mgf1107notes4 - FAIR DIVISION Example: Two legislative...

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FAIR DIVISION Example: Two legislative aides for a congressman are assigned to research 12 proposed bills. The congressman gives each aide 6 bills to work on. Juan complains that the division of the workload is unfair in that several of his bills are much more complex than those given to Patsy. How can they divide the work so that neither of them is envious of the other’s share of the load? Solution: Allow aide Juan to divide the work into two parts so that he thinks the split is fair. Then let Patsy choose which of the two parts she works on. Juan can’t complain because he is the one who made the division. Patsy can’t complain because she made the choice. This method of fair division is called the divide-and-choose method . The individuals among whom the object is being divided are called the players . For the sake of simplicity and uniformity, mathematicians often state fair division problems of this type in terms of cutting a cake. In other words, how can Juan and Patsy divide a cake into two pieces so that neither is envious of the other? After working together for a few weeks, Juan and Patsy decide to go out on a date. They go to the county fair and win a cake that is half chocolate and half vanilla. To Juan chocolate and vanilla are equal in value; he has no preference for one over the other. Patsy, on the other hand, loves chocolate and despises vanilla. To her the value of the cake is concentrated entirely in the chocolate half. The vanilla half has zero value. Since this is their first date, we will assume neither one of them knows anything about the other’s likes and dislikes. They decide to use the divide-and-choose method. Juan volunteers to go first and be the divider. He divides the cake as shown. It is now Patsy’s turn and she obviously chooses the piece in the lower left. Notice that Juan gets a share that to him is worth exactly one-half, while Patsy ends up with a share that to her is worth much more than one-half. This example shows that there is a definite advantage to being the chooser in the divide-and-choose method. The simplest way to handle this problem is to randomly choose who gets to be the divider and who gets to be the chooser by, say, tossing a coin. Can we find a cake-division scheme that works for three players? Have one person move a knife over the top of the cake in such a way that the amount of cake on one side of the knife starts at zero and continually increases. As soon as one of the players thinks at least one-third of the cake is on that side of the knife, that player yells “cut.” A slice is made and given to that player. How do we divide the remainder between the other 2 players? (The divide-and-choose method.) In the case that 2 or all 3 yell cut at the same time, that slice is given to any one of them. What about 4 players?
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mgf1107notes4 - FAIR DIVISION Example: Two legislative...

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