Recap
Economics 409
Spring
2016
Centipede
Game
Prof. Haile
Problem Set 4
Due March 1
Imperfect-Information Extensive-Form Games
Perfect Recall
Centipede Game
1. Consider the extensive form game between players 1 and 2 below. At each decision
node, one player can take the action “Agree” or “Disagree.” If this player Agrees, the
game continues and the potential payo¤s to both players grow. If a player Disagrees, the
game ends, with payo¤s as shown in the extensive form. Find the unique subgame perfect
equilibrium of this game.5 Reasoning and Computing with the Extensive Form
1q A
2q A
1q A
2q A
1q A
D
D
D
D
D
(1,0)
(0,2)
(3,1)
(2,4)
(4,3)
(3,5)
Figure 5.9 The centipede game
2. Suppose two players play the following prisoner’s dilemma stage game in every period
Play
a fun
t = this
1 : : : Tas
. You
cangame...
think of this as two …rms choosing either a High price or Low price.
Player 2
H
L
Player 1 L (11,0)
(1,1)
H (10,10)
(0,15)
a. If T is …nite, what is the unique subgame perfect equilibrium? Why?
b. If T is in…nite and players discount the future using discount factor , is the equilibrium
in part a still a subgame perfect equilibrium? Explain.
c. If T is in…nite, what is the smallest value of such that each player can obtain a payo¤
of 10 each period in a subgame perfect equilibrium?
d. Now suppose that the stage game is played in period 1 and that, conditional on the
Extensive Form Games:
Induction in
andaImperfect
Information
Gamesis probability 1 p that
Lecture
10, Slidegame
9 is played
gameBackward
being played
given period
t, there
the stage
in period t + 1 and probability p that the repeated game ends after period t: For simplicity,
assume now that …rms do not discount future payments (i.e., that = 0). Show that for p
su¢ ciently small, each player can obtain the payo¤ 10 in a subgame perfect equilibrium.
place. In other words, you have reached a state to which your analysis has given a
probability of zero. How should you amend your beliefs and course of action based
on this measure-zero event? It turns out this seemingly small inconvenience actually
aises a fundamental problem in game theory. We will not develop the subject further
here, but let us only mention that there exist different accounts of this situation, and
hey depend on the probabilistic assumptions made, on what is common knowledge (in
particular, whether there is common knowledge of rationality), and on exactly how one
evises one’s beliefs in the face of measure zero events. The last question is intimately
elated to the subject of belief revision discussed in Chapter 2.
Imperfect-information extensive-form games
1
Up to this point, in our discussion of extensive-form games we have allowed players to
3. Now consider a richer speci…cation of the same game in which …rms can choose any
quantity, not just “high”or “low.”Suppose there are two identical …rms with cost functions
C(q) = 2q in an industry with inverse demand given by the equation P = 50 4Q; where Q
is the market quantity.
a. Suppose …rms compete only once by choosing quantities simultaneously. What is the
Nash equilibrium?
b. If both …rms produce the same quantity qi ; which value of qi maximizes joint pro…ts?
c. Now suppose that …rms choose quantities once a year and discount the future at
an annual rate of 5%. Can joint pro…t-maximization (as derived above) be achieved in a
subgame perfect equilibrium?
4. Suppose an industry consists of 2 …rms that compete by choosing quantities in each period
t = 1; 2; : : : . Inverse demand in the industry is given by the linear equation P = 50 Q2 :
Marginal cost is equal to zero.
a. Suppose all …rms discount future pro…t using the per-period discount factor . How
large must be for there to be a subgame perfect equilibrium in which the industry makes
the same pro…t a monopolist would?
b. Suppose now that …rms set quantities only in odd periods (t = 1; 3; 5; : : : ): How large
must be for industry to make the pro…t would if there were a monopolist? Provide some
intuition for the di¤erences from your answers in part a.
2