Simon Fraser University
Fall 2014
Econ 302 (D100) Quiz Solution
Instructor: Songzi Du
Thursday October 2 2014, 8:30 – 9:20 AM
1. (10 points) Consider the following game: Player 1 and 2 simultaneously guess an
integer between 1 and 3. If the numbers coincide, both win the amount they guess. If not,
each loses the amount of his guess.
(i) Write down the payoff matrix.
(ii) 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.
(iii) Find the pure-strategy Nash equilibria.
(iv) Find the symmetric, mixed-strategy Nash equilibrium.
Solution:
(i) Payoff matrix:
1
2
3
1, 1
-1, -2
-1, -3
2 -2, -1
2, 2
-2, -3
3 -3, -1
-3, -2
3, 3
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(ii) All strategies survive.
(iii) The pure-strategy Nash equilibria are (1, 1), (2, 2) and (3, 3).
(iv) Suppose that player 2 guesses 1 with probability p, 2 with probability q, and 3 with
probability 1 − p − q. For 1, 2 and 3 all to be player 1’s best responses, we must have:
p − (1 − p) = −2(1 − q) + 2q
p − (1 − p) = 3(1 − p − q) − 3(p + q)
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The solution to the above equations is p = 5/22 and q = 4/11.
Therefore, the symmetric, mixed-strategy Nash equilibrium is to guess 1 with probability 5/22, 2 with probability 8/22, and 3 with probability 9/22.
2. (20 points) The market demand function for a homogeneous good is P (Q) = 9 − Q.
There are two firms: firm 1 has a constant marginal cost of 1 for producing/selling each unit
of the good, and firm 2 has a constant marginal cost of 5.
(i) The two firms compete by simultaneously setting their prices (p1 and p2 ), and the firm
with the lowest price captures the whole market (if the two firms set the same price,
they split the market evenly). Assume the price that each firm sets must be in the
increments of 0.01: for example, 1.98, 1.99, 2, 2.01, 2.02, ...
Calculate a Nash equilibrium (p1 , p2 ) in this game, such that each pi is not a weakly
dominated strategy. Explain why it is a Nash equilibrium.
(ii) Now assume 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 a Nash equilibrium (q1 , q2 ) in this game.
Solution:
(i) The monopolist price of firm 1 is 5, which is solved from:
max(9 − Q)Q − Q =⇒ Qm = 4, Pm = 5.
Q
Here are two Nash equilibria in undominated strategy: (1) p1 = 4.99 and p2 = 5; (2)
p1 = 5 and p2 = 5.01. (Only one is needed as an answer to this question; in fact, p1 = 5
and any p2 > 5 is also a Nash equilibrium, with the same outcome as (2).)
(Add explanations here.)
(ii) For best response to q2 , firm 1 solves:
max(9 − q1 − q2 )q1 − q1 =⇒ FOC: 9 − q2 − 2q1 − 1 = 0 ⇔ q1 =
q1
2
8 − q2
.
2
For best response to q1 , firm 2 solves:
max(9 − q1 − q2 )q2 − 5q2 =⇒ FOC: 9 − q1 − 2q2 − 5 = 0 ⇔ q2 =
q2
4 − q1
.
2
The solution to the above two equations are q1 = 4 and q2 = 0, which is the Nash
equilibrium.
Note that this is a rare instance when the Cournot and Bertrand models make similar
prediction.
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