Simon Fraser University
Spring 2015
Econ 302 D200 Final Exam
Instructor: Songzi Du
Tuesday April 21, 2015, 12 – 3 PM
Write your name, SFU ID number, and tutorial section number on both the exam
booklets and the questionnaire. Hand in both the exam booklets and the
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.
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NE = Nash equilibrium, SPE = subgame perfect equilibrium, PBE = perfect
Bayesian equilibrium
1. (10 points) Consider the following simultaneous-move game.
W
X
Y
Z
A
0, 2
4, 1
4, 0
4, 0
B
1, 0
2, 1
2, 0
7, 0
C
0, 1
2, 3
3, 1
3, 2
D
0.35, 1
3, 3
4, 2
3, 1
(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.
(ii): Find all NE (pure and mixed) in this game, and for each NE that you find, calculate
the two players’ expected payoffs in the equilibrium.
2. (15 points) Consider a game in which there is a prize worth $30. There are three
players: Alice, Bob, and Clair. Each can buy a ticket worth $15 or $30 or not buy a ticket at
all. They make these choices simultaneously and independently. Then, knowing the ticketpurchase decisions, the game organizer awards the prize. If no one has bought a ticket, then
prize is not awarded. Otherwise, the prize is awarded to the buyer of the highest-cost ticket
if there is only one such player or is split equally between two or three if there are ties among
the highest-cost ticket buyers. (For example, if Alice buys $15 ticket, Bob buys $30 ticket
and Clair does not buy, then Bob gets the prize of $30; if Alice, Bob and Clair all buy $15
ticket, then each gets a prize of $10.) Suppose the final payoff of a player is the prize that
he gets minus the money he pays.
(i) Find all pure-strategy NE of this game (remember, a pure-strategy profile is of the
form: Alice chooses X, Bob chooses Y, and Clair chooses Z, not necessarily distinct choices).
(ii) Each player has a weakly dominated strategy. What is it? (A strategy is weakly
dominated if there is another strategy that is always weakly better (weakly better = either
the same or strictly better).) Consider the remaining two pure strategies. Suppose each
player uses the same mixed strategy over these two pure strategies. Calculate this symmetric
mixed-strategy NE. What is a player’s expected payoff in this mixed strategy equilibrium?
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3. (10 points) Consider the price competition between two firms with differentiated
products (for example, iPhone and Samsung Galaxy). Firm 1 and 2 simultaneously set
p1 and p2 , where pi is the price of firm i’s product. The demand for firm 1’s product
is q1 = 22 − 2p1 + p2 (the demand for firm 1’s product increases with p2 (the competitor’s
price) and decreases with p1 ). The demand for firm 2’s product is similarly q2 = 22−2p2 +p1 .
The payoff of firm i is pi qi − 10qi , i = 1, 2 (each firm produces its product at a constant
marginal cost of 10).
Calculate the Nash equilibrium (p1 , p2 ). What are the quantities (q1 , q2 ) at this equilibrium?
4. (15 points) Player 1 and 2 are bargaining about the division of $100 over two rounds.
In round 1, Player 1 proposes p1 ∈ {9, 44, 94, 100} in which he gets p1 and player 2 gets
100 − p1 ; Player 2 responds yes or no to p1 : if yes this division is implemented and the game
ends. If no, the game proceeds to round 2, in which Player 2 proposes p2 ∈ {0, 45, 90}, where
he gets p2 and player 1 gets 90 − p2 (by going to round 2 there is a delay cost of 10); Player
1 responds yes or no to p2 : if yes this division is implemented, and if no each player gets 0.
The game ends after round 2.
(i) Draw the game tree. (ii) Find and report two distinct pure-strategy SPE.
5. (15 points) Consider the following game between a firm (Player 1) and a worker
(Player 2). Nature first chooses the type of the firm, which is high type with probability 0.5
and is low type with probability 0.5. The firm (of either type) chooses to offer a job to the
worker or not offer a job. If no job is offered, the game ends and both players receive 0. If the
firm offers a job, the worker either accepts or rejects the offer (without observing the type of
the firm). The worker’s acceptance of the offer brings a payoff of 2 to the firm; the worker’s
rejection brings −1 to the firm. Rejecting the offer yields 0 to the worker. Accepting the
offer of a high-type firm yields 2 to the worker, and accepting the offer of a low-type firm
yields −1 to the worker.
(i) Draw the game tree. (ii) Find all pure-strategy PBE (if any exists), and explain.
Include conditional beliefs (for example, P(H | Offer) ≤ 1/2) in your description of PBE.
6. (10 points)
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In the above game, player 1 is either a cooperative (C) type or an ordinary (O) type.
Player 1 and Player 2 take turns to decide whether to invest on a motorcycle. After both
have invested, player 1 decides whether to share the motorcycle with player 2 (self = not
share). Find all pure-strategy PBE (if any exists), and explain. Include conditional beliefs
(for example, P(C | Invest) ≤ 1/2) in your description of PBE.
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