4. (6 points) (Buying and selling) Two house owners, players 1 and 2, are trying to sell their houses. There are three potential buyers, players 3,4 and 5 . Each buyer is only interested in buying one house. Players 1 and 2 value their houses at 60 and 80 units, respectively. The buyers' valuations are given in the following table. If a buyer and a seller match and the valuation of the buyer is greater than or equal to that of the seller, then the worth of the coalition is the valuation of the buyer. (i) Determine the characteristic function of the corresponding game. (The worth of any one member coalition is zero. The worth of any coalition consisting of only sellers or only buyers is zero. The worth of a coalition consisting of one seller and one buyer is zero if the seller's valuation is higher than the buyer's valuation, otherwise it is the buyer's valuation. The worth of a coalition consisting of at least one seller and at least one buyer is the maximum willingness to pay by the buyers for the objects, provided that at least one buyer's valuation is greater than or equal to the corresponding valuation by the seller.) (ii) Obtain the Shapley value. (iii) Find the core. [Hint: If a set has n1 elements, then the number of subsets which has exactly m elements, 0mn, is C(n,m)=n!/(m!(nm)!)] 4. (6 points) (Buying and selling) Two house owners, players 1 and 2, are trying to sell their houses. There are three potential buyers, players 3,4 and 5 . Each buyer is only interested in buying one house. Players 1 and 2 value their houses at 60 and 80 units, respectively. The buyers' valuations are given in the following table. If a buyer and a seller match and the valuation of the buyer is greater than or equal to that of the seller, then the worth of the coalition is the valuation of the buyer. (i) Determine the characteristic function of the corresponding game. (The worth of any one member coalition is zero. The worth of any coalition consisting of only sellers or only buyers is zero. The worth of a coalition consisting of one seller and one buyer is zero if the seller's valuation is higher than the buyer's valuation, otherwise it is the buyer's valuation. The worth of a coalition consisting of at least one seller and at least one buyer is the maximum willingness to pay by the buyers for the objects, provided that at least one buyer's valuation is greater than or equal to the corresponding valuation by the seller.) (ii) Obtain the Shapley value. (iii) Find the core. [Hint: If a set has n1 elements, then the number of subsets which has exactly m elements, 0mn, is C(n,m)=n!/(m!(nm)!)]