Arthell, Coby, and Tamika will divide their timeshare vacations. They have 30 days in Ma...

Arthell, Coby, and Tamika will divide their timeshare vacations. They have 30 days in March, 30 days in July, and 30 days in December. The division will be made using the continuous envy-free division method. Their preferences are given in the following table, where larger numbers indicate a greater preference.

a. Suppose Arthell is the divider, and she creates the following vacations:

Vacation I: 4 March days, 10 July days, 12 December days

Vacation II: 5 March days, 5 July days, 15 December days

Vacation III: 21 March days, 15 July days, 3 December days

Verify that each vacation represents one-third of the value of the time shares according to Arthell’s values.

b. Coby will be the chooser, so he evaluates the vacations from part (a). Which vacation will he believe has the greatest value? Which vacation will he believe has the second-greatest value?

c. Coby trims the greatest-valued vacation to match the value of the second-greatest-valued vacation. One way he can do it is to trim 6 days in July and 5 days in March. Verify that this trimming yields two vacations of equal value, and find one other way in which he could trim the vacation.

d. Suppose the vacation is trimmed as suggested in part (c). Tamika, Coby, and Arthell, in that order, will take turns selecting vacations. Which vacation will each one pick and why?

e. Who will the new divider and the new chooser be?

f. Suppose the new divider creates the following three vacations from the excess.

Which vacation will the chooser select, and how will the remaining vacations be chosen?

g. Describe the final division of the vacations and explain why it is both proportional and envy-free.

h. Reconsider part (f). Realistically, the days would be divided so that each person would get a whole number of days rather than a fractional part of a day. Consider the following rounded number of days:

Vacation A: 2 March days and 2 July days

Vacation B: 2 March days and 2 July days

Vacation C: 1 March day and 2 July days

Which vacation will the chooser select and how will the remaining vacations be chosen? Is the final division of days envy-free? Explain.