Boğaziçi University
EC 306 Problem Set 1
1. Suppose there are three players, A, B, and C, who play the following two-stage sequential game:
First stage: Players A and B play the following game:
(A)/(B)
U
D
L
(1,2)
(2,1)
R
(2,1)
(1,2)
Second stage: the game ends unless the outcome of the game above is (U, R) or (D,L), otherwise players
B and C learn about the actions chosen in the first stage and play the game below.
(B)/(C)
c
d
a
(0,3)
(2,5)
b
(5,1)
(1,7)
The overall payoff for a player is the sum of the payoffs, if any, from both stages.
(a) Represent the game above in extensive form.
(b) Find the set of SPNE (Pure+Mixed).
2. Consider a duopoly market with two firms, Firm L and Firm F. They compete according to Stackelberg
Game. First, Leader firm, Firm L, chooses its quantity supply, QL , then Follower firm, Firm F, observes
this quantity and chooses its quantity supply. Next market price is determined by the price function
P = 60 − 2Q where Q is the total market supply. Assume that both firms have same cost function
C(Q) = 8Q.
(a) Find Backward Induction strategies.
(b) Find the Backward Induction outcome (profits).
(c) Show that there is a pure strategy Nash equilibrium in which Firm L produces 0.
(d) If the equilibrium you find in part c a Backward Induction solution, prove it. Otherwise, explain the
problem with the Nash equilibrium you find in part c.
3. Consider a market with an Incumbent firm, Firm I, and a potential entrant, Firm E. Firm I’s cost function
is CI (Q) = cI ∗ Q where cI ∈ {4, 32} and Firm E’s cost function is CE (Q) = 8 ∗ Q. Values of cI are equally
likely and cost functions, possible values of cI and probabilities are common knowledge. In the market,
price is determined by the price function P = 64 − Q where Q is the total supply in the market. They play
a two-period game. In the first period, nature picks a marginal cost for Incumbent firm, Firm I observes
that value cI and chooses a quantity supply Q1I ∈ {16, 30} and make some profit. At the end of this period
Firm E observes chosen quantity but not the marginal cost and decides either to enter or to stay out. If
Firm E chooses to enter the market then pays the cost of $200, enters the market and learns the true value
of cI . In the second period if there are two firms in the market, first Firm I chooses its quantity supply, Q2I ,
next Firm E observes that and chooses its quantity Q2E (they play a Stackelberg Game), otherwise Firm I
chooses a quantity supply Q2I and make some profit. The payoffs are total net profits they earn at the end
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of the second period, i.e. uI = πI1 (cI , Q1I ) + πI2 (cI , Q2I , Q2E ), uE (In) = πE
(Q2I , Q2E ) − 200, uE (Out) = 0.
(a) What would Firm I choose as Q2I if Firm E stays out? Find two values for Firm I (one for high cost
and one for low cost).
(b) What would be the best response function of Firm E, Q2E (Q2I ) if Firm E enters?
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(c) What would Firm I choose as Q2I in subgames where Firm E enters (Backward induction strategy)?
What about Firm E? Find two values for each firm (one for high cost Firm I and one for low cost
Firm I).
(d) Given that they choose Q2I , Q2E you found in part a,b and c in the second period, draw a reduced
game for the first period.
(e) Find the set of pure strategy Perfect Bayesian Equilibrium such that they choose Q2I , Q2E you found
in part a,b and c in the second period.
4. Consider the following game of incomplete information. First, Nature picks a type for Player 1 (H or L,
equally likely). Then, only Player 1 observes Nature’s choice and they start playing a game. The extensive
form representation is given below:
(2, 0)
L
P1
U
(3, 2)
D
(1, 0)
U
(1, 0)
D
(1, 1)
R
(q)
H
P2
L
(2, 0)
L0
P1
R0
(1 − q)
Find the set of Perfect Bayesian Equilibria of the game above.
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