Assume you have your car broken down just before the weekend. You value your weekend trip as much as v and if you have to stay home you get the zero utility. There are two dealerships in your town. At the beginning of the day they simultaneously choose a price for repair. Dealers know that when you come to one of them and observe the price, you can always call to another dealer to make an inquiry about his price. The call is costless. The other dealer, however, can be occupied for this day. Assume, this happens with probability ? ? (0, 1) which is a common knowledge (but the dealers do not know whether the other dealer is occupied or not). Assume zero repair cost for the dealer.

1. Show that there is no pure strategy equilibrium of the game.

2. Show that the upper bound of the support of mixed strategies equals to v.

3. Derive the mixed strategy equilibrium of the game. What happens as ? ? 1 and as ? ? 0?

The Model n identical firms produce and sell homogeneous good with marginal production costs c. There is a unit mass of consumers. Each consumer has a valuation of the good 1/ > c now it is the same for all consumers! Fraction A of consumers are informed. These consumers know both the prices and buy at the lowest price. If the prices are equal they split evenly. Fraction 1 A of consumers are uninformed, they just buy the good at the closest store. We assume that they split evenly among the firms. What is the equilibrium price(s) in this model? What are the options left??? There is nothing better than to try mixed strategies! We look at a symmetric candidate equilibrium, in which all firms play a mixed strategy F (p) Note, that the supports of this distributions can include only prices between c and v. The profit of firm one if it charges p comes from two sources: ' Uninformed consumers: 1g)'(p c) ' Informed consumers. if it charges the lowest price. Because each other firmj chooses prices according to F(p) the probability that we have a lower price is 1 F(p). This must hold for all firms. Therefore. our expected profit from uninformed is A[1 F(p)]"'1(p C)- 0 The total expected profit is m(p):A[1F(p)r-1