Simon Fraser University
Spring 2015
Econ 302 D200 Midterm Exam
Instructor: Songzi Du
Tuesday March 10, 2015, 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 simulataneous-move game: Alice secretly chooses a
number a from the set {1, 2, 3}, and Bob tries to guess Alice’s number (let b be Bob’s guess).
If Bob guessed correctly (a = b), he gets two dollars from Alice (leading to payoffs (−2, 2)
for Alice and Bob); if Bob guessed incorrectly (a 6= b), he pays Alice one dollar (leading
to payoffs (1, −1)). Write the payoff matrix, find the pure and mixed Nash equilibria, and
calculate players’ expected payoffs in the equilibrium.
Solution:
The payoff matrix is:
b=1 b=2 b=3
a=1
−2, 2
1, −1
1, −1
a=2
1, −1
−2, 2
1, −1
a=3
1, −1
1, −1
−2, 2
Clearly, there is no pure-strategy Nash equilibrium.
For the mixed strategy Nash equilibrium, suppose Alice chooses 1 with probability p,
chooses 2 with probability q, and chooses 3 with probability 1 − p − q. For all 1, 2 and 3 to
be Bob’s best responses, we must have:
2p − (1 − p) = 2q − (1 − q),
2p − (1 − p) = 2(1 − p − q) − (p + q).
(1)
The solution to the above equations is p = 1/3 and q = 1/3. So in equilibrium Alice chooses
1, 2 and 3 with equal probability of 1/3.
Likewise, in equilibrium Bob must also guess 1, 2 and 3 with equal probability of 1/3.
1
Substituting p = 1/3 and q = 1/3 into Equation (1), we see that Alice gets an expected
payoff of 0 in equilibrium. Likewise, Bob also gets an expected payoff of 0 in equilibrium.
2. (10 points) Ten people (suspects) are arrested after committing a crime. The police
lacks sufficient resources to investigate the crime thoroughly. The chief investigator therefore
presents the suspects with the following proposal: if at least one of them confesses, every
suspect who has confessed will serve a one-year jail sentence, and all the rest will be released
with no sentence. If no one confesses to the crime, the police will fully investigate the crime,
and at the end all ten suspects will each receive a ten-year jail sentence.
We model this situation as a simultaneous-move game in which each player has two
actions: confess or not confess, and his payoff is the negative of his years in jail (e.g., a
payoff of −10 if he gets a ten-year sentence).
(i) Find all pure-strategy Nash equilibria, and what is the intuitive meaning of such an
equilibrium? (ii) Find the symmetric mixed-strategy Nash equilibrium, calculate the total
probability that no one confesses in this equilibrium, and what is a player’s expected payoff
in this equilibrium?
Solution:
Part (i), pure-strategy Nash equilibrium:
The only pure-strategy Nash equilibrium is one person confesses while the rest do not
confess. This is because if no one confesses, then any one person has an incentive to deviate
and to confess (1 year is better than 10 years); and if more than one person confesses, then
any one of the confessor has an incentive to deviate and to not confess (0 year is better than
1 year). The intuitive meaning of the pure-strategy Nash equilibrium is that the suspects
coordinate to sacrifice one person for the greater good of the rest.
Part (ii), mix-strategy Nash equilibrium:
Suppose each person confesses with probability p. Let’s look at the incentives of a person
i. For both confessing and not confessing to be best responses for person i, we must have
−1 = −10(1 − p)9 ,
where (1 − p)9 is the probability that all nine other people do not confess, so that is the
probability that person i gets 10 years if he does not confess.
2
So in equilibrium each person confesses with probability
p=1−
1
≈ 0.23.
101/9
The total probability in equilibrium that no one confesses is
(1 − p)10 =
1
≈ 0.077.
1010/9
In equilibrium, each person gets an expected payoff of −1, because he is indifferent
between confessing and not confessing.
3. (20 points) Find and describe the pure-strategy subgame perfect equilibria (if any)
in the following games. Explain the steps that you use to find the SPE.
(i):
3
(ii):
In (ii) there are random moves by Nature at the beginning of the tree, leading to two
nodes of Player 1. You can think of these two nodes of Player 1 as two distinct players, like
the strong and weak types in the Beer-Quiche game.
Solution:
Part (i):
Clearly in any SPE Player 1 must choose C and F when he reaches those decision nodes.
Given this, a strictly dominates b for Player 2. Thus the only SPE is Player 1 choosing
(B, C, F ) and Player 2 choosing a.
Part (ii):
In any SPE Player 1 must choose c, since c strictly dominates d. At Player 2’s information
set, A strictly dominates B. Given A of Player 2, action b is better than action a for Player
1. Thus a SPE is Player 1 choosing (b, c) and Player 2 choosing A.
There is however another SPE: Player 1 chooses (a, c) and Player 2 chooses B. In this
equilibrium, Player 2 is indifferent between A and B, because his information set is never
reached. This equilibrium is less plausible than the first one.
4.
(10 points) The market (inverse) demand function for a homogeneous good is
P (Q) = 20 − 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 1 for producing each unit of the good, and firm
2 has a constant marginal cost of 3. Firm 1 is the industry leader, so it sets its quantity
4
of production first. Firm 2 sets its quantity of production after observing firm 1’s quantity.
Find the subgame perfect equilibrium of this game. And calculate the market price and the
firms’ profits in the equilibrium.
Solution:
In the subgame in which firm 1 sets q1 , the best response of firm 2 can be found by
solving:
max(20 − (q1 + q2 ))q2 − 3q2 .
q2
The first order condition is:
20 − q1 − 2q2 − 3 = 0 ⇐⇒ q2 =
Thus in the SPE, firm 2’s strategy is q2 (q1 ) =
response in the beginning of the game is:
17−q1
.
2
17 − q1
max 20 − q1 +
q1
2
17 − q1
.
2
Given this strategy, firm 1’s best
q1 − q1 .
The first order condition is:
20 − 2q1 − 17/2 + q1 − 1 = 0 ⇐⇒ q1 = 21/2.
Thus, the SPE is firm 1 producing 21/2, and firm 2 producing
17−q1
2
if firm 1 produces
q1 .
In this equilibrium, the price is 20 − 21/2 +
17−21/2
2
= 25/4, firm 1 gets (25/4 − 1) ×
17−21/2
2
≈ 10.56.
A
B
C
A
5, 5
0, 0
11, 0
B
0, 0
1, 1
11, 0
C
0, 11
0, 11
10, 10
21/2 ≈ 55.13, and firm 2 gets (25/4 − 3) ×
5. (10 points)
Suppose the above game is played twice. In each stage the two players move simultane5
ously, 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 (C, C) is played in the
first stage, and report the players’ payoffs in the equilibrium. Explain why it is a SPE.
Solution:
A (symmetric) SPE is both players taking the following strategy: in the first stage play
C; in the second stage, play A if (C, C) was played in the first stage, and play B if (C, C)
was not played in the first stage. In this equilibrium, each player gets 10 + 5 = 15.
Clearly, in the second stage a Nash equilibrium of the stage game is always played. If a
player does not play C in the first stage, then he gets at most 11 + 1 < 15, so he has no
incentive to deviate from C in the first stage. Thus we have a SPE.
6