In class we discuss the assignment game (P, Q, a). In this question we interpret P as a set of bidders and Q as a set of objects. Each object has a reservation price of cj. The value of object j to bidder i is (15} 2 0. A feasible price vector p is a function from Q to R+ such that p( j) : p J- satises pj 2 cj. We will assume that Q contains a null object 0 whose value is 05,0 : 0 for all bidders and whose price p0 is always zero. Then if a bidder is unmatched we will say that she or he is assigned the null object (note that more than one bidder may be assigned to the null object). 1. For a given price vector p, dene the demand set D;( p) for bidder i. (15 Points) 2. A price vector p is called quasi-competitive if there is a matching p. from P to Q such that p(i) : j then j is in D;(p), and if i is unmatched under it then 0 is in D;(p). In other words, at a quasi-competitive prices p each buyer can be assigned to an object in his or her demand set. In this case, p. is said to be compatible with p. The pair (p, u) is a competitive equilibrium if p is quasi-competitive, p is compatible with p, and p j : cj for all j 1MP) In this case we denote (1),\") is a competitive equilibrium and p is called an equilibrium price vector. Show that if (p, p) is a competitive equilibrium then the corresponding payoffs are stable. (15 Points). 3. Describe an algorithm to compute an equilibrium price vector. (20 points) 4. In your previous answer: does the algorithm generate an equilibrium price vector that maxi- mizes total happiness? (10 Points)