Exercise session #7
Lecturer: Dmytro Vikhrov
Date: April 3, 2012
Problem 1 (Bertrand price competition)
Assume the following game: two producers in the market simultaneously name their
prices p1 and p2. Given that MC1 = MC2 = c, define and find the Nash equilibrium.
Problem 2 (Cournot quantity competition)
Given the demand function p(q) and MC = c, firms simultaneously decide on quantities,
q1 and q2.
1. Recall the assumptions on the demand function.
2. Define the Nash equilibrium here.
3. Show that the resulting price p∗
> ppc
, where ppc
is the perfect competition price.
4. Show that q∗
1 + q∗
2 > qm
, where qm
is the monopoly quantity.
5. For p(q) = A − Bq find q∗
1, q∗
2, p∗
, π∗
1 and π∗
2.
Problem 3 (Prisoner’s dilemma)
You are given the following one-shot game:
L R
L
2
2
5
0
R
0
5
1
1
1. Find the Nash equilibrium. Is it the maximum payoffs the players can get?
2. Suppose that this game is played repeatedly. Incorporate the possibility of cooperation
and compute the payoffs from playing (L, L) and (R, R)?
3. Change the payoffs of this game to generate the possibility of retaliation (tit-for-tat
strategy). Under what conditions is playing (L,L) sustainable when the game is
played infinitely?
Problem 4 (Repeated interaction in duopoly)
Find such values of σ, for which playing the Nash reversion strategy in Problem 2 is
sustainable for both players.
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