Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
ECON0027: GAME THEORY
1 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 ill-
defined, 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.
ECON0027 Coursework 1 TURN OVER
(a) Suppose that cA > 0. Find an equilibrium
(b) Now suppose cA is random: cA = 0 with probability 1−β and cA = c > 0 with probability
β ∈ (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 − β and cA = c > 0 with probability β.
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 commis-
sioner 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).