Simon Fraser University
Spring 2016
Econ 302 (D200) Quiz Solution
Instructor: Songzi Du
Thursday February 4 2016, 8:30 – 9:20 AM
1.
(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
Z
A
4, 2
3, 5
3, 4
5, 3
B
3, 3 10, 4
2, 3
3, 2
C
3, 6
1, 4
2, 2
3, 10
D
3, 5
6, 3
4, 2
3, 11
Solution:
Part (i) ISD:
1. First iteration: C is strictly dominated by A; Y is strictly dominated by X.
2. Second iteration: D is strictly dominated by 1/2A + 1/2B.
3. Third iteration: W and Z are strictly dominated by X.
4. Fourth iteration: A is strictly dominated by B.
The strategies that survive ISD: {B} for player 1 and {X} for player 2.
Part (ii) Nash equilibrium:
Since one strategy survive ISD for each player, there is only one Nash equilibrium (in
pure and mixed strategy): (B, X). The players get 10 and 4, respectively.
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2. (10 points) The market demand function for a homogeneous good is Q(P ) = 50−P/2.
There are two firms producing this good: firm 1 has a total cost of C1 (q1 ) = q12 ; and firm 2
has a total cost of C2 (q2 ) = 4q2 , where q1 and q2 are the quantity of production for firm 1
and 2, respectively.
Suppose that the two firms compete by simultaneously setting their quantities (q1 and
q2 ), and the price of the good is determined by the market demand function given the total
quantity. Calculate the Nash equilibrium (q1 , q2 ) in this game, and calculate the price in this
equilibrium.
Solution: omitted; see solution to homework or past exams. (The Nash equilibrium is
q1 = 10.4 and q2 = 18.8; the equilibrium price is 41.6.)
3. (10 points) You and a friend are in an Italian restaurant, and the owner offers both of
you a free eight-slice pizza under the following condition. Each of you must simultaneously
announce how many slices you would like; that is, each player i, (i = 1, 2) announces his/her
desired amount of pizza, 0 ≤ si ≤ 8 (si has to be an integer). If s1 + s2 ≤ 8 then the players
get their demands (and the owner eats any leftover slices). If s1 + s2 > 8, then the players
get nothing. Suppose a player’s payoff is the number of slices that he/she gets. For example,
if s1 = 3 and s2 = 4, then player 1 gets 3 and player 2 gets 4; if s1 = 3 and s2 = 6, then each
player gets 0.
(i) Find all pure-strategy Nash equilibrium (s1 , s2 ) in this game.
(ii) Is the following a mixed-strategy Nash equilibrium: player 1 announces s1 = 4 with
probability 1/2 and announces s1 = 5 with probability 1/2, and player 2 announces s2 = 4
with probability 1/2 and announces s2 = 3 with probability 1/2? Why or why not?
Solution:
Part i: pure-strategy NE: (0, 8), (1, 7), (2, 6), (3, 5), (4, 4), (5, 3), (6, 2), (7, 1), (8, 0),
and (8, 8).
Part ii: this is not a mixed-strategy NE, because the only best response of player 2 is
s2 = 3 to player 1’s mixed strategy stated here.
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