Question

2. (The Dynamic Model of the Market for Lemon) Consider the dynamic version of the model of the market for lemon with a buyer (player 1) and a seller (player 2). The seller owns a car whose quality is low (L), medium (M), or high (H), each with equal probability (1/3). The seller knows the quality of the car, but the buyer does not. The seller assigns value 0 to low-quality car, value 3 to a medium-quality car and value 6 to a high-quality car. (You can interpret that one unit of value means $1,000.) Whatever the car's quality, the buyer's value for the car is 2 higher than the seller's. (That is, the buyer assigns value 2 to low-quality car, value 5 to a medium-quality car, and value 8 to a high-quality car) To begin the interaction, the buyer chooses p E (1,4,7) representing an offer to buy the car at price p (i.e., the buyer chooses one of these three prices). Upon receiving the offer, the seller decides either to accept or reject the offer. If the seller accepts the offer, then the car is traded at price p; in this case, the buyer's payoff is his value for the car minus the price and the seller's payoff is the price minus her value for the car. If the seller rejects the offer, then trade does not occur, and both players receive a payoff of zero. Note that, only the first mover, the buyer, has uncertainty about the quality of the car in this game. (a) Represent this game as a general extensive form game (b) Find all pure-strategy perfect Bayesian equilibria of this game

Answer 1

a) To summarise the facts in the question please refer the below table which would give a good understanding.

a) To summarise the facts in the question please refer the below table which would give a good understanding. Based on the table the extensive form of the game will be as below. b) The pure strategy under Perfect Bayesian equilibria of the game is shown in the extensive form below. Please refer the table below as other possibilities have also been added. Since the seller would have been in advantage if buyer makes the first move. On gaining one more point on each value buyer's gain would increase by 1 and seller would be in loss. The perfect equilibria will be at (1,1,1),(1,1,1) which is an optimum gain for both buyer and seller.

Based on the table the extensive form of the game will be as below.

b) The pure strategy under Perfect Bayesian equilibria of the game is shown in the extensive form below. Please refer the table below as other possibilities have also been added.

Since the seller would have been in advantage if buyer makes the first move. On gaining one more point on each value buyer's gain would increase by 1 and seller would be in loss.

The perfect equilibria will be at (1,1,1),(1,1,1) which is an optimum gain for both buyer and seller.