PROBLEM SET 4
1. Consider a repeated game in which the stage game in the following figure is played in each of
two periods and there is no discounting.
L
8,8
9,0
0,0
U
C
D
M
0,9
0,0
1,3
R
0,0
3,1
3,3
Fully describe a subgame perfect equilibrium in which the players select (U,L) in the first period.
2. Consider the following game. Nature selects A with probability ½ and B with probability ½. If
Nature selects A, then players 1 and 2 interact according to matrix “A.” If nature selects B, then
the players interact according to matrix “B.”
MATRIX A
X
Y
Z
V
6,0
0,0
5,1
W
4,1
0,1
3,0
MATRIX B
X
Y
Z
V
0,0
6,0
5,1
W
0,1
4,1
3,0
a) Suppose that, when the players choose their actions, the players do not know which
matrix they are playing. That is, they think that with probability ½ the payoffs are as in
matrix A and with probability ½ the payoffs are as in matrix B. Write the normal-form
matrix that describes this Bayesian game. What is the strategy profile that it is played?
b) Now suppose that, before the players select their actions, player 1 observes nature’s
choice, but player 2 does not. What is the Bayesian Nash Equilibrium of this game now?
3. Consider the following stage game:
A
B
X
5,6
8,2
Y
0,0
2,2
a) Find all Nash equilibria of this game;
b) Consider the two-period repeated game in which this stage game is played twice and the
repeated game payoffs are simply the sum of the payoffs in each of the two periods. Is
there a subgame perfect equilibrium of this repeated game in which (A,X) is played in the
first period? If so, fully describe the equilibrium. If not, explain why.
4. Consider the following game of incomplete information. Nature selects the type (c) of player
1, where c=2 with probability 2/3 and c=0 with probability 1/3. Player 1 observes c, but player 2
does not. Then the players make simultaneous and independent choices and receive payoffs as
described by the following matrix.
A
B
X
0,1
1,0
Y
1,0
c,1
a) Draw the normal-form matrix of this game;
b) Compute the Bayesian Nash equilibrium.
5. Consider the following extensive-form game of incomplete information. There is a firm and a
worker. In this game, Nature first chooses the type of the firm (player 1). With probability p, the
firm is of high quality (H) and, with probability 1-p, the firm is of low quality (L). The firm
chooses either to offer a job to the worker (O) or not to offer a job (N). If no job is offered, the
game ends and both parties receive 0. If the firm offers a job, then the worker either accepts (A)
or rejects (R) the offer. The worker’s effort on the job brings the firm a profit of 2. If the worker
rejects an offer of employment, then the firm gets a payoff of -1 (associated with being jilted).
Rejecting an offer yields a payoff of 0 to the worker. Accepting an offer yields the worker a
payoff of 2 if the firm is of high quality and -1 if the firm is of low quality. The worker does not
observe the quality of the firm directly.
a) Is there a separating PBNE in this game? If so, specify the equilibrium under what conditions
it exists. If not, briefly demonstrate why.
b) Is there a pooling PBNE in which both types of firms offer a job? If so, specify the
equilibrium and explain under what conditions it exists. If not, briefly demonstrate why.
c) Is there a pooling PBNE in which neither type of firm offers a job? If so, specify the
equilibrium and explain under what conditions it exists. If not, briefly demonstrate why.
d) Is there a semi-separating PBNE? If so, specify the equilibrium and explain under what
conditions it exists. If not, briefly demonstrate why.
6. Jack and Jill play a game. First, each flips a coin. After seeing their own coins (but not each
others’), each player (separately) says either “Red” or “Black”. If they name opposite colors,
then the Black-sayer gets $4 and the Red-sayer gets nothing. If both say Black, then they both
get either $5 (if both flipped heads) or $10 (otherwise). If they both say Red, then they both get
either nothing (if both flipped heads) or $20 (otherwise). Assume both players play optimally. If
Jack flips heads, what is the probability that he says “Black”? What if Jack flips tails?
7. There are two players, a plaintiff and a defendant in a civil suit. The plaintiff knows whether
or not he will win the case if it goes to trial, but the defendant does not have this information.
The defendant knows the plaintiff knows who would win, and the defendant has prior beliefs of
1/3 that the plaintiff will win; these prior beliefs are common knowledge. If the plaintiff wins, his
payoff is 3 and the defendant’s is -4; if the plaintiff loses, his payoff is -1 and the defendant’s is
0. The plaintiff can ask for a settlement of m1 = 1 or m2 = 2. If the defendant accepts a
settlement offer of m, the plaintiff’s payoff is m, and the defendant’s is −m. If the defendant
rejects the settlement offer, the case goes to court. List all perfect Bayesian equilibria strategy
profiles.
8. There are two types of player 1, real man and wimp, which Nature chooses with probability
0.9 and 0.1, respectively. Both players know this probability but only player 1 observes Nature’s
move. Player 1 is sitting in a bar when player 2 walks in. Player 2 is the rowdy type and wants to
pick a fight, but he’s also a coward, so he only wants to fight a wimp. Player 2 has two actions,
fight (F) and not fight (N) but before he makes the choice, he observes player 1’s behavior.
Player 1 can choose to drink beer (B) or eat quiche (Q) and obtains one unit of utility if he
consumes his most preferred breakfast. Real men prefer beer and wimps prefer quiche. Player 2’s
payoff does not depend on player 1’s breakfast and is 1 if it fights the wimp or if he avoids
fighting the real man, and zero otherwise. Find all perfect Bayesian equilibria of this game.
9. Analyze the evolutionary dynamics for the following two-species games:
X
Y
X
6,6
8,1
Y
1,8
4,4
X
Y
X
5,3
1,2
Y
0,1
4,3
X
Y
X
4,1
0,5
Y
2,3
5,0
10. Analyze the evolutionary dynamics for the following one-species, three-phenotype game:
X
Y
Z
X
2,2
1,1
0,2
Y
1,1
4,4
3,2
Z
2,0
2,3
6,6