Simon Fraser University
ECON 302
Spring 2016
S. Lu
Problem Set 7
Due on March 14, at 2:30pm
1. Suppose that GT consists of the following stage game repeated T times, where T is an
arbitrary positive integer. Before each stage, both players observe the history of play.
Top
Middle
Bottom
Left
2,2
0,1
-10,-5
Centre
1,0
0,0
-10,-5
Right
-5,-10
-5,-10
-10,-10
Find every SPE of GT, and carefully justify your answer.
2. Consider the following version of the prisoner’s dilemma, repeated infinitely, with discount
factor 𝛿 ∈ (0,1):
Cooperate
Defect
Cooperate
4,4
6,-1
Defect
-1,6
0,0
a) What is the grim trigger strategy?
b) For what values of 𝛿 does there exist a SPE where both players play the grim trigger
strategy?
c) What is the tit-for-tat strategy?
d) [OPTIONAL] For what values of 𝛿 does there exist a NE where both players play the titfor-tat strategy?
3. Consider an oligopoly with 2 firms, each with constant marginal cost and no fixed costs.
Firm 1 has marginal cost 1, while firm 2 has marginal cost 2. The market demand is
P = 6 – 0.1Q. The firms compete by picking quantities, and this game is repeated indefinitely.
Both firms have discount factor 𝛿.
a) Find each firm’s profit from the one-period Cournot outcome (𝑞1𝐶 , 𝑞2𝐶 ).
b) Suppose that the firms are not allowed to share profits. Is there an SPE where the oneperiod maximum joint profit is realized in each period? Explain.
c) Consider the following strategy profile: “In period 1, firm 1 plays q1* and firm 2 plays
q2*. Thereafter, if (q1*, q2*) has been played in every previous period, then it is played
again. Otherwise, (𝑞1𝐶 , 𝑞2𝐶 ) is played.” Write down the conditions on q1* and q2* so that
this strategy profile is a SPE. The only variables in your answer should be q1*, q2* and 𝛿.
4. Consider the following stage game, played twice, with “discount” factor 𝛿 ≥ 0 (for this
question, do NOT make the usual assumption that 𝛿 ≤ 1; that is, players might value stage 2
payoffs more than stage 1 payoffs here):
Left
Right
Top
2,2
1,0
Bottom
1,0
3,3
For what values of 𝛿 does there exist a SPE where (Top, Left) occurs with probability 1 in
stage 1? Answer the same question for each of the three other stage game outcomes.
[Hint: As usual, start by finding the possible stage 2 outcomes. For each player, which one is
the best reward? The worst punishment?]