Two stores, Allfoods (A) and Barks&Sensor (B), are located on the two opposite ends of a straight road. The length of the road is 1 mi. The road is populated by a continuum of consumers (a unit mass) that are distributed uniformly along this road (alternatively, you may assume that there is a single consumer living on the road, but the location of the consumer is uniform random and is not observed by A and B). In order to buy from a store, each consumer has to travel to that store and back. The cost of a return trip is equal to the distance from consumer's home to the store in miles. The stores sell identical nondivisible products and each consumer wants to buy at most one unit of this product. The value of the product for the consumer is v. The cost of producing one unit of the product is ci for store i {A, B}. The stores maximize profits and the consumers maximize the value of the product net of expenses.

(a) Assume that ci = 0 for store i {A, B}, and that v = (since the value is illdefined, you can assume that the consumers minimize expenses, rather than maximize value). Suppose the two stores set their prices simultaneously (the consumers can see the prices before they decide which store to attend). Formalize this as a game. Suggest an equilibrium notion. (b) Find an equilibrium. (c) Now assume that v is finite. Find an equilibrium.

2 Consider the setup described in Question 1. Assume that the stores set their prices sequentially: First, store A announces its price publicly. Then, store B reacts to store A's price by setting its own. Assume that ci = 0 for store i {A, B}, and that v = .

(a) Formalize this as a game. Suggest an equilibrium notion.

(b) Find an equilibrium.

(c) Which of the two stores has an advantage over the other? Explain the nature of this advantage. Compare your findings with the results from a standard Stackelberg model of oligopoly. Compare this equilibrium to the one you found in 1(b) and explain the differences.

3 Consider the setup described in Question 1 and assume that cB = 0, and that v = . Assume that stores set prices simultaneously

(a) Suppose that cA > 0. Find an equilibrium

(b) Now suppose cA is random: cA = 0 with probability 1 and cA = c > 0 with probability (0, 1). Assume that store A privately observes the realization of cA. Formalize this as a game and suggest an equilibrium notion.

(c) Find an equilibrium under the assumptions stated in 3(b). Explain the difference between this equilibrium and the ones you found when solving questions 1(b) and 3(a).

4 Consider the setup described in Question 1 and assume that cB = 0, and that v = . Suppose cA is random: cA = 0 with probability 1 and cA = c > 0 with probability . Assume that store A privately observes the realization of cA. Also, assume that stores set prices simultaneously.

(a) Suppose that before the prices are set, a state commissioner visits store A. The commissioner examines the costs of sourcing the product. If the costs are zero, the commissioner produces and publicly shares a report certifying that the costs are indeed zero. If the costs are not zero, the commissioner does not produce a report. Formalize this as a game and find an equilibrium.

(b) Now suppose that firm A can privately bribe the commissioner to convince her not to issue the public report. The size of the bribe is b. Formalize this as a game and suggest an equilibrium notion.

(c) Find an equilibrium under the assumptions stated in 4(b).