Three plumbers, Moe Howard (1), Larry Fine (2), and Curly Howard (3), work at the same company…

Three plumbers, Moe Howard (1), Larry Fine (2), and Curly Howard (3), work at the same company and at the same times. Their houses are located as in the figure and they would like to carpool to save money.1 Once they reach the expressway, the distance to the company is 12 miles. They drive identical Chevy Corvettes so each has the identical cost of driving to work of 1 dollar per mile. Assume that for any coalition, the route taken is always the shortest distance to pick up the passengers (doubling back may be

necessary to pick someone up). It doesn’t matter whose car is used. Only the direction from home to work is to be considered. Shemp is not to be considered in this part of the problem.

(a) By considering  is the cost of driving to work if S forms a carpool, what is the characteristic function of this game.

(b) Find the least core.

(c) Assuming that Moe is the driver, how much should Larry and Curly pay Moe if they all carpool?

(d) Find the Shapley allocation of savings for each player.

(e) The problem is the same as before but now Shemp Howard (4) wants to join the plumbers Howard, Fine and Howard at the company. Answer all of the questions posed above.