Tom, Jack, and Peter are identical triplets, and they each took summer jobs. Tom works at a fast-food restaurant 30 hours each week. Jack works at a video store 30 hours each week. Peter works in an auto parts store 27 hours each week. Because they are identical triplets, they decide to share the three jobs. Tom feels that he would much prefer to work at the auto parts store. His restaurant to video store to auto parts store preference ratio is 1 to 2 to 3. Jack prefers the fast-food restaurant and video store twice as much as the auto parts store. Peter does not have a preference and thinks he would like each job equally.
a. If Tom is the first divider and Jack is the first chooser, find a proportional, envy-free division of the jobs.
b. Demonstrate that the division from part (a) is proportional by comparing the value each person places on his portion to the value he thinks is fair.
c. Demonstrate that the division from part (a) is envy-free by comparing the value each person places on his own portion to the value he places on the portions of each of the other triplets.
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