Consider the following game representing the market for vintage handbags. There are two players: a seller and a buyer. The handbag for sale can either be real, and presumably of high quality, or fake. The seller knows the whether the bag is real but the buyer does not. The probability that the bag is real is a and is known to both players. Using this probability, Nature determines the type of the bag. The seller then decides whether to put the bag up for sale and, if so, what price to set (which can be any positive integer). If the bag is for sale, the buyer observes the price and decides whether or not to buy it. For each type, a seller's strategy tells her whether to put the bag up for sale and at which price. A buyer's strategy tells him whether to accept or decline each possible price (in the event that the bag is for sale). If the bag is sold, the seller's payoff is the price paid. If the bag is not sold, then the seller's payoff is the value they attach to the bag, given in the table below. If the buyer purchases the bag, the buyer's payoff is the value of the bag to them, minus the price paid. The value of the bag to the buyer for each type is given in the table below, but this true value is only revealed after purchase. If they do not buy the bag, then their payoff is zero. Value to Buyer Value to Seller Fake 100 10 Real 14,100 5,000 (a) Show that no perfect Bayesian equilibrium exists in the present set-up where both types of bags are sold, and the real bag is sold for a higher price. (b) At what value of a would a pooling equilibrium exist? Interpret your result. Do you believe that this value of a is feasible in this market? Discuss why or why not? (c) Suppose that = .5. Formally and intuitively describe the separating perfect Bayesian equilibrium. Interpret your result. (d) Suppose = .5. The seller can pay some amount x to have the legitimacy of the bag confirmed. What is the highest amount x the seller would be willing to pay?