Diluted Happiness: Consider a relationship between a bartender and a cus tomer. The bartender serves bourbon to the customer and chooses x E [0, 1 which is the proportion of bourbon in the drink served, while 1-x is the pro- portion of water. The cost of supplying such a drink (standard 4-ounce glass is cx, where c 0. The customer, without knowing x, decides on whether or not to buy the drink at the market price p. If he buys the drink his payoff is vx p, and the bartender's payoff is p cx. Assume that v c and all pay offs are common knowledge. If the customer does not buy the drink he gets 0 and the bartender gets -cx. Because the customer has some experience, once the drink is bought and he tastes it, he learns the value of x, but this is only after he pays for the drink.

A. Find all the Nash Equ.

B. Assume the customer is in town for 10 days, each player tries to maximize the non-discounted sum of his stage payoffs. Find all the subgame-perfect equ.

C. Now assume it is an infinite game, where the discount rate is between 0 and 1, what is the range of prices p (expressed in parameters of the problem) for which there exists a subgame-perfect equilibirium in which every day the bartender chooses x=1 and the customer chooses to buy