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Universidad Carlos III de Madrid
GAME THEORY
Problem set on
repeated games and Bayesian games
- Consider the following game in the normal form:
Player 2
C N P
Player 1
C 6, 6 0, 7 0, 0
N 7, 0 3, 3 0, 0
P 0, 0 0, 0 1, 1
a) Find all the pure strategy Nash equilibria.
Suppose now that the game from part a is played twice. Before playing the second time,
the players observe what happened in the first time the game was played, so they can
make their strategies for the second round contingent to what occurred in the first round.
The final payoffs are the sum of the payoffs from each round.
b) Find a subgame perfect equilibrium where the players play (C,C) in the first
round.
- In the telecommunications market of a country there are two Firms that face a market
demand of
P ( q
a
+ q
b
) = 160 - q
a
b
The two firms have the same costs of production C(q) = 40q.
a) Find the Nash equilibria in this game when the two firms simultaneously select
their quantities of production just one time.
b) Find a discount rate and a subgame perfect Nash equilibrium such that the firms
collude (that they obtain the maximum possible joint benefit) if the game is
repeated infinitely.
- In the following game, the government has to decide between high taxes T or low
taxes t while the householders choose between keeping high savings S or low savings s.
The payoff matrix is given by:
S s
T 6, 6 3, 3
t 8, 3 4, 4
a) If the taxes policy and the saving decisions are taken once a year, and the
government term is four years, what is the subgame perfect Nash equilibrium of
this game?
b) If the government and the householders live forever, what is the minimum
discount factor such that the payoffs (6,6) may result from a subgame perfect
Nash equilibrium? Give a strategy profile that supports this equilibrium.
- There are two firms who are repeating infinitely the game of Bertrand duopoly. In
each period the firms set their prices simultaneously; the demand for the product of
Firm i is 𝑎 − 𝑝
𝑖
if 𝑝
𝑖
𝑗
, is 0 if 𝑝
𝑖
𝑗
and is (𝑎 − 𝑝
𝑖
)/ 2 if 𝑝
𝑖
𝑗
; their marginal
costs are 𝑐 < 𝑎. Show that if the discount rate is 𝛿 ≥ 1 ⁄ 2 then there is a subgame
perfect equilibrium in trigger strategies that allows the monopoly price to be
maintained.
- Calculate all of the pure strategy Bayesian Nash equilibria of the following static
Bayesian game. It is determined randomly whether the payoffs of the players, A and B ,
are those of game 1 or of game 2, with the probability of each being equal. Player A is
informed which of the games, 1 or 2, was chosen, but Player B does not know which of
the games is being played. Player A chooses x or y ; simultaneously, Player B chooses m
or n.
Game 1 Game 2
m n m n
x 1, 1 0, 0 x 0, 0 0, 0
y 0, 0 0, 0 y 0, 0 2, 2
- There are two suspects who face the problem of the prisoners’ dilemma, with the
added complication that neither suspect knows if the other is a man of honor. Say it is
known with certainty that Suspect 1 is not a man of honor, but it is not clear whether or
not Suspect 2 is also. If Suspect 2 is not a man of honor, the payoffs have their usual
form in this game:
Suspect 2
Confess Not to Confess
Suspect 1 Confess 1, 1 15, 0
Not to Confess 0, 15 10, 10
On the other hand, if Suspect 2 is a man on honor, then he prefers to spend years in jail
before he would rat on his colleague. Moreover, even Suspect 1 would feel bad
betraying someone so honorable. For these reasons, if Suspect 2 is a man of honor the
payoffs are:
Suspect 2
Confess Not to Confess
Suspect 1 Confess 1, 1 5, 20
Not to Confess 0, 15 10, 30
Denote the probability that Suspect 2 is a man of honor by p.
a) Identify the strategies that are strictly dominant for Suspect 2 in this game of
imperfect information.
b) Identify the Nash equilibria in this game for each p.
- A firm (Player 1) is already established in a market and must whether or not to
construct a new factory. The potential benefits of this action depend on whether another
firm (Player 2) enters or does not enter the market. Player 2 is uncertain of the cost
faced by Player 1 of constructing the factory, which Player 2 believes may be high or
Other Exercises
- Consider the following game in the normal form:
A B C
A 3, 3 x , 0 - 1, 0
B 0, x 4, 4 - 1, 0
C 0, 0 0, 0 1, 1
a) Find the Nash equilibria in pure strategies for the different values of x.
Suppose now that the game is played twice. After playing the first time, the players
observe what happened before playing the second time. The final payoffs are the sum of
the payoffs from each round.
b) For x = 5 is ( B , B ) a Nash equilibrium for the game when it is played just once?
Is there a subgame perfect Nash equilibrium where ( B , B ) is played in the first
round? If yes, write the strategies for both players for such equilibrium. If no,
explain why not, using the definition of subgame perfect Nash equilibrium.
c) For x = 7 is there a subgame perfect Nash equilibrium where ( B , B ) is played in
the first round? If yes, write the strategies for both players for such equilibrium.
If no, explain why no.
- Given the following Normal form game:
L R
A 1, 1 8, 0
B 0, 5 3, 3
Find the smallest discount factor necessary to get average payoffs equal to (3,3) in a
SPNE if the game is repeated an infinite number of times. Describe the strategies that
allows us to sustain such an equilibrium
- Carlos decided to have lunch at the restaurant Casa Pepe. The waiter of the
restaurant has to decide whether to attend Carlos with a good service or with a bad
service. Carlos, after observing the service received from the waiter, decides if he tips
the waiter or not. The waiter likes to receive a tip, but he has a cost to provide a good
service. Carlos, on the other hand, likes to receive a good service, although he does not
like to tip waiters. Each one of them wants to maximize his own expected value.
Suppose that the only possible tips are 2 or 0 euros. For Carlos, a good service is worth
6, while a bad service is worth nothing. For the waiter, a good service costs 1, while a
bad service has no cost.
a) Draw the extensive form of this game.
b) Which are the pure strategies for Carlos?
c) Is there a Nash equilibrium where Carlos pays the tip only if he receives a good
service, and the waiter gives a good service? Explain.
d) Represent this game in the normal form.
e) Which of the following phrases is correct?:
i) For the waiter, to give a good service is a dominant strategy.
ii) Never give a tip, no matter the quality of the service, is a dominant strategy
for Carlos.
iii) For Carlos, it is better to give a tip if the service is good, and do not if it is
bad.
iv) None of the above is correct.
f) Find the pure strategy Nash equilibrium.
g) Does this equilibrium (from part f) result in a Pareto efficient allocation? If not,
indicate a strategy profile that results in a Pareto superior allocation.
Suppose now that Carlos goes to this restaurant every week, and he is always served by
the same waiter. Each one maximizes expected value, and no one discounts the future
(i.e., one euro today is worth exactly the same as one euro in the future). In that case,
they could reach the following verbal agreement: the waiter starts providing a good
service, and will keep doing so in the future if he always receives a tip. If in some week
Carlos does not give a tip, the waiter will never give a good service again. And Carlos
will always give tips, as long as he receives a good service. If the waiter fails once, and
give him a bad service, Carlos will stop tipping the waiter forever. If everybody knows
that the waiter is leaving the restaurant the first day of next month, will them follow the
agreement? Find the subgame perfect Nash equilibrium.
- Calculate all of the Bayesian Nash equilibria in pure of the following static Bayesian
game:
- Player 1 can choose between two actions A and B. Player 2 can choose between
two actions, I and D.
- The payoffs depend on the players types. Player 1 has just one type and this is
known by Player 2.
- Player 2 can be of either type x or y. Player 2 knows her own type but Player 1
does not know for certain the type of Player 2.
- Player 1 thinks that Player 2 is Type x with probability 2/3, and Type y with
probability 1/3.
- The payoffs are those that are given in the game that is randomly determined.
Type x Type y
I D I D
A 4, 1 3, 3 A 3, 6 1, 3
B 3, 6 2, 3 B 1, 1 5, 3
- The game of chicken will be familiar to those who have seen West Side Story or
Rebel without a cause. One simplified version of this game is as follows. Two players
drive there cars at each other at full speed. They can take one of two possible actions,
swerve aside or continue straight on. The payoffs are those following.
James Dean
Continue Swerve
Bad guy
Continue - 3, - 3 2 , 0
Swerve 0, 2 1, 1
