ECON 4415 - Player 1 is a monopolist supplier of oil

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
B
N P1 type
In
0, -1
2, 1 I
Out
2, 0
3, 0 1\2
B
N P1 type
In
3, -1
2, 1 II
Out
5, 0
3, 0 Q3.
Consider the following game between two players. Player 1 can be either type A or type B.
Player 2 can be either type X or type Y. The payoffs for each combination of types are given
below.
1A\2X
C
N C
1, 1
0, 7 N
1, 2
0, 8 1A\2Y
C
N C
2, 9
1, 9 N
3, 6
0, 9 1B\2X
C
N C
1, 2
2, 2 N
2, 3
3, 3 1B\2Y
C
N C
0, 7
5, 2 N
4, 7
5, 0 The probability nature chooses each combination of types is given in the following table:
1\2
A
B X Y
?
5
12 ? ? 1
4
1
3 5
where ? ? (0, 12
). a) Use Bayes’ Rule to find each of the following consistent beliefs (as functions of ? where
necessary):
i) p1 (X|A)
ii) p1 (X|B)
iii) p2 (A|X)
iv) p1 (A|Y )
b) Does any type have any strictly dominated actions? If so, list them.
c) Find all BNEs, including in mixed strategies. 2 Q4.
Consider the following game of incomplete information:
Player 2 can be one of two types, A or B. If player 2 is of type A, each of which results in a
different set of payoffs. The probability that player 2 is of type A is ?. The payoffs are:
P1 type B
1\2
C
N
C
1, 0 0, 3
N
0, 1 3, 0 P1 type A
1\2
C
N
C
1, 3 2, 0
N
0, 0 0, 1 In this game, player 2 will move first, after which player 1 will observe player 2’s action and then
choose her own action.
(a) Draw an extensive form for the game, including the relevant information sets, and player 1’s
beliefs. Note that ‘Nature’ first chooses whether player 2 is of type A or B.
(b) Find an example of a separating pure-strategy Perfect Bayesian Nash equilibrium and one
example of a pooling pure-strategy PBE. For each of these equilibria, remember to explicitly
state player 1’s updated beliefs, both on and off the equilibrium path (denote these µC for
the C signal and µN for the N signal, for uniformity). Show your working in full. Q5.
The following 3-player game is known as ‘Selten’s horse’:
B 1 2 A C (2, 0, 1) R F (3, 3, 7) E 3
L 1 D 3
L (4, 2, 2) (3, 4, 2) R L (2, 1, 1) (4, 2, 4) R
(5, 4, 3) (a) Draw the associated normal form of this game, and use it to find all pure strategy Nash
equilibria.
(b) For each of the NEs you have found, specify consistent beliefs for player 3, then determine
whether the NE and its beliefs constitute a PBE. 3 Q6.
The owner of a firm is deciding how much to pay an employee, but she cannot observe how much
effort he exerts. He can exert two levels of effort — high (eH ) or low (eL ). If he exerts high
effort, then the firm earns a revenue (R) of 200 with probability 0.8 and a revenue of 100 with
probability 0.2. If he exerts low effort, then the firm earns a revenue of 200 (good revenue) with
probability 0.3 and a ?
revenue of 100 (bad revenue) with probability 0.7. The employee’s utility
is given by U (w, e) = w ? d(e), where w is the wage paid to him by the owner, and d(e) is the
disutility of effort, which is 3 for high effort and 0 for low effort. The owner’s utility is given by
the firm’s profits, V (R, w) = R ? w.
(a) Suppose that the owner cannot condition the wage on observed revenue. Explain why she
will not be able to induce the employee to exert high effort.
(b) Now suppose that the owner can condition the employee’s wage on the revenue observed.
Suppose that she gives the employee a wage of zero if she observes bad revenue. What is
the smallest wage that she could give the employee if she observes good revenue in order to
induce him to exert high effort?
(c) Verify that the expected payoff to the owner from inducing high effort is higher than her
expected payoff from inducing low effort, so that the wage structure from part (b) can be
part of a Nash equilibrium where high effort is induced.
(d) Suppose that as in part (b), the owner can condition the employee’s wage on the revenue
observed, but minimum wage legislation forces her to pay a wage of at least 25, regardless
of the level of revenue. Now what is the smallest wage that she could give the employee if
she observes good revenue in order to induce him to exert high effort?
(e) Can the wage structure from part (d) be part of a Nash equilibrium where high effort is
induced? Explain.
(f) Find the largest minimum wage for which there exists a high-effort equilibrium. What wage
is given to the employee if good revenue is observed in this case? 4