Diluted Happiness:
Consider a relationship between a bartender and a customer. The bartender serves bourbon to the customer and chooses x ∈ [0, 1], which is the proportion of bourbon in the drink served, while 1- x is the proportion 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 payoffs 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 equilibria of this game.
b. Now assume that the customer is visiting town for 10 days, and this "bar game" will be played on each of the 10 evenings that the customer is in town. Assume that each player tries to maximize the (nondiscounted) sum of his stage payoffs. Find all the subgame-perfect equilibria of this game.
c. Now assume that the customer is a local, and the players perceive the game as repeated infinitely many times. Assume that each player tries to maximize the discounted sum of his stage payoffs, where the discount rate is δ ∈ (0, 1). What is the range of prices p (expressed in the parameters of the problem) for which there exists a subgameperfect equilibrium in which every day the bartender chooses x = 1 and the customer buys at the price p?
d. For which values of δ (expressed in the parameters of the problem) can the price range that you found in (c) exist?