ECON 2070 SUMMER SEMESTER, 2020
Practice Questions for Online Test 4
Problem 1
Two firms compete in quantities. The aggregate inverse demand is given by
P = 100− 4(Q1 +Q2),
where Q1 is the quantity of output produced by Firm 1, and Q2 the quantity of
output produced by Firm 2. Firm 1 has a constant marginal cost of 8 per unit of
output, Firm 2 a constant marginal cost of 4. Neither firm has fixed costs.
(A) Suppose Firm 1 and Firm 2 simultaneously choose Q1 and Q2, respectively.
That is, Firm 1 and Firm 2 are in Cournot competition. Write down the Nash
equilibrium.
Solution: The Nash equilibrium is (Q∗
1
, Q∗
2
) = (22
3
, 25
3
). (For details on how
to solve such a problem, see e.g. Tutorial 6 Question 1.)
(B) Suppose Firm 1 chooses its quantity first, and then Firm 2 chooses its quan-
tity, after having observed Firm 1’s choice of quantity. Write down the sub-
game perfect equilibrium.
Solution: The subgame perfect equilibrium is (Q∗
1
, Q∗
2
(Q1)) where Q
∗
1
= 11
and
Q∗
2
(Q1) =
{
12− 1
2
Q1 if Q1 ≤ 24
0 if Q1 > 24
.
(For details on how to solve such a problem, see the beginning of Lecture
11.)
(C) How much does Firm 1’s profit change if moving from the simultaneous
move game described in (A) to the sequential move game described in (B)?
Solution: Firm 1’s profit increases from pi1(
22
3
, 25
3
) ≈ 215.11 to pi1(11, 6.5) =
242.
Problem 2
Consider the following extensive form game G.
1
ECON 2070 SUMMER SEMESTER, 2020
RL
M
1, 1
1.1
c
0, 1
a
3, 0
b
2.1
ed
2, 1
2.2
X
1, 1
W
0, 3
1.2
Z
2, 2
Y
1, 0
1.3
(A) How many subgames are there?
Solution: 5
(B) List all strategies of player 1.
Solution: LWY , LWZ, LXY , LXZ,MWY ,MWZ,MXY ,MXZ,RWY ,
RWZ, RXY , RXZ
(C) List all strategies of player 2.
Solution: ad, ae, bd, be, cd, ce
(D) Construct a normal form representation of G.
Solution:
ad ae bd be cd ce
LWY 3, 0 3, 0 0, 3 0, 3 0, 1 0, 1
LWZ 3, 0 3, 0 0, 3 0, 3 0, 1 0, 1
LXY 3, 0 3, 0 1, 1 1, 1 0, 1 0, 1
LXZ 3, 0 3, 0 1, 1 1, 1 0, 1 0, 1
MWY 1, 1 1, 1 1, 1 1, 1 1, 1 1, 1
MWZ 1, 1 1, 1 1, 1 1, 1 1, 1 1, 1
MXY 1, 1 1, 1 1, 1 1, 1 1, 1 1, 1
MXZ 1, 1 1, 1 1, 1 1, 1 1, 1 1, 1
RWY 2, 1 1, 0 2, 1 1, 0 2, 1 1, 0
RWZ 2, 1 2, 2 2, 1 2, 2 2, 1 2, 2
RXY 2, 1 1, 0 2, 1 1, 0 2, 1 1, 0
RXZ 2, 1 2, 2 2, 1 2, 2 2, 1 2, 2
2
ECON 2070 SUMMER SEMESTER, 2020
(E) Find all (pure strategy) Nash equilibria of G.
Solution: As we can see from the underlined best responses in the normal
form representation, the (pure strategy) Nash equilibria of G are (RWY, bd),
(RWY, cd), (RWZ, be), (RWZ, ce), (RXY, bd), (RXY, cd), (RXZ, be),
(RXZ, ce).
(F) Find all subgame perfect equilibria of G.
Solution: The SPE of G are (RXZ, be) and (RXZ, ce).
(G) Find a Nash equilibrium in G that leads to a different payoff profile than
every subgame perfect equilibrium. Find a decision node at which a player
makes a non-credible threat in this Nash equilibrium.
Solution: E.g., in (RWY, bd), player 1 makes the non-credible threat Y at
node 1.3.
Problem 3
RL
1.1
r
2, 2
l
1, x
2.1
r
1, 1
l
x, 0
2.2
(A) Identify all values of x for which (L, lr) is a SPE.
Solution: x ≥ 2
(B) Identify all values of x for which this game has a unique SPE.
Solution: x < 2
Problem 4
Consider the following extensive form game (with perfect information):
• There is a Son (Player 1) and a Father (Player 2)
• At the initial node 1.1, Son chooses whether to Obey (O) or Rebel (R).
• If Son chooses O the game ends with Son, Father payoffs equal to (1, 3).
3
ECON 2070 SUMMER SEMESTER, 2020
• If Son choosesR, then we reach node 2.1. Father chooses whether to Punish
(P ) or be Lenient (L).
• If Father chooses L the game ends with Son, Father payoffs equal to (3, 0).
• If Father chooses P , then we reach node 1.2 and Son chooses whether to
Accept (A) or Fight (F ).
• If Son chooses A the game ends with Son, Father payoffs equal to (2, 1).
• If Son chooses F the game ends with Son, Father payoffs equal to (1, 0).
Write down the payoffs of both Son and Father in the subgame perfect equilib-
rium.
Solution: The Son, Father payoffs in the subgame perfect equilibrium are
(2, 1).
Problem 5
At time zero a manager makes an offer w0 to a worker, which the worker may
accept or reject. If she accepts, then the game ends and the payoffs received by the
manager and the worker are 1−w0 and w0, respectively. If the worker rejects the
offer, he makes a counteroffer w1 at time one. If the manager accepts, the game
ends. The payoffs to the manager and the worker are now equal to δ(1−w1) and
δw1, respectively, where δ ∈ (0, 1) is a common discount factor. If the manager
rejects, then the game also ends and both the manager and the worker each receive
a payoff δ
2
.
Write down a SPE strategy of the manager.
Solution: A SPE strategy of the manager is to offer w0 =
δ
2
at t = 0, and to
accept an offer w1 at t = 1 if w1 ≤
1
2
and reject it otherwise.
Problem 6
(In this and all other questions in which a game matrix is given, Player 1
chooses the row, Player 2 chooses the column, and if there is a Player 3, she
chooses the matrix.)
Suppose the following normal form game is played twice. Players observe the
actions chosen in the first period prior to the second period. Each player’s total
payoff is the sum of her payoffs in the two periods. Let x > 0.
4
ECON 2070 SUMMER SEMESTER, 2020
A B C
A 1, 10 0, 0 0, 0
B 2, 0 x, x 0, 0
C 0, 0 0, 0 10, 1
Consider the following strategy: Play A in period 1, play C in period 2 if the
action profile in period 1 is (A,A), otherwise play B. Which of the following is
true?
• Both players following the strategy is a SPE for all values of x > 0.
• Both players following the strategy is a SPE for all values of x ≤ 11.
• Both players following the strategy is a SPE for all values of x ≤ 9.
• Both players following the strategy is a SPE for all values of x ∈ [9, 11].
Problem 7
(In this and all other questions in which a game matrix is given, Player 1
chooses the row, Player 2 chooses the column, and if there is a Player 3, she
chooses the matrix.)
Suppose the following stage game is repeated infinitely:
C D
C 2, 2 0, 5
D 5, 0 1, 1
Let uti be the payoff to player i in period t. Player i (i = 1, 2) maximizes her
average discounted sum of payoffs, given by (1 − δ)
∑
∞
t=1
δt−1ut
i
, where δ is the
common discount factor of both players. Suppose the players try to sustain (C,C)
in each period by the trigger strategy: (a) Play C in the first period; in any other
period (b) play C if both players have played C in all previous periods, (c) play
D otherwise. Write down the minimal discount factor δ for which the adoption of
above trigger strategy by both players constitutes a subgame perfect equilibrium
of the infinitely repeated game.
Solution: δ = 0.75. You can calculate this from equation (3) of the Repeated
Games notes.