Simon Fraser University
Spring 2014
Econ 302 D200 Midterm Exam Solution
Instructor: Songzi Du
Monday March 10, 2014, 8:30 – 10:20 AM
The brief solutions suggested here may not have the complete explanations necessary for
full marks.
1. (10 points) Consider the following game.
W
X
Y
Z
A
2, 1
0, 3
2, 0
2, 0
B
1, 2 3, 1
1, 1
2, 1
C
1, 4
1, 0
4, 3
3, 3
D
0, 1
2, 1
2, 3
4, 0
Part i: Find the strategies that survive iterative deletion of strictly dominated strategies
(ISD). For each strategy that you delete, write down the strategy that strictly dominates it.
Part ii: Find all Nash equilibria (pure and mixed) in this game, and for each Nash
equilibrium that you find, calculate the two players’ expected payoffs in the equilibrium.
Solution:
Part i, Iterated deletion of strictly dominated strategies (ISD):
Round 1: Z is strictly dominated by W .
Round 2: D is strictly dominated by 0.6B + 0.4C.
Round 3: Y is strictly dominated by W .
Round 4: C is strictly dominated by 0.5A + 0.5B.
Strategies that survive ISD: {A, B} for Player 1 and {W, X} for Player 2.
Part ii, Nash equilibria:
Clearly, there is no pure-strategy Nash equilibrium.
Suppose that Player 1 plays pA + (1 − p)B and Player 2 plays qW + (1 − q)X.
1
For both A and B to be Player 1’s best responses, we must have:
2q = q + 3(1 − q),
i.e.,
3
q= .
4
For both W and X to be Player 2’s best responses, we must have:
p + 2(1 − p) = 3p + (1 − p),
i.e.,
1
p= .
3
Thus, Player 1 playing 1/3A + 2/3B and Player 2 playing 3/4W + 1/4X is a Nash
equilibrium. In this equilibrium, Player 1 gets 2q|q=3/4 = 3/2, and Player 2 gets p + 2(1 −
p)|p=1/3 = 5/3.
2. (10 points) Find and describe the pure-strategy subgame perfect equilibria (SPE) in
the following game. Explain the steps that you use to find the SPE.
(This is a three-player game: the payoffs of (1, 3, 1) (given by actions a, d and f ) mean
that player 1 gets 1, player 2 gets 3, and player 3 gets 1.)
1
a
c
b
2
2
e
d
3
2
3
d0
e
d
3
3
e0
3
3
f
g
f
g
f
g
f
g
f0
g0
f 00
1
3
1
3
0
0
3
0
0
1
1
3
0
1
1
1
3
2
1
2
3
0
1
1
−1
1
1
2
0
0
0
1
0
g 00
2
0
1
Solution:
First, any SPE must have Player 2 playing d0 and Player 3 playing f 0 and g 00 . This means
that Player 1 gets −1 from c in any SPE.
2
Thus in any SPE Player 1 will play a. Then it is clear that there are two pure-strategy
Nash equilibria given a: (d, f ) and (e, g).
Therefore, there are two SPE’s:
1. Player 1 plays a, Player 2 plays d and d0 , and Player 3 plays f , f 0 and g 00 .
2. Player 1 plays a, Player 2 plays e and d0 , and Player 3 plays g, f 0 and g 00 .
3. (10 points) The market demand function for a monopolist’s product is P (q) = 20 − q.
The monopolist’s cost function is C(q) = 2q 2 , where C(q) is the cost to produce q units of
the product. Assume that the government imposes a sales tax of 50%, and the sales tax
does not impact the monopolist’s cost. Calculate the monopolist’s price and quantity, and
calculate the corresponding consumer’s price.
Solution:
The monopolist gets 1/(1 + 0.5) = 2/3 of the price paid by consumer. Thus, he solves:
2
max (20 − q)q − 2q 2 .
q
3
The first order condition is:
2
(20 − 2q) − 4q = 0,
3
i.e.,
q = 5/2,
which is the monopolist’s quantity of production.
The consumer’s price is P (5/2) = 35/2, and the monopolist’s price is 2/3 × 35/2 = 35/3.
4. (10 points) Consider a market setting with three firms. Firm 2 and 3 are already
operating as monopolists in two different industries (they are not competitors). Firm 1 must
decide whether to enter firm 2’s industry and thus compete with firm 2, or enter firm 3’s
industry and thus compete with firm 3. Production in firm 2’s industry occurs at zero cost,
whereas the cost of production in firm 3’s industry is 2 per unit. Demand in firm 2’s industry
is given by p = 9 − Q, whereas demand in firm 3’s industry is given by p0 = 14 − Q0 , where
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p and Q denote the price and total quantity in firm 2’s industry and p0 and Q0 denote the
price and total quantity in firm 3’s industry.
The game runs as follows: First, firm 1 chooses between E 2 and E 3 . (E 2 means “enter
firm 2’s industry” and E 3 means “enter firm 3’s industry.”) This choice is observed by firm
2 and 3. Then, if firm 1 chooses E 2 , firm 1 and 2 competes in quantity (simultaneously
setting their production quantities q1 and q2 ); in this case firm 3 is the monopolist in his
industry. On the other hand, if firm 1 chooses E 3 , then firm 1 and 3 competes in quantity
(simultaneously setting their production quantities q10 and q30 ); in this case firm 2 is the
monopolist in his industry.
Calculate and report the subgame perfect equilibrium (SPE) of this game. Note that the
SPE involves the strategies of all three firms.
Solution:
Consider in general the subgame with n firms (where n is either 1 or 2), demand p = a−Q
(where a is either 9 or 14), and cost c per unit (where c is either 2 or 0).
Fix a symmetric strategy profile in which every firm produces q units.
The profit of any firm (say firm i) when he produces qi and the others each produce q is:
Πi (qi ) = qi (a − (qi + (n − 1)q)) − cqi .
Given q, firm i wants to maximize Πi (qi ). The first order condition of firm i with respect
to qi is
a − (n − 1)q − 2qi − c = 0,
which gives firm i’s best-response to q of others:
a − (n − 1)q − c
.
2
At a symmetric Nash equilibrium, we must have qi = q, i.e.,
qi =
q=
a − (n − 1)q − c
,
2
or equivalently,
q=
a−c
.
n+1
4
Firm i gets
Πi (q) = q(a − nq) − cq|q=(a−c)/(n+1) =
(a − c)2
(n + 1)2
in equilibrium.
Therefore, Firm 1 gets (9/3)2 = 9 from E 2 and (12/3)2 = 16 from E 3 . Thus firm 1
chooses E 3 .
In summary, the subgame perfect equilibrium is: firm 1 chooses to enter firm 3’s industry
and produces q1 = (14 − 2)/3 = 4 (if firm 1 enters firm 2’s industry he produces q1 =
(9 − 0)/3 = 3); firm 2 produces q2 = (9 − 0)/3 = 3 if firm 1 enters his industry and produces
q2 = (9 − 0)/2 = 4.5 if firm 1 enters firm 3’s industry; firm 3 produces q3 = (14 − 2)/3 = 4 if
firm 1 enters his industry and produces q3 = (14 − 2)/2 = 6 if firm 1 enters firm 2’s industry.
5. (10 points) The following (stage) game is repeated twice:
L
M
R
U
8, 8
0, 9
0, 0
C
9, 0
0, 0
3, 1
D
0, 0
1, 3
3, 3
In each stage the two players move simultaneously, and in the second stage the players
observe the actions taken in the previous stage. Assume that the final payoff of a player is
the sum of his payoffs from the two stages.
Find and describe a subgame perfect equilibrium (SPE) in which (U, L) is played in the
first stage. Explain why it is a SPE.
Solution:
Clearly, (D, M), (C, R) and (D, R) are all pure-strategy Nash equilibria of the stage
game.
Thus, the following is a SPE: play (U, L) in stage 1; if player 1 did not play U in stage
1, then play (D, M) in stage 2; if player 2 did not play L and player 2 played U in stage 1,
then play (C, R) in stage 2; if player 1 played U and player 2 played L in stage 1, then play
(D, R) in stage 2.
6.
(10 points) Player 1 and player 2 simultaneously make demands m1 and m2 , where
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m1 and m2 are real numbers between 0 and 1. If m1 + m2 ≤ 1 (compatible demands, given
that the surplus to be divided equals 1), then player 1 obtains the payoff m1 and player 2
obtains m2 . On the other hand, if m1 + m2 > 1 (incompatible demands), then both players
get 0. Find and describe all pure-strategy Nash equilibria of this game. Explain your answer.
Solution:
A pure-strategy Nash equilibrium is any (m1 , m2 ) such that m1 + m2 = 1, m1 ≥ 0 and
m2 ≥ 0. Formally, the set of pure-strategy Nash equilibrium is {(m1 , m2 ) : m1 + m2 =
1, m1 ≥ 0, m2 ≥ 0}.
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