Simon Fraser University
Spring 2013
Econ 302 Quiz — Solution
Instructor: Songzi Du
Section D200
Thursday Feb. 7, 2013, 8:30 – 9:20 AM
• Name:
• SFU ID number:
• Tutorial section number:
1. This is a closed-book exam.
2. You may use a non-graphing calculator.
3. There is no separate exam booklet. Write your solution in the space following each
question. There are also some blank pages in the back.
4. Show your work! Partial credits are given. Answers without proper explanation/calculation
will be penalized.
5. We will accept a request for regrade only if the solution is written with a pen.
6. The quiz has 30 points (10% of the class grade).
7. Stay in your seat if less than 20 minutes remain in the exam. You may leave early if
you finish 20 minutes before the exam is over.
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1. (10 points) Recall the traveler’s dilemma from class: an airline has lost two suitcases
belonging to Alice and Bob. There are identical antiques in both suitcases. To determine
the amount of compensation, the airline comes up with the following scheme: Alice and Bob
each writes a claim value, simultaneously: 2 ≤ vA , vB ≤ n, where n is the maximum possible
value (which is common knowledge), vA is the claim of Alice, and vB is the claim of Bob;
the claim values vA and vB must be integers.
If both Alice and Bob claim the same value (vA = vB ), each of them gets that amount.
If Alice claims a smaller value (vA < vB ), Alice gets vA + 2, and Bob gets vA − 2. If Bob
claims a smaller value (vB < vA ), Alice gets vB − 2, and Bob gets vB + 2.
Assume n = 5. Part i: Write down the normal form of this game. Part ii: Find the set
of strategies that survive iterative deletion of strictly dominated strategies (ISD). For each
strategy that you delete, you must write down the strategy that strictly dominates it.
Answer:
Normal form:
$5
$4
$5 5, 5 2, 6
$4 6, 2 4, 4
$3 5, 1 5, 1
$2 4, 0 4, 0
$3
1, 5
1, 5
3, 3
4, 0
$2
0, 4
0, 4
0, 4
2, 2
Iterated deletion of strictly dominated strategies (ISD):
First iteration — $5 is strictly dominated by 0.9($4) + 0.1($2) for both Alice and Bob.
Second iteration — $4 is strictly dominated by 0.5($3) + 0.5($2) for both Alice and Bob.
Third iteration — $3 is strictly dominated by $2 for both Alice and Bob.
Strategies that survive: $2 for both Alice and Bob.
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2.
(10 points) 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 Nash equilibria (pure and mixed) in this game, and for each Nash
equilibrium that you find, find the two players’ expected payoffs in the equilibrium.
W
X
Y
A
1, -3
0, 2
3, -4
B
0, 4
1, -5
0, 3
C
0.3, 10
0.2, 10
10, 7
D
0.5, 12
0.1, 12 6, 12
Answer:
Iterated deletion of strictly dominated strategies (ISD):
First iteration — D is strictly dominated by 0.4A + 0.6C.
Second iteration — Y is strictly dominated by W .
Third iteration — C is strictly dominated by 1/2A + 1/2B.
Strategies that survive: {A, B} for player 1 and {W, X} for player 2.
Nash equilibrium:
A Nash equilibrium cannot involve strategies that are eliminated in ISD. Clearly there is
no pure-strategy Nash equilibrium.
For the mixed-strategy Nash equilibrium: suppose that player 1 plays pA + (1 − p)B, and
player 2 plays qW + (1 − q)X. Then, we must have the following equilibrium conditions:
q =1−q
and
−3p + 4(1 − p) = 2p + (−5)(1 − p).
Therefore, the mixed-strategy Nash equilibrium is (9/14A + 5/14B, 1/2W + 1/2X). Substitute these solved probabilities into the equilibrium conditions reveals that in this mixedstrategy Nash equilibrium, player 1 gets an expected payoff of 1 × 1/2 + 0 × 1/2 = 1/2, and
player 2 gets (−3) × 9/14 + 4 × 5/14 = −1/2.
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3. (10 points) Assume that there are two firms in the smart-phone market: Apple (with
product iPhone) and Samsung (with product Galaxy). The demand for iPhone is
qa = 400 − 2pa + ps ,
where pa is the price of iPhone, ps is the price of Samsung Galaxy, and qa is the quantity
of iPhone demanded given these prices. Notice that the demand for iPhone decreases with
its own price, but increases with its competitor’s price. Likewise, the demand for Samsung
Galaxy is
qs = 400 − 2ps + pa ,
where qs is the quantity of Samsung Galaxy demanded given the prices of iPhone and Galaxy.
Assume that Apple has a constant marginal cost of 10 for producing each unit of iPhone,
and Samsung also has a constant marginal cost of 10 for producing each unit of Galaxy. And
assume that Apple and Samsung compete by simultaneously setting their prices. Calculate
the Nash equilibrium of this game, and the quantities in this equilibrium. (Hint: symmetry.)
Answer:
Suppose that the two firms follow the symmetric strategy of pricing at p.
Given that the other firm sets the price p, the profit of (say) Apple is from pricing at pa
is:
Πa (pa ) = (400 − 2pa + p)pa − 10(400 − 2pa + p).
Clearly, Apple wants to maximize Πa (pa ) over pa , fixing p. His first order condition is:
400 − 4pa + p + 20 = 0.
At a symmetric Nash equilibrium, we must have pa = p. Therefore, the first order
condition becomes:
400 − 4p + p + 20 = 0,
i.e., p = 140 at the symmetric Nash equilibrium.
Apple’s demand in this equilibrium is qa = 400 − 140 = 260, and Samsung’s demand in
this equilibrium is qs = 400 − 140 = 260.
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