Simon Fraser University
Spring 2015
Econ 302 D200 Midterm Exam
Instructor: Songzi Du
Tuesday March 10, 2015, 8:30 – 10:20 AM
Write your name, SFU ID number, and tutorial section number both on the exam
booklets and on the questionnaire. Hand in both the exam booklets and this
questionnaire. But note that only the exam booklets are graded.
• Name:
• SFU ID number:
• Tutorial section number:
General instructions
1. This is a closed-book exam: no books, notes, computer, cellphone, internet, or other
aids. A scientific, non-graphing calculator is allowed.
2. If you use decimals in calculation, keep two decimal places.
3. Write clearly. Illegible answers will receive no credit.
4. Show your work! Partial credits are given. Answers without proper explanation/calculation
will be penalized.
5. A request for regrade can be accepted only if the exam is written with a pen.
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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.
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?
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.
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(i):
(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.
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
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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.
5. (10 points)
A
B
C
A
5, 5
0, 0
11, 0
B
0, 0
1, 1
11, 0
C
0, 11
0, 11
10, 10
Suppose the above game is played twice. 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 (C, C) is played in the
first stage, and report the players’ payoffs in the equilibrium. Explain why it is a SPE.
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