P1 has an old furniture which he wants to get rid of and two buyers, P2 and P3, wish to purchase the furniture from him. The furniture has no value to P1 but P2 values it at 5 units and P3 at 7 units. One or both buyers may make a bid for the furniture, and if both makes a bid, the furniture goes to P3 since he values the furniture most.

Let () be the total profit that all members in K, where K is a subset of the set of players, make through this sale. Thus, for example, ({1, 2}) = 5 since if P1 sold the furniture at price p, then he makes a profit of p and P2 makes a profit of 5 - p. Note that for a profit to have been made, a sale must have taken place; thus, for example ({2, 3}) = 0 since no sale is made between the buyers.

(a) Express () as a function of K and prove that it is a characteristic function.

(b) Is the game essential? Explain.

(c) Reduce the game into one in 0-1 form and use this new game to determine if this game has core. Sketch the core if it is non-empty. (d) Determine the Shapley value of each player.