Question

Consider the used car Lemons game played in class. Suppose there are two kinds of used cars, good and bad. The current owner (seller) values good cars at S10,000 and bad cars at S6,000. She knows if the car is good or bad. The dealer (buyer) values good cars at $12,000 and bad cars at $7,000. The buyer's payoff is his value of the car minus the price paid. The seller's payoff is the price she receives minus her value of the car. Finally, assume the dealer thinks a car is good with probability p and thinks bad cars occur with probability 1-p 

2a) What is the type space of this game? 

2b) Give an example of an outcome from a separating equilibrium 

2c) Give an example of an outcome from a pooling equilibrium 

2d) Find the Bayesian Nash equilibrium of this game assuming p-0.5 

2e) Find the smallest value of p such that a pooling Bayesian equilibrium exists.

Answer 1

Ans 2 a The type space of the game will be 2*2 matrix with the respective price for seller and buyer for both type of cars.

Ans b Suppose if the buyer has fixed that he would be buying the car at 7000 dollars then the selllers who have cars at good condition won't be willing to sell their cars because they wanted to get 10000 dollars for their cars at good condition.Thus this is will create a seperate equilibrium.

Ans c In pooling equilibrium both type of owner would be willing to sell their cars.So suppose if the buyer is willing to pay 12000 dollars then owners for both type of cars would be willing to sell their cars.

Ans e Expected Value Buyer willing to give is p*12000 + (1-p)*7000 = 7000 + 5000p

For pooled equilibrium this should be greater than 10000 therefore p > 3/5

So smallest value of p is 3/5.