, and Chrissy jointly own a house. They no longer want to be roommates and decide to usethe method of sealed bids, with the understanding that one of them will get the house and the other two will get cash.Jack bids $180,000, Janet bids $120,000, and Chrissy bids $150,000. After the final s ettleme nt, how much money doJanet and Chrissy get from Jack for their one-third s hare of the house?12.5.2 After this lecture, you should be able to...• carry out the method of sealed
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