Simon Fraser University
Fall 2015
Econ 302 D200 Final Exam
Instructor: Songzi Du
Wednesday December 16, 2015, 8:30 – 11:30 AM
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. There is a shared internet connection which has a maximum capacity of 100. There
are n = 10 users of this connection, and user i uses xi units of data, where 0 ≤ xi ≤ 100.
P
(xi can be a real number, not necessarily an integer.) If ni=1 xi ≥ 100, then the connection
P
is jammed and each user gets zero payoff. If ni=1 xi < 100, then each user i gets a payoff of
P
xi (100 − ni=1 xi ). Suppose that each user chooses his/her xi simultaneously. (Recall that
Pn
i=1 xi = x1 + x2 + · · · + xn .)
P
P
P
i. (5 points) The total payoff of all users is ( ni=1 xi )(100 − ni=1 xi ). Let y = ni=1 xi .
What value of y maximizes the total payoff?
P
ii. (10 points) Calculate the symmetric NE (in pure strategy) such that ni=1 xi < 100.
Would everyone be better off (relative to the NE outcome) if xi = y/n, where y is
determined in part (i), and why or why not?
2.
(10 points) Consider a new version of rock-paper-scissors game which has the
following payoff matrix (player 1 has a big advantage if he uses Rock and player 2 uses
Scissors):
Rock
Scissors
Paper
Rock
0, 0
30, -30
-10, 10
Scissors
-10, 10
0, 0
10, -10
Paper
10, -10
-10, 10
0, 0
As before each player chooses his action simultaneously. Find the mixed-strategy NE
(this equilibrium is asymmetric: the two players use different mixed strategies). What are
players’ expected payoff in this equilibrium?
3. There are 4 houses in a neighborhood. For a resident in a house, the cost of installing
an alarm system is 13, and the cost of his/her house being burglarized is 50. If n (0 ≤ n ≤ 4)
houses have the alarm, then these n houses will not be burglarized (each gets a payoff of
−13), while for the remaining 4−n houses each will be burglarized with probability 1/(4−n)
(each gets a payoff of −50/(4 − n)).
2
Suppose the residents sequentially decide whether or not to install the alarm system:
resident 1 moves first and either installs the alarm or does not; after observing what resident
1 did, resident 2 decides; after observing what resident 1 and 2 did, resident 3 decides; after
observing what resident 1, 2 and 3 did, resident 4 decides.
i. (7 points) Draw the game tree (write the first payoff for resident 1’s payoff, the second
payoff for resident 2’s payoff, and so on).
ii. (8 points) Find and describe the SPE in pure strategy.
4. A pharmaceutical company (player 1) introduces a new cold medicine. The medicine
may either be highly effective (H type) or have little effect (L type). The company knows
the type of the drug, but the consumer (player 2) does not know the type (he/she only
knows the medicine is H type with probability 0.3 and is L type with probability 0.7). The
company can choose either to advertise the drug (action A), at a cost of c, or not to advertise
(action N ), which costs nothing. The consumer decides whether or not to buy the drug after
observing whether the company advertised the drug.
The consumer gets a payoff of 1 if he buys a highly effective drug, -1 if he buys the
little-effect drug, 0 if he does not buy.
If the drug is highly effective, then the company gets 100 if the consumer buys the drug
(i.e., making money from repeated purchases); if the drug has little effect, then the company
gets 10 if the consumer buys the drug (i.e., making money from one time purchase). The
company gets 0 if the consumer does not buy. The net payoff of the company is the payoff
from sales minus the advertisement cost. For example, if the drug has little effect but the
company advertises it, and the consumer buys, then the net payoff of the company is 10 − c.
i. (4 points) Draw the game tree.
ii. (6 points) Suppose c = 20, find all pure-strategy PBE (if any exists).
iii. (10 points) Suppose c = 5, find all pure-strategy PBE (if any exists). Find a mixedstrategy PBE in which the L-type company randomizes between advertising and not
advertising.
In part (ii) and (iii), include conditional beliefs P(H | A) and P(H | N ) in your description
of PBE.
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