08 Jun Q1 Consider the following game played by tw
Question
S4415D Game Theory, Problem Set 4
Q1 Consider the following game played by two players: Player 1 is a monopolist supplier of oil. Player
2 owns a factory with an oil powered generator. The two players play a simultaneous game in
which Player 1 decides whether to raise or not raise the price of oil (R or NR, respectively), and
Player 2 decides whether to build or not build (B or NB, respectively) a new plant that uses
solar energy instead of oil. We assume there is incomplete information, as only Player 2 knows
the cost of building the plant, which can be high (H) or low (L). The payoff of each scenario is
summarized in the following payoff matrix:
Low Costs, L
1\2
B
NB
R
-1, 3 1, 2
NR 0, 5 0, 3 High Costs, H
1\2
B
NB
R
-1, 0 1, 2
NR 0, 2 0, 3 Player 1 does not know the cost, but believes with probability ? that the true cost is H.
a) Model this static Bayesian game as a dynamic game with imperfect information (i.e. state
the players, their (pure) strategies and draw the extensive form game, carefully labeling the
payoffs and denoting information sets).
b) Are there any “pooling” Bayesian Nash Equilibria in pure strategies? That is, are there any
BNE in which Player 2 plays the same strategy regardless of what the cost of building the
plant is? If so, for what values of ? will these equilibria hold?
c) Are there any “separating” Bayesian Nash Equilibria in pure strategies? That is, are there
any BNE in which Player 2 plays one strategy when the cost of building the plant is high,
and a different strategy when the cost of building the plant is low? If so, for what values of
? will these equilibria hold? 1 Q2.
Consider the following duopoly game. Firm 1 is the incumbent and decides whether to build (B)
a new plant or not (N). Firm 2 is the potential entrant and does not know how costly it is for the
incumbent to build the new plant. Suppose that there is probability ? that P1 has high building
costs (type I). Payoffs for each type are as given below. Find all Bayesian Nash equilibria of this
game, including those in mixed strategies.
1\2
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