Four doctors, Moe, Larry, Curly, and Shemp,1 are in partnership and cover hours for each other. At most one doctor needs to be in the office to see patients at any one time. They advertise their office hours as follows

A coalition is an agreement by one or more doctors as to the times they will really be in the office to see everybody’s patients. The characteristic function should be the amount of time saved by a given coalition. Note that v(i) = 0, and v(1234) = 13 hours. This problem is an example of the use of cooperative game theory in an scheduling problem and is an important application in many disciplines.

(a) Find the characteristic function for all coalitions.

(b) Find X1, X2. When you get to X2, you should have the fair allocation in terms of hours saved.

(c) Find the exact times of the day that each doctor will be in the office according to the allocation you found in X3.