171111 entrepreneurial economics 3

Entrepreneurial Economics #3
Organized by NUS MBA Entrepreneurship Club
BIZ 1-0301 11th November (Sat), 9:50am – 1:00pm
Topics to be covered:
• Market Theory – Price Discrimination
• Game Theory – from the beginning
Speaker:
• Kyoichiro Tamaki
• Ryosuke ISHII

Ryo’s Part
• Game Theory
• Information
• Pre-Nash
• Nash Equilibrium

Game Theory
from the
beginning.

What is Game Theory?
Game Theory is a simplification of the World

How Simplify?
Only three parameters!
• Player – Player A vs Player B, U.S.A vs NorthKorea
• Strategy (option) – left or right? , Threat or Conciliation?
• Payoff – how much gain/loss ?

Notation – Payoff Matrix
Option α Option β
Option 1 (𝐴1, 𝐵 𝛼) (𝐴1, 𝐵 𝛽)
Player A
Player B

Notation – Payoff Matrix
Option α Option β
Player A
Player B
When A selected Option 1 and B selected Option α,
The Payoff will be 𝑨 𝟏 𝒇𝒐𝒓 𝑨, 𝑎𝑛𝑑𝑩 𝜶 𝒇𝒐𝒓 𝑩

Example – The Price Wars
Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B

The types of Game Theory
and information

Information of Game Theory
Perfect / Imperfect information
• Perfect Information
• Each player knows full history of the play of the game thus far.
• Dynamic game (=Sequential game) and also could observe other players’ move.
• Chess, Go

Perfect / Imperfect information
• Perfect Information
• Each player knows full history of the play of the game thus far.
• Dynamic game (=Sequential game) and also could observe other players’ move.
• Chess, Go
• Imperfect Information
• Each player knows parts of history
• has no information about the decisions of others
• Simultaneous moving game (= static games)
• Rock – Paper – Scissors, Sealed bid auction
• Tennis (professional level = high speed!)
• Dynamic (=Sequential) game but cannot observe other players’ move.
• 麻将(麻雀, Mah-jong), Texas hold’em (don’t know other player’s private 牌/cards)

Complete / Incomplete information
• Complete Information
• Every player knows the rules and the structure of the game
• Which means, every player knows
• Who is the Players
• Possible strategies that players can choose
• Payoff
• Knows the other players also knows the rule
Option α Option β

• Complete Information
• Every player knows the rules and the structure of the game
• Which means, every player knows
• Who is the Players
• Possible strategies that players can choose
• Payoff
• Knows the other players also knows the rule
• Incomplete information
• Not complete information situation
• There are information asymmetry about the rule/payoff
• You are penalized from your professor on the exam,
but you don’t know why you lose mark.

Information
Perfect
-knows full history /decision
Imperfect:
Simultaneous games
Complete:
Knows the
rules, payoff
Tennis, soccer (amateur-level)
Not interesting game!
Just select an obvious best!
Rock - Paper – Scissors
Tennis, soccer (pro-level)
Sealed bid auction
Incomplete Price negotiation of used cars Hiring talents

[advanced]Information of Game Theory
Information
Perfect
-knows full history
Imperfect:
Simultaneous games
Complete:
Knows the
rules, payoff
Sealed bid auction
Introducing probability p, any incomplete game can be re-written to complete and imperfect information game.

Imperfect and complete
information game

Imperfect and complete information game
Imperfect and complete information game should be
Static Game (=Simultaneous Game)

In the Imperfect and complete information game
We Know these parameters!
• Player
• Strategy
• Payoff
But we don’t know the other player’s decision before you made a decision.
Also, this is non-cooperative game.
So, How to analyze game and to optimize your strategy? – before you know
your competitor’s move.
Option α Option β

How to analyze game and to optimize your strategy?
– Find an Equilibrium!
The types of Equilibrium:
• Pre- Nash
• Dominant Strategy Equilibrium ← We will start here
• Iterated Dominance Equilibrium
• Maxmin Strategy Equilibrium
• Nash
• Post-Nash
• Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)

How to analyze and optimize? – Fix the opponent.
Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other player’s
strategy.
Step1: We are Store A!
We have only 2 options:
Price↑ or ↓

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
Step2: Fix B’s strategy
as Price ↑

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
as Price ↑
Step3: Compare only A’s
Payoff. And decide which
one would A choose.
A would choose Price↓
If A knows B choose ↑

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
as Price ↑
Pay off. And decide which
one would A choose.
Step4: Move to B’s
next strategy. Price ↓

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
one would A choose.
If A knows B choose ↓

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
one would A choose.
If A knows B choose ↓
So, A will choose Price↓ regardless B is ↑ or ↓
This is called “Dominant Strategy”

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
Step6: Change Player.
We are Store B!

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
Step6: Change Player. We are Store B!
Step 7: Fix A’s strategy as Price ↑
Step 8: Compare only B’s Payoff.
B would prefer Price↓

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
Step 9: Fix A’s strategy as Price ↓

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
Step 9: Fix A’s strategy as Price ↓
So, B will choose Price↓ regardless A is ↑ or ↓
This is called “Dominant Strategy”

Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
strategy.
So, for both A and B,
Price ↓ is a dominant strategy.
This is called
“dominant-strategy equilibrium”
As a result, they will be worse off by
playing “non-cooperative game”.
“Not Efficient.”
If they are playing cooperative game,
They could acquire (+ +,+ +), but
Keep on eyes not to cheat each other.

Dominant Strategy
• Dominant strategy is a best strategy for a Player
No matter what the other player does.
• In this setting, every player has a “strictly dominant strategy”
• Strategy X Strictly dominant Y means 𝑋𝑖 > 𝑌𝑖
• Strategy X Weakly dominant Y means 𝑋𝑖 ≥ 𝑌𝑖
Strategy C Strategy D
Strategy A 2, 3 4, 3
Strategy B 1, 4 3, 3
Strategy A ______ly dominant B
Strategy C ______ly dominant D
player 2
player 1

Dominant Strategy
• Strategy X weakly dominant Y means 𝑋𝑖 ≥ 𝑌𝑖
As player 1’s perspective,
Strategy A strictly dominant B
If you fix the opponents strategy
as C, strategy A=2 > B=1
Also, fix that as D,
Strategy A=4 > B=3
player 1

Dominant Strategy
As player 2’s perspective,
Strategy C Weakly dominant D
If you fix the opponents strategy
as A, strategy C=3 = B=3
Also, fix that as B,
Strategy C=4 > D=3

Dominated Strategy
• Dominated strategy is a inferior strategy for a Player
which rational player would not choose.
• Strategy Y is Strictly dominated by X means 𝑋𝑖 > 𝑌𝑖
• Strategy Y is weakly dominated by X means 𝑋𝑖 ≥ 𝑌𝑖
Strategy B is ______ly dominated by A
Strategy D is ______ly dominated by C
player 2
player 1

Dominated Strategy
• Dominated strategy is a inferior strategy for a Player
which rational player would not choose.
• Strategy Y is Strictly dominated by X means 𝑋𝑖 > 𝑌𝑖
• Strategy Y is weakly dominated by X means 𝑋𝑖 ≥ 𝑌𝑖
Strategy B is strongly dominated by A
Strategy D is weakly dominated by C
player 2
player 1

Dominant Strategy
• Every player has a strictly dominant strategy, the outcome of the
game is called a “dominant-strategy equilibrium”
• Both Player worse off by this Dominant strategy.
• Such case we call “Prisoners’ Dilemma”
Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −

Example : Find a Dominant Strategy Equilibrium :
Apologize Break up
Apologize 0, 0 -10, +1
Break up +1,-10 -3, -3
You
Your boy/girl friend
Suppose you and your boy/girl friend are relatively new relationship.
You got a big fight last night.
Both of you have 2 strategies: Apologize or Break up.
This is non-cooperative and simultaneous move.
Where is a Dominant Strategy Equilibrium ?
[2min - Workshop]
• Find your partner. (2 ppl 1 team)
• Decide who will do first (Player1).
• Explain how to solve it (process)
• Player2 should do feedback to the
Player1 for Player1 could explain
better (on the exam paper).

Example2 : Find a Dominant Strategy Equilibrium
Ad No Ad
Ad 3, 3 13, -2
No Ad -2, 13 8, 8
Suppose you are Japan Tobacco and your opponent is Phillip Morris.
This is an advertisement wars. Advertisement cost you much. If only your
competitor do not Ad, you gain much profit from Ad.
• This is non-cooperative and simultaneous move.
Where is a Dominant Strategy Equilibrium? How to get out of the Dilemma?
[2min - Workshop]
• Player2 Explain how to solve it
(process)
• Player2 should be better to
explain utilizing your own
feedback to Player 1.

Example2 : Find a Nash Equilibrium:
Ad No Ad
Ad
Nash
3, 3
13, -2
No Ad -2, 13 8, 8
Thanks to Government regulation for Non-Ad policy for tobacco industry,
They could get out of Prisoners’ Dilemma and they are better off!!!
Government regulation

• Pre- Nash
• Dominant Strategy Equilibrium
• Iterated Dominance Equilibrium← What’s NEXT
• Nash
• Mixed Strategy
• Post-Nash

Iterated Dominance Equilibrium
• Dominant strategy was about “which strategy will be played for sure”
• But NOT every time dominant strategy equilibrium exists.
• So, let’s think about “which strategy will NOT be played for sure”
• Which is Dominated Strategy.
• If we find dominated strategy, we could eliminate the option(s).
• After few times do the elimination(=iterated), we will find a
equilibrium which is called “Iterated Dominance Equilibrium”

How to find and eliminate?
Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Same as before, let fix the other player’s strategy one by one and
compare Your strategy.
Step1: We are Player A!
Select Top, Middle, or
Bottom.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Select Top, Middle, or
Bottom.
So, we will see only
Player A’s payoff

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step2: Fix your rival’s
strategy. Say, Left.
- Top and Middle are
worse than Bottom.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step2: Fix rival’s strategy.
Step3: change rival’s STR.
Say, Center.
- Top and Bottom are
worse than Middle.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Say, Right.
- Top and Bottom are
worse than Middle.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step4: Find a dominant or
Dominated strategy.
In this case,
No dominant STR.
And the Top STR is
dominated strategy.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Dominated strategy.
In this case,
No dominant STR.
And the Top STR is
dominated strategy.
So Eliminate it.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step5:
Neither Middle / Bottom
are Dominated/Dominant
STR.
So Move to Player B.
As Player B,
We should focus only on
Player B’s payoff

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step6:We are Player B!
Select Left, Center, or
Right.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step6: We are Player B!
Step7: Fix Rival’s strategy.
Say, Middle.
-Right is better than
Left and Center.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step8: Change Rival’s STR.
Say, Bottom.
-Right is better than
Left and Center.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Dominated strategy.
In this case,
Right is a Dominant STR
For player B.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Dominated strategy.
In this case,
Right is a Dominant STR
For player B.
So, eliminate others.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step9: Find a STR.
Step10: As a Player A,
Select the Best: Middle.

Left Center Right
Top 3, 6 7, 4 10, 1
Middle 5, 1 8, 2 14, 6
Bottom 6, 0 6, 2 8, 5
Player A
Player B
Step11:
Finally we find
iterated dominance
equilibrium

Exercise: Find an iterated dominance equilibrium
東
east
南
south
西
west
北
north
Top 1,1 1,2 5,0 1,1
Middle 2,3 1,2 3,0 5,1
Bottom 1,1 0,5 1,7 0,1
Player A
Player B
[5min - Workshop]
• Find your partner. (2 ppl 1 team)
• Decide who will do first (Player1).
• Both player solve it individually.
(2 min)
• Explain how to solve it (process)
(2 min)

東
east
南
south
西
west
北
north
Top 1,1 1,2 5,0 1,1
Middle 2,3 1,2 3,0 5,1
Bottom 1,1 0,5 1,7 0,1
Player A
Player B
[+3min - Workshop]
• Change your role!
• Player2 Explain how to solve it
(process – 2min)
• Player2 should be better to
explain utilizing your own
feedback to Player 1.
• Player1 give Player2 a quality
feedback.
• High challenge! High support!

東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
let’s turn in Player A’s shoes.

東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
Bottom is dominated STR.

東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
Bottom is dominated STR.
So, Eliminate it.

東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
As Player B,
西 – west and 北 – north are also
dominated.
so,

東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
As Player B,
西 – west and 北 – north are also
dominated.
so, delete it.
(But we still cannot say east or south
is dominant. So left it.)

東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
As Player A again,
Top is weakly dominated.

東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
As Player A again,
Top is weakly dominated.
So eliminate it.

東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
In this situation, A has no choice but
Middle STR.
So, as Player B, you could choose
Proper one.
Comparing
東east = 3
南south = 2
And choose 3.
So, (Middle, East) is
Iterated dominance equilibrium.

Tips… (another way to get answer)
東
east
南
south
西
west
北
north
Top 1, 1 1, 2 5, 0 1, 1
Middle 2, 3 1, 2 3, 0 5, 1
Bottom 1, 1 0, 5 1, 7 0, 1
Player A
Player B
You can start from Player B’s viewpoint, and eliminate north at first, because
north is inferior strategy which means north is always dominated by west.
We are comparing
north
, 1
, 1
, 1
And others. Especially south.
south
, 2
, 2
, 5
And we know north is dominated by
south.

• Pre- Nash
• Maxmin Strategy Equilibrium ← Let’s talk about it!
• Nash
• Mixed Strategy
• Post-Nash

Maxmin Strategy Equilibrium
• What we already done:
• We dealt with 2 strategy: Dominant and Dominated.
• Mainly focus on how much payoff we could earn, how to maximize your profit.
• Remember we eliminate dominated as Dominated STR has lower profit always.
• Also suppose all players are Rational and No mistakes.
• Remember we eliminate dominated and we adopt dominant for sure.
• What is MAXMIN?
• Sometimes people want to avoid risk, rather Gain or Payoff.
• So, MAXMIN STR. is a strategy which select a strategy:
• Max(Min(Strategy1), Min(Strategy2), Min(Strategy3)… Min(Strategy 𝑛), )

Maxmin Strategy Equilibrium
• So, MAXMIN STR. is a strategy which select a strategy:
• Max(Min(Strategy1), Min(Strategy2), Min(Strategy3)… Min(Strategy 𝑛), )
a b c d
Min
of A
Top 1, 1 1, 2 5, 0 1, 1 1
Middle 2, 3 -100, 2 3, 0 5, 1 -100
Bottom 1, 1 0, 5 1, 7 0, 1 0
Player A
Player B
This Top STR.
has Maximum
payoff among
each strategy’s
Minimum payoff.
So, MAXMIN.

How to do MAXMIN? And how different?
Let’s compare 2 strategies: Dominant STR and MAXMIN STR.
STR B1 STR B2
STR A1 10, 4 8, 15
STR A2 -10,5 20, 10
5 min for
① Dominant (or iterated dominance)
Strategy:
② MAXMIN Strategy:
(2 min)
• Player1 explain how to solve it (process)
(2 min)
• Player2 should do feedback to the Player1 for
Player1 could explain better (on the exam
paper).

STR B1 STR B2
STR A1 10, 4 8, 15
STR A2 -10,5 20, 10
① Dominant(or iterated dominance)
Strategy:
Player A has no dominant Strategy.
For Player B, STR B2 is dominant STR.
Given the Player B’s STR as B2,
Player A will choose A2.
So, Dominant STR is (A2, B2)

STR B1 STR B2
STR A1 10, 4 8, 15
STR A2 -10,5 20, 10

As the Graph Shows,
For A:
MIN(A1)=8
MIN(A2)= –10
MAX(MIN(A1), MIN(A2))→ A1
For B:
MIN(B1)=4
MIN(B2)=10
MAX(MIN(B1), MIN(B2))→B2
So, MAXMIN STR.= (A1,B2)
STR B1 STR B2
Min
of A
STR A1 10, 4 8, 15 8
STR A2 -10,5 20, 10 -10
Min of B 4 10

• Pre- Nash
• Nash
• Mixed Strategy
• Post-Nash
Before taking break:
• Form a team of 4 people
• Every player recall the “Pre-Nash” part
and remember you thought “it’s
important”(2min)
• Each player share 1 point above to the
team. Player 1 → 2 → 3 →4.
• Repeat the share 3 times. So you will get
12 points from this 10 min.

• Pre- Nash
• Nash
• Nash Equilibrium ← So finally, “Beautiful Mind”
• Mixed Strategy
• Post-Nash

“Nash Equilibrium”
What is a Nash equilibrium?
• No firm wants to change its strategy, given what the other firm is doing.
• So a Nash equilibrium is when no firm wants to change/deviate, given
what the other firms are doing.
• It's basically when, given what all the other firms are doing, you are
happy with what you're doing.
• If every players’ Dominant Strategy intersects, it is Nash.
In Nash Equilibrium, You could think:
• This is a best response to others’ strategy.
• The game is stable
• Everybody's satisfied and the market can roll forward.

“Nash Equilibrium”
What is a Difference?
• Dominance STR.
• Search out equilibrium by “take dominance” or “eliminate dominated”
• You are already given the strategy 𝐴𝑖 , 𝐵𝑗, 𝐶 𝑘, … . .
• Your strategy is the best strategy for everyone if you fix others’ strategy.
• So, some payoff matrix could have more than 1 Nash Equilibrium.
• Nash does not mean “the best” strategy bundle.

Nash Equilibrium
Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
Equilibrium = the point at which all of the players are satisfied.
Remember the example of
Dominance Strategy.
We concluded (Price↓, Price↓) is
“dominant-strategy equilibrium”.
This is Also called “Nash Equilibrium”
Because Given the situation,
No player want to change
Each of strategies.

Nash Equilibrium
Price ↑ Price ↓
Price ↑ + +,+ + − −,++ +
Price ↓ ++ +,− − −, −
Store A
Store B
Here is not a Nash
For Player A,
Given the Player B’s strategy (P↑)
Player A want to change strategy.
So, this is not a Nash Equilibrium.

Nash Equilibrium finding
• So, How to find a Nash Equilibrium?
東
east
南
south
西
west
北
north
Top
Middle
Bottom
Player A
Player B If you do not want to use
your brain resource, but
have a lot of time,
You can check
Whether Nash or Not
Cell by Cell
Remember, each player can
only select his/her STR.

• So, How to find a Nash Equilibrium? More efficient?
東
east
南
south
西
west
北
north
Top 1,1 1,2 5,0 1,1
Middle 2,3 1,2 3,0 5,1
Bottom 1,1 0,5 1,7 0,1
Player B Step1: Find out “The Best
Response” given the other
players’ strategy.
Suppose you are Player A,
and Player B
select STR 東(east).
Player A

東
east
南
south
西
west
北
north
Top 1,1 1,2 5,0 1,1
Middle 2,3 1,2 3,0 5,1
Bottom 1,1 0,5 1,7 0,1
Suppose you are Player A,
and Player B
select STR 東(east).
Given, the situation, “2” is
the best strategy.
Next, Change B’s STR.
Player A

東
east
南
south
西
west
北
north
Top 1,1 1,2 5,0 1,1
Middle 2,3 1,2 3,0 5,1
Bottom 1,1 0,5 1,7 0,1
Player A
So , these are A’ s best
responses given the each of
B’s STR.

東
east
南
south
西
west
北
north
Top 1,1 1,2 5,0 1,1
Middle 2,3 1,2 3,0 5,1
Bottom 1,1 0,5 1,7 0,1
Player A
Player B
Step2: Find out B’s “The
Best Response” given the
A’s strategy.
Suppose you are a Player B,
and Player A select
“Top” STR.
Given the situation,
(Top, south) is a best
response for B.

東
east
南
south
西
west
北
north
Top 1,1 1,2 5,0 1,1
Middle 2,3 1,2 3,0 5,1
Bottom 1,1 0,5 1,7 0,1
Player A
Step2: Find out B’s “The
Best Response” given the
A’s strategy.
We could do same thing.
Player B

東
east
南
south
西
west
北
north
Top 1,1 1,2 5,0 1,1
Middle 2,3 1,2 3,0 5,1
Bottom 1,1 0,5 1,7 0,1
Player A
Step3: Find Nash.
So, if the cell becomes
(Player A, Player B)
This is a Nash.
In this case, 2 Nash EQs.
Player B

Exercise on Nash Equilibrium finding
eat run study Drink
Top 3,5 7,5 3,6 1,-3
Bottom 2,6 8,7 2,5 5,-5
Right 6,9 0,5 1,4 0,-3
Left 1,9 1,3 2,5 2,1
Player A
Player B
[5 min - Workshop]
(2 min)
• Player2 explain how to solve it
(process)
(2 min)

eat run study Drink
Top 3,5 7,5 3,6 1,-3
Bottom 2,6 8,7 2,5 5,-5
Right 6,9 0,5 1,4 0,-3
Left 1,9 1,3 2,5 2,1
Player B
As Player A, let’s fix B’s STR one by
one. And check it red.
We can ignore B’s payoff now.Player A

eat run study Drink
Top 3,5 7,5 3,6 1,-3
Bottom 2,6 8,7 2,5 5,-5
Right 6,9 0,5 1,4 0,-3
Left 1,9 1,3 2,5 2,1
Next, as Player B,
Deal A’s STR as given.
And select a best response
We can ignore A’s payoff now.
Player A
Player B

eat run study Drink
Top 3,5 7,5 3,6 1,-3
Bottom 2,6 8,7 2,5 5,-5
Right 6,9 0,5 1,4 0,-3
Left 1,9 1,3 2,5 2,1
These three are (pure STR) Nash
Equilibrium.
Player A
Player B

eat run study Drink
Top 3,5 7,5 3,6 1,-3
Bottom 2,6 8,7 2,5 5,-5
Right 6,9 0,5 1,4 0,-3
Left 1,9 1,3 2,5 2,1
Player A
Player B
If one strategy is dominated, this
strategy must not Nash Equilibrium
Because this cannot be a best
response.
So you can start from eliminate
dominated strategy.
As Player B, Drink is dominated.

eat run study Drink
Top 3,5 7,5 3,6 1,-3
Bottom 2,6 8,7 2,5 5,-5
Right 6,9 0,5 1,4 0,-3
Left 1,9 1,3 2,5 2,1
Player A
Player B
If one strategy is dominated, this
strategy must not Nash Equilibrium
Because this cannot be a best
response.
So you can start from eliminate
dominated strategy.
As Player B, Drink is dominated.
After eliminated Drink, Player A also
could eliminate Left as a dominated
STR.
But after this, we could not eliminate.
So, start same as before.

eat run study Drink
Top 3,5 7,5 3,6 1,-3
Bottom 2,6 8,7 2,5 5,-5
Right 6,9 0,5 1,4 0,-3
Left 1,9 1,3 2,5 2,1
Player A
Player B
And finally, same as before.

Exercise Nash
You and your rival could select 0~4 integer. And if only $𝑁 − $𝑀 = 1,
you could earn the $N where N is a integer you selected and M is your rival’s.
How many Nash Equilibrium do we have?
0 1 2 3 4
0
1
2
3
4

Exercise Nash
you could earn the $100 where N is a integer you selected and M is your rival’s.
0 1 2 3 4
0 0, 0 0, 1 0, 0 0, 0 0, 0
1 1, 0 0, 0 1, 2 0, 0 0, 0
2 0, 0 2, 1 0, 0 2, 3 0, 0
3 0, 0 0, 0 3, 2 0, 0 3, 4
4 0, 0 0, 0 0, 0 4, 3 0, 0
Suppose, you are Player A.
Fix, your opponent’s strategy.
Given the B’s strategy above, Search out the best
response, among your strategies.
In this case, when B select STR B=1,
Your best response will be STR A=2
So, color it.

Exercise Nash
you could earn the $100 where N is a integer you selected and M is your rival’s.
0 1 2 3 4
0 0, 0 0, 1 0, 0 0, 0 0, 0
1 1, 0 0, 0 1, 2 0, 0 0, 0
2 0, 0 2, 1 0, 0 2, 3 0, 0
3 0, 0 0, 0 3, 2 0, 0 3, 4
4 0, 0 0, 0 0, 0 4, 3 0, 0
2 Nash Equilibriums.

• Pre- Nash
• Nash
• Mixed Strategy ← Every payoff matrix has at least one Nash!
• Post-Nash

Nash Equilibrium in Mixed Strategy
What is an Nash Equilibrium in Mixed Strategy?
Pure
strategy
Paper Scissors Rock
Paper 0, 0 -1, 1 1, -1
Scissors 1, -1 0, 0 -1, 1
Rock -1, 1 1, -1 0, 0
In the repeated game, if you fix your strategy, such as “you always use paper”.
You surely lose. So we need to “Mix Strategies.” and find a proper probability to
mix the strategies.
Mixed
Strategy
Paper
q1
Scissors
q2
Rock
1-q1-q2
Paper
p1
0, 0 -1, 1 1, -1
Scissors
P2
1, -1 0, 0 -1, 1
Rock
1-p1-p2
-1, 1 1, -1 0, 0

Mixed Strategy
This payoff matrix below represents the probability (%) of successfully – get
a score for kicker, and keep a goal for goalkeeper, on a penalty kick.
Guard
Left
Guard
Right
Kick
Left
58%, 42% 95%, 5%
Kick
Right
93%, 7% 70%, 30%
In this case, There are no (Pure
Strategy) Nash Equilibrium.
But there are MAXMIN strategy.

Mixed Strategy
This payoff matrix below represents the probability (%) of successfully – get
a score for kicker, and keep a goal for goalkeeper, on a penalty kick.
Guard
Left
Guard
Right
Min of
kicker
Kick
Left
58%, 42% 95%, 5% 58%
Kick
Right
93%, 7% 70%, 30% 70%
Min of
keeper
7% 5%
From the MAXMIN strategy, it is
better
-for kicker to select “kick right”
-for keeper to select “guard left”
So the MAXMIN Equilibrium is
(Right, Left).
But if do so, kicker always win.
So keeper changes to right and
your MAXMIN payoff will be 70%.
Or, if you fix your strategy, your
rival always gain from your
inflexibility.

Mixed Strategy
So, let’s “mix” your strategy.
To mix your strategy it is important (i) randomize, and (ii) maximize your
payoff or minimize rival’s payoff.
Guard
Left
Guard
Right
Kick
Left
58%, 42% 95%, 5%
Kick
Right
93%, 7% 70%, 30%
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
There are no (Pure Strategy) Nash Eq.

Mixed Strategy
Exercise for understanding:
Suppose you are kicker, calculate your “minimum payoff” if 𝑝 = 0.5, 0.4
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
kicker
keeper
[5 min - Workshop]
(2 min)
(2 min)
Player2 could explain better (on the exam paper).

Mixed Strategy
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
kicker
keeper
If p = 0.5 how should we think?
1. Fix the rival’s strategy again.
So the keeper select left.
2. Kicker’s payoff will be
58% × 0.5 + 93% × 0.5＝75.5％
3. Next, change rival’s strategy.

Mixed Strategy
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
kicker
keeper
If p = 0.5 how should we think?
58% × 0.5 + 93% × 0.5＝75.5％
3. Next, change rival’s strategy. Say keeper → Right.
95% × 0.5 + 70% × 0.5＝82.5％
4. So if keeper select left always, your MIX strategy will
result in 75.5%.
5. This is better than MAXMIN Strategy(70%)
Mix Strategy Left Right
𝑃 = 0.5 75.5% 82.5%

Mixed Strategy
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
kicker
keeper If p = 0.4 how should we think?
58% × 0.4 + 93% × 0.6＝79％
3. Next, change rival’s strategy. Say keeper → Right.
95% × 0.4 + 70% × 0.6＝80％
4. So if keeper select left always, your MIX strategy will
result in 79%.
𝑃 = 0.5 75.5% 82.5%
𝑃 = 0.4 79% 80%

Mixed Strategy
In this case, the most awful thing is your opponent could predict your action.
So, Find a P where: make the opponent feel indifferent whether s/he select
Left or Right
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
kicker
keeper
𝑃 = 0.5 75.5% 82.5%
𝑃 = 0.4 79% 80%
𝑃 = 𝑋 =Right =Left
1. If Keeper’s strategy is Left, your payoff is:
58% × 𝑝 + 93% × (1 − 𝑝)
2. If Keeper’s strategy is Right, your payoff is:
95% × 𝑝 + 70% × (1 − 𝑝)
3. So, the best mix strategy is when
58% × 𝑝 + 93% × 1 − 𝑝 = 95% × 𝑝 + 70% × 1 − 𝑝
↔ 𝑝 =
23
60

Mixed Strategy
In this case, the most awful thing is your opponent could predict your action.
So, Find a P where: make the opponent feel indifferent whether s/he select
Left or Right
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
kicker
keeper
𝑃 = 0.5 75.5% 82.5%
𝑃 = 0.4 79% 80%
𝑝 =
23
60
79.6% 79.6%

Mixed Strategy
Suppose you are keeper. Find your “q”
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
kicker
keeper
Mix Strategy 𝑞 = ?
Left
Right
[5 min - Workshop]
• Both player solve it individually.(2 min)
(2 min)
Player1 could explain better (on the exam paper).

Mixed Strategy
Suppose you are keeper. Find your “q”
Left
q
Right
1-q
Left
p
58%, 42% 95%, 5%
Right
1-p
93%, 7% 70%, 30%
kicker
keeper 1. Fix the rival’s strategy again.
So the kicker select left.
2. Keeper’s payoff will be
42% × 𝑞 + 5% × (1 − 𝑞)
3. Next, change rival’s strategy. Say kicker → Right.
7% × 𝑞 + 30% × (1 − 𝑞)
4. The mix probability that Kicker feel indifferent is:
42% × 𝑞 + 5% × 1 − 𝑞 = 7% × 𝑞 + 30% × (1 − 𝑞)
5. Solve for 𝑞 and we will get
Mix Strategy 𝑞 =
5
12
Left 20.42%
Right 20.42%

Mixed Strategy
The result is:
Mix Strategy 𝑞 =
5
12
Kicker Left 20.42%
Kicker Right 20.42%
Mix Strategy Keeper Left Keeper Right
𝑝 =
23
60
79.6% 79.6%
Keeper’s payoff
Kicker’s payoff
This is zero-sum game, so kicker’s payoff + keeper’s payoff = 100%

Game Theory in dynamic(=sequential)
setting and Strategic Move:
Perfect but incomplete information game

[advanced]Information of Game Theory
Information
Perfect
-knows full history
Imperfect:
Simultaneous games
Complete:
Knows the
rules, payoff
Sealed bid auction

“Dynamic Game”
(Sequential Game)
What is a Dynamic Game?
• Like Chess, first, Player A move
• And then, Player B move
• (then, Player A move again)
So, your action today influence other
players’ behavior tomorrow.
• Remember the topic we already
dealt was “Static” = Simultaneous
game.

“Dynamic Game”
Also called, extensive form.
Different timing of strategic choice
Multiple decision points

Notation – Game Trees / extensive form
Morning You
Wake up!
Sleep
forever
Wake up!
Sleep
forever
(before, morning)
(10, 0) (0, 10)(-2, -5)(8, -1)
Nodes: decision point
Actions: you could choose
Strategies: whole game plan.
In this case, 4 strategies we have.
Path: a sequence from start
to end.
Not every path is an
equilibrium path.
You before sleep
Set Alarm Don’t set Alarm

Morning You
Wake up!
Sleep
forever
Wake up!
Sleep
forever
(before, morning)
(10, 0) (0, 10)(-2, -5)(8, -1)
Information set
You before sleep

What is a difference of these two “information set”?
Morning You should select your action,
Without knowing what option You Before sleep select.
Say, “I forgot set alarm or not. Should I wake up now…?”
Morning You
Wake up!
Sleep
forever
Wake up!
Sleep
forever
(10, 0) (0, 10)(-2, -5)(8, -1)
You before sleep
Morning You
Wake up!
Sleep
forever
Wake up!
Sleep
forever
(10, 0) (0, 10(-2, -5)(8, -1)
You before sleep
You know you set alarm or not.
And you will decide wake up or not.

So, what does it mean?
Player n+1 will explain to Player n.
You
Your Rival
Rock PaperScissors
R PS R PSR PS

This is cheating Rock Paper Scissors.
So you will always win!!
You
Your Rival
Rock PaperScissors
R PS R PSR PS
You
Your Rival
Rock PaperScissors
R PS R PSR PS
This is normal Rock Paper Scissors.

Can you tell why this is prohibited?
If this is information set, we need same action each node.
(if you select S, it means you ignore the possibility of the right node.)
You
Your Rival
Rock PaperScissors
R PS R PR PS

Subgame
Subgame: A smaller part of the whole game starting from any one node and continuing to
the end of the entire game, with the qualification that no information set is subdivided.
(That is, it consists of a singleton decision node owned by a player and all subsequent
parts of the original game.)
So, How many subgame do we have?
You
Your Rival

Subgame
Subgame: A smaller part of the whole game starting from any one node and continuing to
the end of the entire game, with the qualification that no information set is subdivided.
(That is, it consists of a singleton decision node owned by a player and all subsequent
parts of the original game.)
You
Your Rival
So we are all set, about notation!

How to find an Equilibrium on Dynamic Game?
HIJACKER
Success! Give in Explode
(100, -50) (-∞, -∞)(-100, 1M)
PILOT
Obey Hijacker Resist Hijacker
Think Backward.

How to find an Equilibrium on Dynamic Game?
HIJACKER
Success! Give in Explode
(100, -50) (-∞, -∞)(-100, 1M)
PILOT
Obey Hijacker Resist Hijacker
Think Backward.
Focus this subgame first,
(We assume Hijacker is enough rational.)
And we will think it “Backward” which means,
We will think the subgame first.
So, Give in = (100, −50) > Explode = (−∞, −∞)
So, pilot should only compare the result of the subgame
Success (−100, −1𝑀) and, Give in (100, −50)
So, PILOT should resist.
This way of thinking is called Backward induction:
ruling out the actions that players would not play if
they were actually given a chance to choose.
subgame

Reference
NUS MBA BMA5001 Lecture Note 8-13 by Professor Jo
Principles of Microeconomics (Mankiw's Principles of Economics)
MITx: 14.100x Microeconomics
https://en.wikiquote.org/wiki/Greg_Mankiw#Ch._1._Ten_Principles_of_Economics
Amazon.com
https://www.youtube.com/watch?v=ErJNYh8ejSA
The Art of Strategy
http://ykamijo.web.fc2.com/lecture1.html

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