ECON 2112 T1 2024: Problem Set 2
Due: 5pm, March 1 (Friday), 2024
• Please write your name and ID on the first page of your assignment.
• Please show your work for all parts, unless the part is worth 0.5 points.
• Please submit your assignment as a pdf document.
Exercise 1. [3 points] Airbus and Boeing - A and B hereafter - are two major players in the
market for aircrafts. B is deciding whether to enter a new market. If B stays Out, B receives 0
and A bags $10 million in profit. If B enters, i.e., if B stays In, A can either start a Price war or
Accommodate. If A chooses Accommodate, each receives $3 million. Else, if A chooses Price
war, each loses $1 million (i.e., think of this as −1).
(i) (1 point) Write down the extensive-form game. [Note: It is sufficient to draw a game
tree specifying players’ names at the decision nodes, choices in the branches, and
payoffs at the terminal nodes]
(ii) (1 point) Solve the game using backward induction and write down the associated (a)
pure strategy profile, and (b) behavioural strategy profile
(iii) (1 point) Now consider the normal-form game where payoffs are as stated above but
A and B are moving simultaneously. Draw the payoff matrix with A being the row
player and B is the column player and find all Nash equilibria (in both pure and mixed
strategies).
Exercise 2. [3 points] Consider the following perfect information game:
RL
1
y
0, 0
x
1, 1
1
z
1, 1
w
0, 0
2
(i) (1 point) Solve the game using backwards induction and write down the behavioural
strategy profile and the pure strategy profile (that conforms with backward induction)
(ii) (0.5 points) Now consider the normal form representation of the game. Draw the
payoff matrix with player 1 as the row player and player 2 as the column player.
(iii) (0.5 points) State all Nash equilibria in pure strategies.
(iv) (1 point) Are all Nash equilibria in (iii) admissible? Explain.
1
2Exercise 3. [3 points]Solve the following perfect information game using backwards induc-
tion. As a part of your answer, you need to state what each player chooses at each node and
why.
ba
1
d
0, 2
c
2
1
2
3, 0
1
2
4, 2
N
f
1, 1
e
2
h
3, 0
g
1
mj
k
4, 3
2
on
1, 2
1
2
3
1, 3
1
3
4, 6
N
q
3, 2
p
2, 5
1
Exercise 4. [3 points] Two companies are selling software which are imperfect substitutes of
each other. Let p1 and x1 denote the price and the quantity sold of software 1. Similarly, let p2
and x2 denote the price and the quantity sold of software 2 respectively. Demand for x1 and x2
are respectively given by
x1 = 90 −
p1
2
+
p2
3
x2 = 90 −
p2
2
+
p1
3
Each company has incurred fixed cost for designing their software and writing the programs,
but the cost of selling to an extra user is zero. Therefore each company will maximize its profits
by choosing the price that maximizes its total revenue (which is same as its total profit in this
case)
(i) (0.5 points) Write payoff functions for the two companies [note: these will be in terms
of p1 and p2]
(ii) (1 point) Suppose company 2 chooses p2 = 90 with probability 0.5 and p2 = 180 with
probability 0.5? What is the best response for company 1?
(iii) (1.5 points) Suppose company 1 locks in its price first. Company 2 sets price after
company 1 has locked in its price. What price will company 1 choose to maximize its
revenue?
Exercise 5. [not for submission] Recall Traveller’s Dilemma. Consider a slight variation of
the game where the airline manager asks Lucy to say a number first and Pete can go next. Pete
will know what Lucy says when he states the price of the gift. Assume both Pete and Lucy are
interested in maximizing just their own payoffs.
• What number will Lucy say? What about Pete?
3• What will Lucy and Pete get?
Briefly explain your response to (i) and (ii). [Hint: One way to think about it is that suppose
Lucy says L ∈ {2, 3, ..., 100}. What is Pete’s best response? Anticipating that what will Lucy
say.]
Exercise 6. [Not for submission] Three oligopolists operate in a market with inverse demand
given by
P(Q) = 24 − Q,
where Q = q1 + q2 + q3 and qi is the quantity produced by firm i. Each firm i’s cost function is
given by C(qi) = 6qi. Each firm’s profit is given by
pii = P(Q)qi − 6qi
The firms choose their quantities dynamically as follows: (1)First, firm 1, who is the industry
leader, chooses q1 (2) Subsequently, firms 2 and 3 observe 1 and then simultaneously choose
q2 and q3 respectively.
• Find the unique subgame-perfect Nash equilibrium (SPNE) of this game.
• How much profit does each firm earn in SPNE?