mgf1107lecture8 - MGF 1107 EXPLORATIONS IN MATHEMATICS...

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MGF 1107 – EXPLORATIONS IN MATHEMATICS LECTURE 8 Ex. (MLP Q.4.) Martha and Nick buy a 14 inch sub for $18, and divide it using the divider-chooser method. The first four inches are ham, the next six are turkey, and the last four are beef. Martha likes ham twice as much as turkey, and likes turkey and beef equally. Nick likes beef twice as much as ham, and likes ham and turkey equally. How would they divide the sub if Martha divides, and which piece would Nick choose? Let x be the value to Martha of one inch of turkey (or beef) sub. Then the value of the combined length of the turkey and beef parts is 10x. While the ham part is only 4 inches long, she likes ham twice as much, so in terms of x, the value of the ham part is 4 x 2x = 8x. So the entire sub is worth 18x, and since it is 18 inches long, that means x = 1, meaning every inch of turkey (or beef) is worth $1, and consequently every inch of ham is worth $2. Now fair value is clearly $9 (half of $18), so she needs to split the sub in such a way that each piece is (to her) worth $9. The ham part is worth $8 (4 inches worth $2 each), so she needs another $1 worth of sub. Hence she moves the knife another inch to the right (since each inch of turkey is worth $1), and then cuts. So s 1 is [0,5] and s 2 is [5,14]. Nick would obviously choose s 2 , as he really likes beef, and s 2 contains all the beef.
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The Lone Chooser Method The lone chooser method, developed by Arlington Fink in 1964, is in some ways the opposite of the lone divider method from Lecture 7, as here we have once chooser, and N-1 dividers among the N players. Note: While the method below can be generalized to any number of players, we will just consider the case where N=3. i) One person is randomly selected to be the chooser (C), with the other two being dividers (D 1 and D 2 ). ii) D 1 and D 2 divide S using the divider-chooser method. iii) D 1 and D 2 then split the piece obtained in the previous step into three pieces of equal value (in their eyes). iv) The chooser then picks the best subshare from each divider (so two pieces in total). v) The dividers then keep their two remaining pieces. Ex. Three people, Angela, Boris, and Carlos are dividing a cake that is half vanilla-half strawberry. Their valuations of each flavor are given in the table below. Vanilla (V
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mgf1107lecture8 - MGF 1107 EXPLORATIONS IN MATHEMATICS...

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