Simon Fraser University
Fall 2014
Econ 302 D100 Final Exam
Instructor: Songzi Du
Monday December 8, 2014, 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.
5. A request for regrade can be accepted only if the exam is written with a pen.
1
NE = Nash equilibrium, SPE = subgame perfect equilibrium, PBE = perfect
Bayesian equilibrium
1. (10 points) Consider the following simultaneous-move game (Table 1).
W
X
Y
Z
A
9, 2
4, 3
1, 2
4, 2
B
3, 2 3, 6
2, 5
7, 5
C
4, 0
7, 1
3, 0
3, 2
D
4, 3
4, 4
9, 3
3, 3
Table 1
Part 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.
Part 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. (10 points) Three firms are considering entering a new market. The payoff for each
, where n is the number of firms that enter. The cost of entering is 62.
firm that enters is 150
n
− 62, and 0 for a firm that does not enter.)
(So the net payoff for a firm that enters is 150
n
Part i: Find all the pure-strategy NE, and calculate the firms’ payoffs in the equilibrium.
Part ii: Find the symmetric mixed-strategy NE in which all three firms enter with the
same probability, and calculate the firms’ payoffs in the equilibrium.
3. (10 points) Find and report all pure-strategy SPE in Figure 1. Explain your answers.
4. (10 points) Two players must choose among three alternatives, a, b, and c. Their
payoffs from each alternative are as follows:
a
b
c
player 1 10
8
-1
player 2
3
1
2
The rules are that player 1 moves first and vetoes one of the three alternatives. Then
player 2 chooses one of the remaining two alternatives.
2
Figure 1
Part i: Draw the game tree.
Part ii: Find and report the pure-strategy SPE. Be sure to report a complete strategy
for each player.
5. (10 points) There are two players. Player 1 is either a high (H) or low (L) type
worker, with probability 0.3 and 0.7 respectively. Player 1 knows his type and chooses to
get an MBA degree (D) or be content with his undergraduate degree (U). Player 2 who is
an employer observes player 1’s degree but not his type, and must decide to assign player 1
to be a manager (M) or a blue-colar worker (B).
Getting an MBA degree has a cost of cH = 3 for a high type worker and cL = 5 for a
low type worker. The market wage for a manager is wM = 10, and the market wage for
a blue-collar worker is wB = 6. The payoff of a high type worker who gets a blue-collar
assignment is wB − cH if he has gotten an MBA and is wB if he has not, and likewise in the
other cases. The employer does not care about the MBA degree per se. The employer’s net
payoff (after paying the wage) depends on the worker’s type and assignment and is given by
the following table:
M
B
H
10
5
L
0
4
Part i: Draw the game tree.
3
Figure 2
Part ii: Find all pure-strategy PBE (if there is any), and explain. Include conditional
beliefs (such as P(H | D) = 1 and P(H | U ) ≤ 1/2.) in your description of PBE.
6. (15 points) Players 1 and 2 put a dollar each in a pot, and player 1 is dealt a card
which is either a king (K) or an ace (A) (with equal probability). Player 1 observes his card
and then decides whether to fold, forfeiting his dollar to player 2, or to bid, proceeding with
the game. If player 1 bids, then without knowing the card of player 1 player 2 can fold and
forfeit his dollar to player 1, or bid, in which case each player must add another dollar to
the pot. After bidding by both players, if player 1 has a king then player 2 wins the pot,
while if player 1 has an ace then player 1 wins the pot. See the game tree in Figure 2.
Part i: Find all pure-strategy PBE, if there is any. Explain your answers.
Part ii: Find a mixed-strategy PBE, with the following steps:
(0) Notice that if player 1 has an ace, then he never folds (i.e., A-Bid); suppose player 1 with
a king bids with probability p, and player 2 bids with probability q, where 0 < p < 1
and 0 < q < 1. We need to calculate p and q that make a PBE.
(1) For player 1 with a king to mix between bidding and folding, he must be indifferent;
what probability q makes player 1 with a king indifferent?
(2) For player 2 to mix between bidding and folding, he must be indifferent; what belief
4
about player 1’s card (conditional on player 1 bidding) makes player 2 indifferent?
(3) What probability p gives (via Bayes’ rule) the conditional belief that makes player 2
indifferent?
(4) Steps 0 – 3 determine the mixed-strategy PBE. What are players 1 and 2’s expected
payoffs in this equilibrium? (For player 1, calculate his expected payoff before he is dealt
the card.) Is the game rigged in a player’s favor?
5