The following problems use the last-diminisher method of fair division.
Three friends, Galen, Leland, and Sandie, each have 15-day vacations scheduled in their time-share vacation homes. One is in the mountains at a ski resort. Another is in an oceanfront condominium. The third is at a cabin in the woods near a lake. They decide to divide the vacations so they each can enjoy all three locations. Galen’s preference ratio for mountain to ocean to woods is 2 to 1 to 2, Leland’s is 3 to 1 to 1, and Sandie’s is 1 to 3 to 2. They will use the last-diminisher method to divide the days in the vacation homes fairly.
a. Using points to represent the value a player places on each vacation day, determine how each player would value the three time-share locations. Also, for each player, determine what point value would constitute a fair share.
b. Suppose Galen is the divider and he creates the following package: 5 days in the mountains, 3 days at the ocean, and 6 days in the woods. Verify that this package represents a fair share according to Galen’s values.
c. The package from part (b) will pass to Leland and then to Sandie. Explain whether either of them will “trim” the package, how the package will be trimmed, and who gets the package or the trimmed package.
d. Determine how the remaining two players can split the time fairly using the divide-and-choose method for two players. Many answers are possible.
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