Simon Fraser University
Fall 2012
Econ 302 Midterm Solution
1.
(10 points) Find all pure-strategy subgame perfect equilibria of the following game.
(Hint: there are two.) Extra credit (5 points): find one (non-pure) mixed-strategy subgame
perfect equilibrium.
1
a
b
(0, 1)
2
c
d
(−1, −1)
1
e
f
(1, −2)
2
g
(0, 0)
h
(2, 0)
Answer:
Pure-strategy SPE #1: Player 1 plays (a, e), Player 2 plays (c, g).
Pure-strategy SPE #2: Player 1 plays (b, f ), Player 2 plays (d, h).
Extra credits: a non-pure mixed-strategy SPE: Player 1 plays (a, e), Player 2 plays (c, 1/2g +
1/2h). Another non-pure mixed-strategy SPE: Player 1 plays (b, f ), Player 2 plays (d, 1/2g +
1/2h). There are many others!
1
2. (10 points) Find all subgame perfect equilibria (both pure and mixed) of the following
game.
1
a
2
e
c
b
d
2
f
e
(2, −2) (4, −4) (1, 0)
2
e
f
2
f
g
(1, 1) (2, −2) (10, 1)
(3, 1)
h
(1, 2)
Answers:
First, in a SPE player 2 must play h (in the information set in which player 1 had played d).
Given this, action a is the dominant action of player 1. And given a by player 1, the best
response of player 2 (in his other information set) is e.
Therefore, the only SPE is a (by player 1) and (e, h) (by player 2).
2
3. (15 points: 5 + 10) There are two players: Alice and Bob. Alice first picks L or R,
which is observed by Bob (and by Alice herself). If L is chosen, then Alice and Bob play a
simultaneous-move Prisoner’s Dilemma game (Table 1). If R is chosen, then Alice and Bob
play a simultaneous-move Battle of the Sexes game (Table 2). (i) Draw the extensive form
of this game. (ii) Find all subgame-perfect equilibria (both pure and mixed).
Table 1: Prisoner’s Dilemma
C
D
C 3, 3 -1, 5
D 5, -1 0, 0
Table 2: Battle of the Sexes
B
H
B
2, 1
-10, -10
H -10, -10
-1, 4
Answers:
Part i:
A
L
R
A
A
C
D
B
B
B
H
B
B
C
D
C
D
B
H
B
H
3
3
−1
5
5
−1
0
0
2
1
−10
−10
−10
−10
−1
4
Part ii:
If L is played by Alice, then the only NE in this subgame is (D, D). If R is played by Alice,
then the subgame has three NE’s: (B, B), (H, H), and (14/25B + 11/25H, 3/7B + 4/7H).
Alice get an expected utility of −34/7 in the mixed-strategy NE.
Therefore, there are three SPE’s:
3
1. Alice plays: (R, D, B); Bob plays (D, B).
2. Alice plays: (L, D, H); Bob plays (D, H).
3. Alice plays: (L, D, 14/25B + 11/25H); Bob plays (D, 3/7B + 4/7H).
4
4. (10 points) Find the strategies that survive iterative deletion of strictly dominated strategies (ISD). Hint: only one strategy of each player survives. What are the Nash equilibria
(pure and mixed) in this game?
D
E
A 1, 0.5
0, 1
B 0, 0.9
1, 1
C 0.4, 1 0.3, 0.3
F
1, 0.7
0, 0.6
0.45, 1
Answers:
First round: delete action C, which is dominated by 1/2A + 1/2B.
Second round: delete D and F, which are dominated by E.
Third round: delete A, which is dominated by B.
ISD strategies: B and E. The unique NE: B and E.
5
5. (10 points) The market (inverse) demand function for a good is P (Q) = 10 − Q, where
Q is the total quantity of the good on the market. There are two firms: firm 1 has a constant
marginal cost of 2 for producing each unit of the good, and firm 2 has a constant marginal
cost of 4. Firm 1 is the industry leader, so it sets its quantity of production first. Firm 2
sets its quantity of production after observing firm 1’s production. Find the subgame perfect
equilibrium in this game.
Answers:
In the subgame (in stage 2) in which firm 1 had produced q1 units, firm 2’s best response is
to solve:
max(10 − (q1 + q2 ))q2 − 4q2 .
q2
The first-order condition is:
10 − q1 − 2q2 − 4 = 0,
or
1
6 − q1
= 3 − q1 .
(1)
2
2
Equation (1) is the equilibrium strategy of firm 2; it specify firm 2’s best-responding action
after every contingency (which is a quantity of firm 1, q1 ).
Given that player 2 is playing according to the strategy (1), firm 1 in the beginning of the
game solves:
1
q1 − 2q1 .
max 10 − q1 + 3 − q1
q1
2
q2∗ (q1 ) =
The first order condition is:
10 − q1 − 3 − 2 = 0,
or
q1∗ = 5.
(2)
Equation (2) is the equilibrium strategy of firm 1, and together with the strategy of firm 2
in (1) they form the subgame-perfect equilibrium of this game.
6
6. (10 points) The following stage game is infinitely repeated:
C
D
C
2, 2
20, -10
D
-10, 20
-1, -1
In each period the two players move simultaneously and observe the actions from the previous
periods. Assume that the final payoff of each player is the time-discounted sum of payoffs
from all periods, with a discount factor of δ ∈ (0, 1). Recall the grim-trigger strategy: each
player plays C unless one of them had played D in the past, in which case each player
plays D. Calculate the range of δ such that the grim-trigger strategy is a subgame perfect
equilibrium of this infinitely-repeated game.
Answers:
δ
2
≥ 20 +
(−1),
1−δ
1−δ
i.e.,
2 ≥ 20(1 − δ) + (−1)δ,
i.e.,
6
δ≥ .
7
7
7. (10 points) Two distinct proposals, A and B, are being debated in the capitol. The
Parliament likes proposal A, and the Prime Minister likes proposal B. The two proposals
are not mutually exclusive; either or both or neither may become law. Thus, there are four
possible outcomes, and the utilities of the Parliament and of the Prime Minster are listed in
the following table.
Parliament
4
1
3
2
A becomes law
B becomes law
Both A and B become law
Neither (status quo prevails)
Prime Minister
1
4
3
2
The game moves as follows. First, the Parliament decides whether to pass a bill and whether
it is to contain A or B or both. If the Parliament does not pass a bill then the status quo
prevails. Otherwise, the Prime Minister sees the bill and decides whether to sign or veto
it. If the Prime Minister signs the bill it becomes the law; if he vetoes the bill then the
status quo prevails. Draw the extensive form for this game, and find the subgame perfect
equilibrium.
Answers:
P
AB
PM
sign
3
3
A
PM
veto
2
2
sign
4
1
not pass a bill
B
2
2
PM
veto
2
2
sign
1
4
veto
2
2
SPE:
If the Parliament passes a bill containing A only, then the Prime Minister vetoes it.
If the Parliament passes a bill containing B only, then the Prime Minister signs it.
If the Parliament passes a bill containing both A and B, then the Prime Minister signs it.
(If the Parliament does not pass a bill, the Prime Minister cannot do anything.)
The Parliament passes a bill containing both A and B.
8