GAME THEORY

“Game” Situation: The game theory is designed to analyze the situation where
people know (1) that what they get is decided not only by his or her own action but
also by other’s actions and (2) that what their action affects what others get as well as
what they get.

Players, Strategy, and payoff
a. players : each decision makers of the game
b. strategy : the plan of actions set by the players
c. payoff : what the players get as the result of the game

Cooperative and Non-cooperative
a. Cooperative Game: A game is played by groups and the group members can
make binding agreement among them.
b. Non-cooperative Game: A game is played by individuals and binding
agreement is not possible among the players.
- From non-cooperative game’s point of view, players cooperate only when it is
better for them individually.
- From now on we will look at only this Non-cooperative cases.

Strategic(Normal) form and Extensive form
 Two ways to describe a game
a. Strategic form : table
(player 1)
Strategy 1
Strategy 2
(player 2)
strategy 1
2,2
10,0
b. Extensive form : tree
strategy 2
0,10
5,5
P1
S1
S2
P2
S1
(2,2)

S2
(0,10)
S1
(10,0)
S2
(5,5)
Equilibrium: Game theory wants to predict the result of the game, in other words,
which strategies the players will choose to play in the game. We call the result that
game theory predicts as “equilibrium of the game”. There have been many
suggestions how to find the equilibrium of a game, and we will see some of the
famous concepts of the equilibrium suggested until now.
1. Equilibrium by Elimination the (strictly) Dominated strategy

Dominant strategy : the strategy that gives the best payoff, whatever the others do.

Dominated strategy : when there is a strategy that always gives a better payoff than a
strategy S, whatever other players do, we call the strategy S as a dominated strategies
 strictly dominant & weakly dominant

One does not have to consider a strictly dominated strategy because the player will
never play that strategy.
S1
2,3
3,1
S1
S2
S2
4,2
1,0
 iterated deletion of dominated strategy
- Once a dominated strategy is eliminated, we have a new game and may find
a new dominated strategy in that new game. This process can go over and over
until there is no more dominated strategy. We call this process as the iterated
deletion of dominated strategy
S1
2,3
3,2
2,2
S1
S2
S3
S2
4,2
1,1
0,0
S3
3,4
4,0
5,1

Equilibrium by (iterated) elimination of the (strictly) Dominated strategy

Conclusion and Problem
(1) If there exists an equilibrium by (iterated) elimination of the (strictly)
dominated strategy, it is very nice. – because it will be very hard for anybody
to disagree with that prediction.
(2) Problem: There are many games where such a dominant equilibrium does not
exist. The following situation where cars drive either on the Right side or on
the Left side is a good example.
Left
Right
Left
1,1
-1,-1
Right
-1,-1
1,1


Example 1
S1
S2
S1
2,3
4,2
S2
3,1
1,2
Example 2 (weakly dominated) – different result depending on the order of
elimination
S1
2,2
1,4
2,1
S1
S2
S3
S2
2,5
3,2
3,1
2. Nash Equilibrium

Consider a case where player 1 has strategies (s11, s12, s13, s14,…) and player 2 has
strategies (s21, s22, s23, s24,…).
Assume that if player 2 chooses strategy s23, the best strategy for player 1 is s12. Also
assume that if player 1 chooses strategy s12, the best strategy for player 2 is s23.
In this case we say that (s12, s23) is a Nash equilibrium.

In other words, at a Nash equilibrium players cannot increase their payoff by choosing
strategies other than what they are supposed to choose in the Nash equilibrium as long
as others follow the Nash equilibrium.

Example 3: Find a Nash equilibrium in the following game.
S1
S2
S3
T1
1,9
8,4
9,1
T2
4,4
5,5
3,4
T3
6,2
0,1
1,7

Example 4 (Multiple equilibria): Gao and Yanfei planed to meet at Shinjuku Station
at 12pm. However, they forgot to decide where in the station they would meet. There
are three possible places, South Exit, West Exit, and East Exit. If they meet, their
utilities will be 1, and if they do not meet their utilities will be 0. Draw the normal
form of this game and find the Nash equilibrium.

Example 5 (Mixed strategy): America sends spies to China every month. There are
two ways for America to send spies to China. One is landing near Shanghai and the
other is landing near Tianjin. China’s only chance to prevent it is to make the police
ships patrol the sea. However, China does not have enough ships to patrol both
Shanghai and Tianjin at the same time.
If American spies go to Shanghai when Chinese police ships are in Shanghai, China
will catch the spies. However, if American spies go to Shanghai when Chinese police
ships are in Tianjin, American spies will land successfully.
(America)
Shanghai
Tianjin
(China)
Shanghai
-10,10
10,-10
Tianjin
10,-10
-10,10
What is the best strategy for America and what is the best strategy for China?

Mixed strategy: Instead of choosing one strategy the players may randomize between
many strategies.

Mixed strategy equilibrium: Nash equilibrium that includes mixed strategies.
Example – Rock, Scissors, Paper.

Exercise: Find a mixed strategy equilibrium in the following games.
(1)
S11
S12
S21
2,2
1,1
S22
0,3
3,0
(2)
R
S
P

R
0,0
-1,1
1,-1
S
1,-1
0,0
-1,1
P
-1,1
1,-1
0,0
Properties of the Nash equilibrium
(1) If we allow the mixed strategy, a Nash equilibrium always exists in any games.
(2) There may be many Nash equilibria. – It is problem because it will be difficult to
predict which equilibrium will be the result.
(3) It is assumed that each player knows other player’s strategy. – Not realistic all the
time.
(4) Some Nash equilibrium does not make sense.
(5) If a dominant equilibrium, it is the only Nash equilibrium of the game. (However,
it is not the case for weakly dominant equilibrium.)

Example 6: Ahad has a daughter and she wants a new bicycle, even though she
already has one. Of course, Ahad (who is a very strict dad) told his daughter that she
does not need a new bike. The daughter said that she would not eat anything and die
by starvation if her father does not buy her a new bike. Of course, Ahad does not want
to see his daughter die. Ahad and his daughter’s payoffs are as the following table.
(Ahad)
Buy Bike
No Bike
(Daughter)
Live
-1,1
0,0
Die
-100,-10000
-99,-10000
(1) Show the normal form of this game.
(2) Draw an extensive form of this game
(3) Find Nash equilibria of this game
(4) Do all the Nash equilibria make sense to you?

Empty Threat
2. Subgame Perfect Equilibrium

Subgame : In an extensive form game, any part of the game that begins with one point
is a subgame.

Subgame perfect equilibrium : A Nash equilibrium of the whole game such that
even when we look at only a subgame of the whole game, it is still a Nash equilibrium
for the subgame.

Property : By using subgame perfect we can avoid Nash equilibria with “empty
threat”.

Example 7: Find a subgame perfect equilibrium in the following game.
P1
Sa
Sb
P1
(2,2)
S1
S2
P2
S1
(3,0)
S2
(0,3)
S1
(0,3)
S2
(3,0)

Example 8: Centipede Game
player1
player2
player1
player2
player1
(5,5)
(1,0)
(0,1)
(3,0)
(2,4)
(6,3)
4. Repeated Game

Example 9: Consider the following game
C
D
C
10,10
20,-5
D
-5,20
0,0
a. What is the Nash equilibrium?
b. If this game is repeated twice, is it possible to achieve (C,C) by Subgame
perfect equilibrium?
c. Will the answer to b. be different if the game is repeated forever?

Example 10:
There is one restaurant. The owner of the restaurant can either use Korean oil
(Haepyo) or low quality Chinese oil. The customer can either come to the restaurant or
not.
(Restaurant)
(Customer)
Come
Not come
Haepyo
10,10
0,-5
Chinese
-5,20
0,0
a. What is the Nash equilibrium of this one-shot game?
b. Can the payoff (10,10) be achieved by SPE, if this game is repeated infinitely?
 Consider this: Why does it seem that crimes happen more often in big cities compared
with small towns?

Example 11: Consider the following game
There are two boxing players, Ali and Muhamad. Ali is about to attack Muhamad and
Muhamad is about to defend it. Assume that Ali’s right punch is much stronger than
his left punch.
(Ali)
(Muhamad)
Right
Left
Right
10,0
-20,20
a. Find the Nash equilibrium of the game.
b. Which punch will Ali use more frequently?

The relationship between different equilibrium concepts
Left
-10,10
10,0
- For any game we can find one or more Nash and Subgame perfect equilibria. However,
there is no guarantee that we can find an equilibrium by elimination of the dominated
strategies.
5. Bargaining Game
- In a separate handouts
6. Principal and Agent Game
- In a separate handouts.
Games with Imperfect or Incomplete Information
O Imperfect information – a player does not know what the other players have done
beforehand.
O Incomplete information – a player does not know other players precise characteristics
(preferences, strategy spaces)
 This distinction may not be as important as it looks, because a game with incompl
ete information may be transformed into one with imperfect information by introd
ucing a new player, “nature”, who chooses the characteristic or the type of each pl
ayer.
Example 1: Assume that people’s personalities totally depend on their blood types. BJ is
not sure if DY’s type is A or B. On the other hand, DY knows that BJ’s type is O.
If DY is of type A, the payoffs are as follows.
BJ
U
D
DY(A)
L
3,1
0,1
R
2,0
4,0
DY(B)
L
3,0
0,0
R
2,1
4,1
If DY is of type B, the payoffs are as follows.
BJ
U
D
Assume that the probability for DY to be type A is 30% and to be type B is 70%. What
will be the result of the game?
O Type contingent strategy – si(ti)
O Bayesian Equilibrium – a set of type contingent strategies such that each player
maximizes her expected utility contingent on her type and other’s type contingent
strategies as given.
Example 2: Consider a duopoly with a Cournot competition. The demand of the market is
Q=2-p, and firm 1’s marginal cost is constant at MC=1 with no fixed cost. On the other
hand, firm 2’s MC can take two values. It can be MC=5/4 with probability 1/2 and
MC=3/4 with probability 1/2. Firm 2 knows its MC but firm 1 does not know firm 2’s
MC (but it knows the probability). What will be he Bayesian equilibrium of the game?
Example 3: Consider a case where two people bid simultaneously for an objective. The
highest bidder gets the object and pays his bid. We will denote the amount that player i
bids as bi. We will also assume that the two players have their own value of the object
and will denote the value of player i as ti. Each player knows her own ti but do not know
the other player’s ti. However, ti’s are random variables that are independently and
uniformly distributed on [0,1]. What will be the Bayesian equilibrium in this case? Please
assume that the player’s bid is strictly increasing as the player’s value increases and also
this bidding function b(ti) is differentiable.
Example 4: There are two firms, firm 1 and firm 2. Firm 1 is currently making products
but firm 2 has not entered the industry yet. When firm 2 enters the market, firm 1 can
fight firm 2 by building a new plant. Firm 1’s cost of building a new plant can be either
zero or three. While firm 1 knows its cost of building, firm 2 does not. The payoffs are as
the following tables.
If firm 1’s cost is high
Firm 1
Build
Don’t
Firm 2
Enter
0,-1
2,1
Don’t
2,0
3,0
Build
Don’t
Firm 2
Enter
3,-1
2,1
Don’t
5,0
3,0
If firm 1’s cost is low
Firm 1
If the probability that firm 1’s cost is high is “p”, what will be the Bayesian equilibrium
of this game?
Example 5: This is the same situation as the example 4 except for that the payoffs are a
little bit different when firm 1’s cost is low.
If firm 1’s cost is high
Firm 1
Build
Don’t
Firm 2
Enter
0,-1
2,1
Don’t
2,0
3,0
Build
Don’t
Firm 2
Enter
1.5,-1
2,1
Don’t
3.5,0
3,0
If firm 1’s cost is low
Firm 1
When the probability that firm 1’s cost is high is “p”, find a Bayesian equilibrium of this
game.
Example 6: T and YR are neighbors. They want to live in a clean environment and want
their street to be clean. The utility from the clean street is same for both at 10. The
problem is someone has to clean the street in order to have a clean street and cleaning the
street is a public good. T’s cost of cleaning the street is ct and YR’s cost is cy. Both ct and
cy can have two values, either 5 or 15. The probability that the cost will be 5 is 2/3.
The payoffs will be as the following table.
T
Clean
Don’t
Find a Bayesian equilibrium of the game.
YR
Clean
10- ct, 10- cy
10, 10- cy
Don’t
10- ct,10
0,0
Dynamic Games with Incomplete Information
Perfect Bayesian Equilibrium
Example 1:
P1
L
R
P2
(2,2)
J
If P1 is type 1: (0,0)
If P1 is Type 2: (1,0)
W
(0,1)
(3,1)
- “(s1(t1)=L, s1(t2)=L) for player 1 and s2=J for player 2” can be a Bayesian equilibri
um. However, this equilibrium has the same problem as a Nash equilibrium whic
h is not a Subgame Perfect Equilibrium has. The problem is that we cannot apply
SPE concept in this case.
Example 2: (Reputation Game) US is considering invading North Korea. North Korea
can “fight” or “surrender,” but it has no chance to win over US once it chooses “fight”
and the war begins. On the other hand, it will be very costly for US to fight North Korea
even though US will win in the end. The payoffs are as follows.
US
Not Invade
Invade
NK
(0, 0)
Fight
Surrender
If NK is sane: (-1,-x)
(1,-2)
If NK is crazy: (-1, 2)
(1,-2)
We will assume that NK can fight US for two days and the payoffs above are for one
day’s fight. If there is a war, US can decide if it will fight the second day after the first
day’s fight. If NK surrender on the first day, it will surrender on the second day, too, and
the payoff for NK will be (-2)+(-2) and payoff for US will be (1)+(1).
What makes this story interesting is that North Korea can be crazy with probability (0.3)
and in that case it will have a strange payoff such that it prefers to fight and defeated
rather than surrender.
(1) What kind result can we expect from this game, if x=5?
(2) If the probability that NK is crazy is (0.6) instead of (0.3) and x=3, what result ca
n we expect? (assume that the discount factor is 1)
(3) If the probability that NK is crazy is (0.3) and x=3, what result can we expect?
O Perfect Bayesian Equilibrium (rough definition)
- A set of strategies and beliefs will be PBE if they satisfy the two conditions below
.
1. After each period the beliefs should be updated according to the Bayes rule (w
henever possible) based on the observations of other players’ action until that
time.
2. At each information set, the strategy should maximize the players’ payoffs giv
en the beliefs calculated by the rule above.
Information Asymmetry and Signaling Games
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? Wh
at 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 pooling equilibrium of the game.
(2) Find out the separating 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.
(3) Find out the separating equilibrium of the game.
(4) Find out the pooling equilibrium of the game.
Example 3: (Intuitive Criterion-Cho & Kreps)
A cowboy named Bill visited a town named Tombstone. Bill entered the only restaurant
in Tombstone and sat down at one of the tables.
All of sudden a cowboy named Jim approached Bill and asked Bill a very rude question.
Jim wants to fight the new cowboy in town, Bill, but do not know how strong Bill is.
Judging from the appearance, the probability that Bill is strong is 90% while the
probability that Bill is weak is 10%. One interesting fact is that a strong man likes whisky
more than beer, while a weak man likes beer more than a whisky.
Before Jim decides whether to fight Bill or not, Bill can order and drink either beer or
whisky. Jim can decide whether to fight or not after observing which beverage Bill orders
and drinks.
The payoffs are as the tables below.
Jim
Jim
fight
No fight
Bill (strong)
beer
-1,0
0,2
whisky
-1,1
0,3
fight
Bill(weak)
beer
1,1
whisky
1,0
No fight
(1)
(2)
(3)
0,3
0,2
Find out a pooling equilibrium where Bill always orders beer regardless of his
type.
In the equilibrium that you find out in (1) a strong Bill has to drink beer. Is it
realistic?
Considering (2) find out a more realistic equilibrium of the game.