Game Exercise
1.
P1
T1
S1
(2,2) (0,10)
S2
T2
PP
S1
T3
P2
S2
(10,0) (5,5)
S3
S4 S5
(6,7) (3,5) (2,11)
(1) Draw the normal form of the extensive form game above.
(2) Find the weakly dominant equilibrium of the game.
2. There are three students, Kim, Lee, and Park, living together at one apartment.
All three like apples and they always keep six apples in the refrigerator. Everyday
evening they check the number of apples and buy apples from E-mart to make it six
again. For example, if they ate four apples today, they will buy four more apples this
evening to make it six again. The apple price is 300 won and everybody pays 1/3,
regardless of whoever ate the apples. In other words, if only Kim ate one apple today,
everybody will pay 100 won to buy one more apple this evening. All three students
have the same utility function. They get the utility same as 400 won from the first
apple they eat. Then, they get the utility same as 200 won from the second apple they
eat. Let’s assume that they cannot eat more than two apples for one day.
* First analyze the case where all three make decisions at the same time without
knowing how many apples others ate. Then, consider the case where Kim makes
decision first and Lee second and Park third knowing how many apples others have
eaten before them.
(1)
(2)
(3)
(4)
Can you draw the extensive form of this game?
Can you draw the normal form of this game? (rather difficult)
What is the dominant equilibrium?
Is the dominant equilibrium Pareto Efficient?
3. Find the (iterated) dominant equilibrium and (mixed strategy) Nash equilibria in
the following games
Game 1
S1
3, 2
1, 1
T1
T2
S2
1, 1
2, 3
Game 2
T1
T2
T3
S1
3,5
2,4
5,3
S2
4,3
6,6
5,5
S3
6,4
4,3
2,1
4. Answer the following questions
Game 3
: For Kamila and Mila there are two narrow roads to go to school, Road 1 (R1) and
Road 2 (R2). If both choose the same road, the road will be crowded, and as a result,
it will take 20 minutes. On the other hand, if they choose different roads, it will take
only 10 minutes to go to school. The utility will be 0, if it takes 20 minutes, and the
utility will be 1, if it takes 10 minutes.
a) Draw this game in a strategic (or normal) form and in an extensive form.
b) Find the Nash equilibrium.
Game 4
Only Russia has enough missiles to destroy US and only US has enough missiles to
destroy Russia. If country A is destroyed by missiles while country B is okay, country
B’s payoff is 2 and country A’s payoff is -1. If both countries are destroyed by each
other, again the payoffs will be 0. If no country is destroyed, the payoffs will be 1 for
both countries. Assume that both countries have only today to shoot the missiles
because some strange electric power from the universe will destroy the missiles
tomorrow. Also, they do not know if the other country shot the missiles until they are
actually hit by the missiles. Once a country is hit by the missiles from the other side,
it is impossible to shoot back its missiles after that.
a) Draw the normal form and the extensive form of this game.
b) Find the dominant and all the Nash equilibria of this game.
Game 5
Think about a similar game as above (game 4). However, now we will assume that
US cannot shoot its missiles first. US can shoot its missiles only when the Russia
shoots first. We will also assume that US has a very good satellite and can detect it if
Russia shoots missiles to US. Even though US cannot stop the missiles, it can shoot
its missiles before the Russian missiles reach US so that both countries will be
destroyed. The payoffs are same as game 4.
a) Draw the normal form and the extensive form of this game.
b) Find the dominant and all the Nash equilibria of this game.
Game 6
Think about a similar game as game 5. The only difference is that now at 12 pm
Tokyo time tomorrow, both countries can shoot their missiles without knowing if the
other country shoots or not. If a country shoots the missiles at 12 pm, the missiles will
arrive the other country at 1pm. However, both countries can see if the other country has
shot the missiles or not at 12:30 pm through their own satellites and they can shoot their
missiles at that time. (If the other country did not shoot at 12 pm, they cannot shoot at
12:30. They can only choose to shoot at 12:30 pm, only if the other country has shot at 12
pm) Again, the countries cannot stop the missiles once they are launched.
a) Draw the normal and the extensive forms of this game.
b) Find all the Nash equilibria of this game.
5. Commitment problem and inflation
Consider a game between a labor union and the government. The labor union should
negotiate and decide the increase of their next year’s wage until December 31st. On the
other hand, the government can choose between two monetary policies, expansionary or
tight on January 1st of each year. An expansionary monetary policy will cause 10%
inflation, while a tight policy will cause 0% inflation. Naturally, the union wants to
increase its wage at least as much as the inflation, because the worker’s real income will
decrease otherwise. However, if the wage increase is higher than the inflation, the firms
may go bankrupt and the union does not want to do that.
On the other hand, the government has two objectives; a low inflation and a high
GDP. (As you must have learned in Macroeconomics course) A high GDP can be
achieved only when the government does expansionary policy while the wage does not
increase because the workers should work even though they are not willing to work at
that wage. Therefore, there are four possibilities. (a) 0% wage increase + expansionary
policy = 10% inflation + high GDP, (b) 0% wage increase + tight policy = 0% inflation +
low GDP, (c) 10% wage increase + expansionary policy = 10% inflation + low GDP, (d)
10% wage increase + tight policy = 0% inflation + very low GDP. Assume that the
government’s and the labor union’s payoffs from the result (a), (b), (c), and (d) are as
follows.
Government
Labor Union
(a)
40
0
(b)
30
10
(c)
20
10
(d)
10
0
(1) Please draw an extensive form and a normal (strategic) form of this game.
(2) Once the union decides on 0% increase of the wage, what will the government
do?
(3) What is the subgame perfect equilibrium of this game? (a), (b), (c), or (d)?
(4) If the government makes an announcement on December 20th whether it will take
an expansionary policy or a tight policy and has to follow the announcement by
law, what will be the game look like this time? Draw an extensive form and a
normal (strategic) form of this new game.
(5) In (4) what do you expects as a result of the game? (a), (b), (c), or (d)?
(6) Compare your answer to (3) and (5). Which is better for the government? What
was the problem in the other case?
6. Exchange rate policy and the trust between the Kazakhstan government and
the Kazakhstan people.
If there is no depreciation, it is better to have Tinge because there is a cost in exchanging
the currencies (e) and also one can get a higher interest rate from Tinge.
Also, if people save money and the government depreciate Tinge, the government gains
by G. However, if the people convert Tinge to Dollar the government will lose the
foreign reserve of R.
We will assume that R>G, and D-e>r.
Government
Depreciate
Don’t depreciate
People
Save in $
+G-R, +D-e
-R, -e
Save in Tinge
+G, + r
0, + r
(1) What will be the equilibrium in this case?
(2) If the government decides whether it will depreciate Tinge or not first and the
people decides after seeing what the government decides, what will the extensive
form of the game look like?
(3) In question (2), what will be the Subgame perfect equilibrium? Is it better for the
government than the equilibrium of question (1)?
7. Mixed strategy (Tax evasion)
Tax payers can evade the tax, and if there is no investigation, it cannot be found and they
will not be punished for it. However, if there is an investigation by the tax collector, it
will be found and punished. On the other hand, it will take some cost (C) to investigate. It
the tax evasion is found, the tax collector can charge a punishment fee (P) in addition to
the tax (T). We will assume that P>C.
The next table shows the payoffs for each case.
Tax collector
Investigate
Do not investigate
(1) Is there any pure strategy equilibrium?
Tax payer
Evade
T+P-C, -T-P
0, 0
Do not Evade
T-C, -T
T, -T
(2) Find the mixed strategy equilibrium of this game.
 Based on your answer to (2) answer the following
(3) If the punishment (P) becomes higher (or severe), will there be more
investigations by the tax collector? Will there be less tax evasions by the tax payer
in this case?
(4) If the investigation cost (C) becomes higher, will there be more investigations?
Will there be more tax evasions?
(5) If the tax (T) increases, what will happen to the investigation and the tax evasion?
8. Mixed strategy equilibria
S1
S2
S3
S1
1,1
0,0
0,0
S2
0,0
2,2
0,0
S3
0,0
0,0
3,3
(1) Find the three pure strategy equilibria of the game.
(2) Find the mixed strategy equilibrium where the players randomize between S1 and
S2.
(3) Find the mixed strategy equilibrium where the players randomize between S1 and
S3.
(4) Find the mixed strategy equilibrium where the players randomize between S2 and
S3.
(5) Find the mixed strategy equilibrium where the players randomize between S1, S2,
and S3.
9. Life time employment
For all the workers there are two periods. First, they are young and then they are old.
Workers are productive when they are young and they can make lots of money for the
company. On the other hand, they are not so productive when they are old and it will be a
burden for the company.
The payoff is as the following table. (Careful! It is not a normal form of the game)
Company
Worker
Young
10,5
0,10
Hire the worker
Fire the worker
Old
-5,10
0,0
The worker can do some business by himself when he is young and is not hired by the
company. However, if he is old he cannot work. On the other hand, if he can work when
he gets old, it is better for him to stay in the company because the life long payoff will be
15, while his life long payoff will be 10 if he begins his own business.
(1) Draw the extensive form of the game.
(2) If the company tells the worker, “Join our company when you are young. Then we
will not fire you when you are old,” will the worker believe that?
(3) What will be the subgame perfect equilibrium in this case?
(4) In that subgame perfect equilibrium, what will be the profit for the company?
What will be the payoff to the worker? Is it Pareto efficient?
(5) Now assume that they company wants to hire new your workers in every period.
Is there any reason the equilibrium may be different from that of (2) now?
10. Tom and Bill are playing a game. In this game Tom holds a certain number of
coins in his hand and Bill is trying to guess if the number of coins in Tom’s hand
is odd or even. If Bill is right, Tom should give Bill 300 yen. If Bill is wrong, Bill
has to pay Tom 500 yen. Tom and Bill pay this twice in this game.
(1) Draw the Extensive form of this game.
(2) How many pure strategies does Bill have in this game?
(3) Find out a Nash Equilibrium
11. The following game is played twice. Find out the SPE of the twice-repeated game.
Also find out a Nash equilibrium which is not SPE.
U
M
D
L
0,0
-6,-4
1,-3
M
-3,1
5,4
4,5
R
-4,-3
4,5
5,4
Game Theory Exercises for Final
Examination
Example 1: Once upon a time there was a queen named Boram(BR). Her neighboring
kingdom was governed by an evil king named Kangnyun(KN), and she had to fight KN
every year to defend her country. Obviously, she would not go to the battle field in
person. Instead she would send one of her generals. This time she would send a general
named Hyoduk(HD). HD wants to win the battle for his queen. The problem is that KN is
very good at fighting. As a result, if HD works hard (strategy H), the probability of
winning is 50%. If HD just does not work hard (strategy N), the probability to win is
40%. The queen can promote or demote HD. If promoted, HD’s utility will increase by
100. If demoted, it will decrease by 100. On the other hand, the queen will have her
utility increased by 4000 if win, and decreased by 2000 if lose. The battle’s result will be
either winning or losing and there is no tie.
On the other hand, HD has family problems. If HD chooses strategy H, his wife will run
away and his children will have problems at school. As a result, his personal cost will be
80. If he chooses strategy N, his wife will not run away but his children will have no
problems at school and his cost will be 0.
(1) Compare the queen’s utility when general HD works hard and not hard. How big
is the difference?
(2) If queen promotes HD when he works hard regardless of the result of the battle a
nd demote him when he does not work hard, which strategy will HD choose? W
hat will be the expected utility of the queen?
(3) Now assume that queen cannot tell if HD works hard or not. As a result, the only
thing that the queen can do is to promote HD when he wins the battle and demot
e him when he loses it. Which strategy will HD choose? What will be the queen’
s utility?
Example 2: Jeonghyun(JH) has just graduated university and went to Makinsey to look
for a job. JH can be either a smart person (type s) or a not so smart person (type n). The
people in Makinsey will produce $10,000,000 for their lives if they are successful, but
produce $0 if they are not successful.
If JH is type s, the probability that she will be successful 80%, while the probability will
be 40% is she is type n. Let’s assume that there are many consulting firms who are
looking for new people, and as a result, JH will get as much as her productivity once
Makinsey is sure about her type. For example, if JH can prove that she is type s,
Makinsey will pay her $8,000,000, but if JH is type n, she will be paid $4,000,000. Also,
assume that the salary should be decided as a fixed amount when JH enters the firm.
On the other hand, for JH entering Makinsey is not the only option. She can open her
own business. If JH is type s, she will get $6,500,000 from her own business. If she is
type n, she will get $3,000,000 from her own business.
The probability that JH is smart is 50%.
(1) Find out the Bayesian equilibrium of the game.
Now assume that professor Hahn opened a new useless course at Yonsei university. The
course’s name is ‘game theory 2’. The course is very difficult to pass but totally useless
in the sense that JH’s productivity will not increase from taking the course. Just one
interesting fact is that it is very difficult for JH to pass prof. Hahn’s course if she is type
n. Actually, if JH is type s, she will pass the course with a cost of $1,000,000. However,
if she is type n she has to suffer so much to pass the course that the cost will be as much
as $5,000,000.
(2) Find out the separating equilibrium of the game.
(3) Find out the pooling equilibrium of the game.
1. Consider a duopoly with a Cournot competition. The demand of the market is Q=
2-p. Both firm 1 and firm 2’s marginal costs can take two values. For firm 1 it ca
n be MC=5/4 with probability 1/3 and MC=3/4 with probability 2/3. For firm 2 it
can be MC=5/4 with probability 2/3 and MC=3/4 with probability 1/3. Each firm
knows its own MC but does not know the MC of the other firm. (But the probabil
ities are known to everyone.) What will be he Bayesian equilibrium of the game?
2. A boy and a girl play a game over a two period. At t=1 the boy asks the girl to go
out for a date and the girl may say “yes” or “no.” If the girl says “no,” the game e
nds there. If the girl says “yes,” they will go out for a date and the boy may be eit
her “nice” or “rude” at the date. At t=2 the boy will ask the girl to marry him and
the girl may say “yes” or “no.” If the girl says “yes,” they will get married. After t
he marriage thy boy may act “nice” or “rude.”
Here, the boy may have two types, “good” and “bad.” Later the payoffs will
make it clear that the good-type boy will choose to act “nice” all the time. On the
other hand, bad-type boy prefers to be “rude” both at the date and in the marriage
life. However, the bad-type boy loves the idea of marriage and may pretend to be
a good-type by acting “nice” at the date. The probability for the boy to be a goodtype is 2/3.
The followings are the payoffs of the players at t=1.
t=1
Girl
Boy (goodtype)
Girl says “no”
0
0
Girl says “yes” and the boy acts “nice” at 10
10
the date
Girl says “yes” and the boy acts “rude” at -10
5
the date
Boy (badtype)
0
-20
20
The followings are the payoffs of the players at t=2.
t=2
Girl
Boy (goodtype)
Girl says “no”
0
0
Girl says “yes” and the boy acts “nice” at 30
30
the marriage
Girl says “yes” and the boy acts “rude” at -100
10
the marriage
Boy (badtype)
0
-100
80
Find out the Perfect Bayesian Equilibrium of the game.
3. (Adverse Selection) KY opened a car insurance company. In this world there are t
wo types of drivers, safe drivers and risky drivers. If the driver is a safe driver, K
Y will get $10 per day for the insurance. On the other hand, if the driver is a risky
driver, KY will charge $20 per day. The problem is that KY cannot tell who is a
safe driver and who is a risky driver.
One day a customer called YR came
to HJ and applied for the car insurance. If YR is a safe driver, he is willing to pay
up to $12. If he is a risky driver, he will be willing to pay up to $24. YR will not
buy the insurance if KY charges more than those amounts. Let’s assume that the
probability that YR is a risky driver is 1/2. First, KY will tell YR how much she
will charge for the insurance and YR can accept the price or reject and leave.
(1) Draw the extensive form of the game.
(2) Find out the Bayesian equilibrium of the game.
(3) Is the Bayesian equilibrium socially optimal?
1. (professor – TA game) Professor Hahn can give a TA scholarship to Choi for max
imum 2 years. At the beginning of each year professor Hahn decides whether he
will give a scholarship to Choi or not. Choi can get a scholarship in t=2, only if sh
e gets it in t=1. Basically, Professor Hahn and Choi will play the following game
twice.
Prof. Hahn
No scholarship
Give scholarship
TA Choi
(0, 0)
Work
Don’t work
(2, X)
(-3, 2)
Choi can be a Hard working type with probably 0.5 and can be a Lazy type with
probability 0.5. Professor Hahn does not know Choi’s type.
If Choi is hard working, it will be X=5 and Choi will always work if she gets a
scholarship.
If Choi is lazy, it will be X=1. There is no discount for t=2.
Find out the (Perfect Bayesian) Equilibrium of the game.
Additional exercises for Game Theory
1. There are two effort level for the agent and they are
. For the agent the cost
for high effort is 500 and the cost for low effort is 0. The reservation utility for
the agent is 500. When the agent is working for the principal, there can be three
levels of profit and they are
. The
principal is risk neutral and the agent is risk averse. The agent’s utility from the
wage, w, would be
. The agent’s total utility would be
where
. When the effort level is high the probabilities that each result would
happen are
. When the effort level is low they are
.
(1) In the ideal case where the principal can observe the effort level of the agent,
what will be the wage system like? In other words, what will be
?
(2) If the principal cannot observe the effort level of the agent, what will be
? And what will be the effort level of the agent in that case? What
will be the profit of the principal?
2. Eric is an owner of a big firm and James applied for that firm. If James is smart,
he will make $10 for Eric and Eric will pay him $7 (so that Eric’s profit from
James will be $3). If James is not-smart, he will make $4 for Eric and Eric will
pay him $3 (so that Eric’s profit from James will be $1). The probability that
James is smart is 1/2. Actually, Eric does not know if James is smart or not and
he will pay the average wage ($7+$3)/2=$5 and the expected profit for Eric
would be $7-$5=$2. Then, professor Hahn opened a “Useless University (=ULU)”
and he gives a degree only to the students who pass his difficult examination. In
order to pass the examination, the students should take professor Hahn’s class
and the tuition is $1 per month. If the student is smart, it takes only 1 month for
him to pass the exam. On the other hand, if the student is not-smart, it takes 5
months to pass the exam. We will assume that Eric can check if James passes the
exam or not, but Eric cannot check how many months James took professor
Hahn’s class. Find out two Perfect Bayesian equilibria of this game, one
separating and one pooling.
3. Two players play the following game for infinite times.
Cooperate
Betray
Cooperate
10, 20
-25, 30
Betray
15, -23
-12, -19
For the player to continue to cooperate what would be the ranges of their discount factor,
, respectively?
1. What is Nash Equilibrium in the following game?
S1
S2
S3
T1
3,5
4,3
6,4
T2
2,4
6,6
4,3
T3
5,3
5,5
2,1
2. Consider a new Rock-Scissors-Paper game. If you win with a Rock you get 1. If
you win with a Scissor you get 2. If you win with a Paper you get 3. This means
that if you lose with a Scissor you get -1, if you lose with a Paper you get -2,
and if you lose with a Rock you get -3. If it is a tie, both players get 0. What is
the Nash Equilibrium of this game?
3. Sarah and Mike are negotiating over a Pizza. The negotiation happens at 12:30,
12:40, and 12:50. At 12:30 and 12:50 Sarah makes an offer and Mike can either
accept it or reject it. At 12:40 Mike makes an offer and Sarah can either accept it
or reject it. If the offer is accepted, the negotiation will stop and the pizza will be
divided as the accepted offer. Assume that the value of the pizza is decreasing as
time passes and the values are 100 at 12:30, 75 at 12:40. And 30 at 12:50. If they
cannot make an agreement by 12:50, both will get 0.
(1) Find out the Subgame Perfect Equilibrium of this game.
(2) Find out a Nash equilibrium that is not a Subgame Perfect Equilibrium of this
game.
4. Victor and Chris are playing a game. Both are supposed to call a number
simultaneously and the number should be bigger than or equal to 0. Assume that
Victor’s number is x and Chris’s number is y. If x>y, Chris wins and Victor
should pay Chris $y. If y>x, Victor wins and Chris should pay Victor $x. If x=y,
both gets $0. Find out a Nash Equilibrium of this game.