Question 13.38

8. Suppose we have three items (, , and ) and three people (Bob, Carol, and Ted). Assume that each person spreads 100 points over the items (as in the adjusted winner procedure) to indicate the relative worth of each item to that person.

Item Bob Carol Ted
40 30 30
50 40 30
10 30 40

For each of the allocations listed below, indicate the following:

  1. Whether or not it is proportional.
  2. Whether or not it is envy-free.
  3. Whether or not it is equitable.
  4. For the ones that are not Pareto-optimal, another allocation that makes one person better off without making anyone else worse off.

565

Allocation 1: Bob gets , Carol gets , and Ted gets . (This is not Pareto-optimal.)

Allocation 2: Bob gets , Carol gets , and Ted gets . (This is not Pareto-optimal.)

Allocation 3: Bob gets , , and . (This is Pareto-optimal; explain why.)

Allocation 4: Bob gets , Carol gets , and Ted gets . (This is Pareto-optimal.)

Allocation 5: Bob gets , Carol gets , and Ted gets . (This is Pareto-optimal.)