Entrepreneurial Economics #3
Organized by NUS MBA Entrepreneurship Club
BIZ 1-0301 11th November (Sat), 9:50am â 1:00pm
Topics to be covered:
⢠Market Theory â Price Discrimination
⢠Game Theory â from the beginning
Speaker:
⢠Kyoichiro Tamaki
⢠Ryosuke ISHII
Ryoâs Part
⢠Game Theory
⢠Information
⢠Pre-Nash
⢠Nash Equilibrium
Game Theoryfrom the beginning.
What is Game Theory?
Game Theory is a simplification of the World
How Simplify?
Only three parameters!⢠Player â Player A vs Player B, U.S.A vs NorthKorea
⢠Strategy (option) â left or right? , Threat or Conciliation?
⢠Payoff â how much gain/loss ?
Notation â Payoff Matrix
Option ι Option β
Option 1 (đ´1, đľđź) (đ´1, đľđ˝)
Option 2 (đ´2, đľđź) (đ´2, đľđ˝)
Player A
Player B
Notation â Payoff Matrix
Option ι Option β
Option 1 (đ´1, đľđź) (đ´1, đľđ˝)
Option 2 (đ´2, đľđź) (đ´2, đľđ˝)
Player A
Player B
When A selected Option 1 and B selected Option Îą,The Payoff will be đ¨đ đđđ đ¨, đđđđŠđś đđđ đŠ
Example â The Price Wars
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
The types of Game Theory and information
Information of Game TheoryPerfect / 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 playersâ move.⢠Chess, Go
Information of Game TheoryPerfect / 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 playersâ 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 â Paper â Scissors, Sealed bid auction
⢠Tennis (professional level = high speed!)
⢠Dynamic (=Sequential) game but cannot observe other playersâ move.
⢠麝ĺ°(éşťé, Mah-jong), Texas holdâem (donât know other playerâs private ç/cards)
Information of Game TheoryComplete / 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 ι Option β
Option 1 (đ´1, đľđź) (đ´1, đľđ˝)
Option 2 (đ´2, đľđź) (đ´2, đľđ˝)
Information of Game TheoryComplete / 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
⢠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 donât know why you lose mark.
Information of Game TheoryComplete / Incomplete information
InformationPerfect
-knows full history /decisionImperfect:
Simultaneous games
Complete:Knows the
rules, payoff
Tennis, soccer (amateur-level)Not interesting game!
Just select an obvious best!
Rock - Paper â ScissorsTennis, soccer (pro-level)
Sealed bid auction
Incomplete Price negotiation of used cars Hiring talents
[advanced]Information of Game TheoryComplete / Incomplete information
InformationPerfect
-knows full history Imperfect:
Simultaneous games
Complete:Knows the
rules, payoffTennis, soccer (amateur-level)
Rock - Paper â ScissorsTennis, soccer (pro-level)
Sealed bid auction
Incomplete Price negotiation of used cars Hiring talents
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 donât know the other playerâs decision before you made a decision.Also, this is non-cooperative game.
So, How to analyze game and to optimize your strategy? â before you know your competitorâs move.
Option ι Option β
Option 1 (đ´1, đľđź) (đ´1, đľđ˝)
Option 2 (đ´2, đľđź) (đ´2, đľđ˝)
How to analyze game and to optimize your strategy?â Find an Equilibrium!
The types of Equilibrium:
⢠Pre- Nash⢠Dominant Strategy Equilibrium â We will start here
⢠Iterated Dominance Equilibrium
⢠Maxmin Strategy Equilibrium
⢠Nash⢠Nash Equilibrium
⢠Post-Nash⢠Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step1: We are Store A!
We have only 2 options:Priceâ or â
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step1: We are Store A!
Step2: Fix Bâs strategyas Price â
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step1: We are Store A!
Step2: Fix Bâs strategyas Price â
Step3: Compare only AâsPayoff. And decide whichone would A choose.
A would choose PriceâIf A knows B choose â
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step1: We are Store A!
Step2: Fix Bâs strategyas Price â
Step3: Compare only AâsPay off. And decide whichone would A choose.
Step4: Move to Bâs next strategy. Price â
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step4: Move to Bâs next strategy. Price â
Step5: Compare only AâsPay off. And decide whichone would A choose.
A would choose PriceâIf A knows B choose â
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step4: Move to Bâs next strategy. Price â
Step5: Compare only AâsPay off. And decide whichone would A choose.
A would choose PriceâIf A knows B choose â
So, A will choose Priceâ regardless B is â or âThis is called âDominant Strategyâ
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step6: Change Player.We are Store B!
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step6: Change Player. We are Store B!
Step 7: Fix Aâs strategy as Price âStep 8: Compare only Bâs Payoff.B would prefer Priceâ
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step6: Change Player. We are Store B!
Step 7: Fix Aâs strategy as Price âStep 8: Compare only Bâs Payoff.B would prefer Priceâ
Step 9: Fix Aâs strategy as Price â
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
Step6: Change Player. We are Store B!
Step 7: Fix Aâs strategy as Price âStep 8: Compare only Bâs Payoff.B would prefer Priceâ
Step 9: Fix Aâs strategy as Price âStep 10: Compare only Bâs Payoff.B would prefer Priceâ
So, B will choose Priceâ regardless A is â or âThis is called âDominant Strategyâ
How to analyze and optimize? â Fix the opponent.
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Each Player could only select a strategy. So, first of all, let fix the other playerâs strategy.
So, for both A and B, Price â is a dominant strategy.This is calledâdominant-strategy equilibriumâ
As a result, they will be worse off by playing ânon-cooperative gameâ.âNot Efficient.â
If they are playing cooperative game,They could acquire (+ +,+ +), butKeep on eyes not to cheat each other.
Dominant Strategy
⢠Dominant strategy is a best strategy for a PlayerNo matter what the other player does.⢠In this setting, every player has a âstrictly dominant strategyâ
⢠Strategy X Strictly dominant Y means đđ > đđ
⢠Strategy X Weakly dominant Y means đđ ⼠đđ
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
⢠Dominant strategy is a best strategy for a PlayerNo matter what the other player does.⢠In this setting, every player has a âstrictly dominant strategyâ
⢠Strategy X Strictly dominant Y means đđ > đđ
⢠Strategy X weakly dominant Y means đđ ⼠đđ
Strategy C Strategy D
Strategy A 2, 3 4, 3
Strategy B 1, 4 3, 3
As player 1âs perspective,Strategy A strictly dominant B
If you fix the opponents strategy as C, strategy A=2 > B=1Also, fix that as D, Strategy A=4 > B=3
player 1
Dominant Strategy
⢠Dominant strategy is a best strategy for a PlayerNo matter what the other player does.⢠In this setting, every player has a âstrictly dominant strategyâ
⢠Strategy X Strictly dominant Y means đđ > đđ
⢠Strategy X weakly dominant Y means đđ ⼠đđ
Strategy C Strategy D
Strategy A 2, 3 4, 3
Strategy B 1, 4 3, 3
As player 2âs perspective,Strategy C Weakly dominant D
If you fix the opponents strategy as A, strategy C=3 = B=3Also, fix that as B, Strategy C=4 > D=3
Dominated Strategy
⢠Dominated strategy is a inferior strategy for a Playerwhich rational player would not choose.
⢠Strategy Y is Strictly dominated by X means đđ > đđ
⢠Strategy Y is weakly dominated by X means đđ ⼠đđ
Strategy C Strategy D
Strategy A 2, 3 4, 3
Strategy B 1, 4 3, 3
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 Playerwhich rational player would not choose.
⢠Strategy Y is Strictly dominated by X means đđ > đđ
⢠Strategy Y is weakly dominated by X means đđ ⼠đđ
Strategy C Strategy D
Strategy A 2, 3 4, 3
Strategy B 1, 4 3, 3
Strategy B is strongly dominated by A
Strategy D is weakly dominated by C
player 2
player 1
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 âstrictly dominant strategyâ
⢠Strategy X Strictly dominant Y means đđ > đđ
⢠Strategy X weakly dominant Y means đđ ⼠đđ
⢠Every player has a strictly dominant strategy, the outcome of the game is called a âdominant-strategy equilibriumâ⢠Both Player worse off by this Dominant strategy.
⢠Such case we call âPrisonersâ Dilemmaâ
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
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.
⢠Player1 should do feedback to the Player2 for Player2 could explain better (on the exam paper).
Example2 : Find a Nash Equilibrium:
Ad No Ad
AdNash3, 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 Prisonersâ Dilemma and they are better off!!!
Government regulation
How to analyze game and to optimize your strategy?â Find an Equilibrium!
The types of Equilibrium:
⢠Pre- Nash⢠Dominant Strategy Equilibrium
⢠Iterated Dominance Equilibriumâ Whatâs NEXT
⢠Maxmin Strategy Equilibrium
⢠Nash⢠Nash Equilibrium
⢠Mixed Strategy
⢠Post-Nash⢠Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
Iterated Dominance Equilibrium⢠Dominant strategy was about âwhich strategy will be played for sureâ
⢠But NOT every time dominant strategy equilibrium exists.
⢠So, letâs think about âwhich strategy will NOT be played for sureâ
⢠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 âIterated Dominance Equilibriumâ
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 playerâs strategy one by one and compare Your strategy.
Step1: We are Player A!
We have only 3 options:Select Top, Middle, or Bottom.
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 playerâs strategy one by one and compare Your strategy.
Step1: We are Player A!
We have only 3 options:Select Top, Middle, or Bottom.
So, we will see only Player Aâs payoff
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 playerâs strategy one by one and compare Your strategy.
Step1: We are Player A!Step2: Fix your rivalâs strategy. Say, Left.
- Top and Middle areworse than Bottom.
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 playerâs strategy one by one and compare Your strategy.
Step1: We are Player A!Step2: Fix rivalâs strategy.Step3: change rivalâs STR.Say, Center.
- Top and Bottom areworse than Middle.
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 playerâs strategy one by one and compare Your strategy.
Step1: We are Player A!Step2: Fix rivalâs strategy.Step3: change rivalâs STR.Say, Right.
- Top and Bottom areworse than Middle.
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 playerâs strategy one by one and compare Your strategy.
Step1: We are Player A!Step2: Fix rivalâs strategy.Step3: change rivalâs STR.Step4: Find a dominant orDominated strategy.
In this case, No dominant STR.And the Top STR is dominated strategy.
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 playerâs strategy one by one and compare Your strategy.
Step1: We are Player A!Step2: Fix rivalâs strategy.Step3: change rivalâs STR.Step4: Find a dominant orDominated strategy.
In this case, No dominant STR.And the Top STR is dominated strategy.So Eliminate it.
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 playerâs strategy one by one and compare Your strategy.
Step5:Neither Middle / Bottomare Dominated/DominantSTR.
So Move to Player B.
As Player B,We should focus only onPlayer Bâs payoff
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 playerâs strategy one by one and compare Your strategy.
Step6:We are Player B!
We have only 3 options:Select Left, Center, or Right.
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 playerâs strategy one by one and compare Your strategy.
Step6: We are Player B!Step7: Fix Rivalâs strategy.Say, Middle.
-Right is better thanLeft and Center.
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 playerâs strategy one by one and compare Your strategy.
Step6: We are Player B!Step7: Fix Rivalâs strategy.Step8: Change Rivalâs STR.Say, Bottom.
-Right is better thanLeft and Center.
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 playerâs strategy one by one and compare Your strategy.
Step6: We are Player B!Step7: Fix Rivalâs strategy.Step8: Change Rivalâs STR.Step9: Find a dominant orDominated strategy.
In this case, Right is a Dominant STR For player B.
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 playerâs strategy one by one and compare Your strategy.
Step6: We are Player B!Step7: Fix Rivalâs strategy.Step8: Change Rivalâs STR.Step9: Find a dominant orDominated strategy.
In this case, Right is a Dominant STR For player B.
So, eliminate others.
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 playerâs strategy one by one and compare Your strategy.
Step6: We are Player B!Step7: Fix Rivalâs strategy.Step8: Change Rivalâs STR.Step9: Find a STR.Step10: As a Player A,Select the Best: Middle.
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 playerâs strategy one by one and compare Your strategy.
Step11:Finally we find
iterated dominanceequilibrium
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
[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)
⢠Player2 should do feedback to the Player1 for Player1 could explain better (on the exam paper).
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
[+3min - Workshop]⢠Change your role!
⢠Player2 Explain how to solve it (process â 2min)
⢠Player2 should be better to explain utilizing your own feedback to Player 1.
⢠Player1 give Player2 a quality feedback.
⢠High challenge! High support!
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
letâs turn in Player Aâs shoes.
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
Bottom is dominated STR.
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
Bottom is dominated STR.So, Eliminate it.
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
As Player B, 輿 â west and ĺ â north are also dominated.
so,
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
As Player B, 輿 â west and ĺ â north are also dominated.
so, delete it.(But we still cannot say east or south is dominant. So left it.)
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
As Player A again, Top is weakly dominated.
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
As Player A again, Top is weakly dominated.So eliminate it.
Exercise: Find an iterated dominance equilibrium
ćąeast
ĺsouth
輿west
ĺnorth
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
Same as before, let fix the other playerâs strategy one by one and compare Your strategy.
In this situation, A has no choice butMiddle STR.
So, as Player B, you could choose Proper one.
Comparing ćąeast = 3ĺsouth = 2And choose 3.
So, (Middle, East) is Iterated dominance equilibrium.
Tips⌠(another way to get answer)
ćąeast
ĺsouth
輿west
ĺnorth
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 Bâs viewpoint, and eliminate north at first, because north is inferior strategy which means north is always dominated by west.
We are comparingnorth, 1, 1, 1
And others. Especially south.south, 2, 2, 5
And we know north is dominated by south.
How to analyze game and to optimize your strategy?â Find an Equilibrium!
The types of Equilibrium:
⢠Pre- Nash⢠Dominant Strategy Equilibrium
⢠Iterated Dominance Equilibrium
⢠Maxmin Strategy Equilibrium â Letâs talk about it!
⢠Nash⢠Nash Equilibrium
⢠Mixed Strategy
⢠Post-Nash⢠Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
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 đ), )
Maxmin Strategy Equilibrium⢠So, MAXMIN STR. is a strategy which select a strategy:
⢠Max(Min(Strategy1), Min(Strategy2), Min(Strategy3)⌠Min(Strategy đ), )
a b c dMinof 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 Maximumpayoff among each strategyâs
Minimum payoff.
So, MAXMIN.
How to do MAXMIN? And how different?Letâs 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 â Dominant (or iterated dominance)
Strategy:⥠MAXMIN Strategy:
⢠Both player solve it individually.(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).
How to do MAXMIN? And how different?Letâs compare 2 strategies: Dominant STR and MAXMIN STR.
STR B1 STR B2
STR A1 10, 4 8, 15
STR A2 -10,5 20, 10
â Dominant(or iterated dominance) Strategy:
Player A has no dominant Strategy.
For Player B, STR B2 is dominant STR.
Given the Player Bâs STR as B2,Player A will choose A2.
So, Dominant STR is (A2, B2)
How to do MAXMIN? And how different?Letâs compare 2 strategies: Dominant STR and MAXMIN STR.
STR B1 STR B2
STR A1 10, 4 8, 15
STR A2 -10,5 20, 10
⥠MAXMIN Strategy:
How to do MAXMIN? And how different?Letâs compare 2 strategies: Dominant STR and MAXMIN STR.
⥠MAXMIN Strategy:As the Graph Shows,For A: MIN(A1)=8MIN(A2)= â10MAX(MIN(A1), MIN(A2))â A1
For B:MIN(B1)=4MIN(B2)=10MAX(MIN(B1), MIN(B2))âB2
So, MAXMIN STR.= (A1,B2)
STR B1 STR B2Minof A
STR A1 10, 4 8, 15 8
STR A2 -10,5 20, 10 -10
Min of B 4 10
How to analyze game and to optimize your strategy?â Find an Equilibrium!
The types of Equilibrium:
⢠Pre- Nash⢠Dominant Strategy Equilibrium
⢠Iterated Dominance Equilibrium
⢠Maxmin Strategy Equilibrium
⢠Nash⢠Nash Equilibrium
⢠Mixed Strategy
⢠Post-Nash⢠Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
Before taking break:
⢠Form a team of 4 people⢠Every player recall the âPre-Nashâ part
and remember you thought âitâs importantâ(2min)
⢠Each player share 1 point above to the team. Player 1 â 2 â 3 â4.
⢠Repeat the share 3 times. So you will get 12 points from this 10 min.
How to analyze game and to optimize your strategy?â Find an Equilibrium!
The types of Equilibrium:
⢠Pre- Nash⢠Dominant Strategy Equilibrium
⢠Iterated Dominance Equilibrium
⢠Maxmin Strategy Equilibrium
⢠Nash⢠Nash Equilibrium â So finally, âBeautiful Mindâ
⢠Mixed Strategy
⢠Post-Nash⢠Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
âNash EquilibriumâWhat 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 playersâ Dominant Strategy intersects, it is Nash.
In Nash Equilibrium, You could think:
⢠This is a best response to othersâ strategy.
⢠The game is stable
⢠Everybody's satisfied and the market can roll forward.
âNash EquilibriumâWhat is a Difference?
⢠Dominance STR.⢠Search out equilibrium by âtake dominanceâ or âeliminate dominatedâ
⢠Nash Equilibrium⢠You are already given the strategy đ´đ , đľđ , đśđ , ⌠. .
⢠Your strategy is the best strategy for everyone if you fix othersâ strategy.
⢠So, some payoff matrix could have more than 1 Nash Equilibrium.
⢠Nash does not mean âthe bestâ strategy bundle.
Nash Equilibrium
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store B
Equilibrium = the point at which all of the players are satisfied.
Remember the example of Dominance Strategy.
We concluded (Priceâ, Priceâ) isâdominant-strategy equilibriumâ.
This is Also called âNash Equilibriumâ
Because Given the situation,No player want to change Each of strategies.
Nash Equilibrium
Price â Price â
Price â + +,+ + â â,++ +
Price â ++ +,â â â, â
Store A
Store BHere is not a Nash
For Player A,Given the Player Bâs strategy (Pâ)Player A want to change strategy.
So, this is not a Nash Equilibrium.
Nash Equilibrium finding⢠So, How to find a Nash Equilibrium?
ćąeast
ĺsouth
輿west
ĺnorth
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.
Nash Equilibrium finding⢠So, How to find a Nash Equilibrium? More efficient?
ćąeast
ĺsouth
輿west
ĺnorth
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 âThe Best Responseâ given the other playersâ strategy.
Suppose you are Player A, and Player B select STR ćą(east).
Player A
Nash Equilibrium finding⢠So, How to find a Nash Equilibrium? More efficient?
ćąeast
ĺsouth
輿west
ĺnorth
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 âThe Best Responseâ given the other playersâ strategy.
Suppose you are Player A, and Player B select STR ćą(east).
Given, the situation, â2â is the best strategy.
Next, Change Bâs STR.
Player A
Nash Equilibrium finding⢠So, How to find a Nash Equilibrium? More efficient?
ćąeast
ĺsouth
輿west
ĺnorth
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 âThe Best Responseâ given the other playersâ strategy.
Suppose you are Player A, and Player B select STR ćą(east).
Given, the situation, â2â is the best strategy.
Next, Change Bâs STR.
Player A
Nash Equilibrium finding⢠So, How to find a Nash Equilibrium? More efficient?
ćąeast
ĺsouth
輿west
ĺnorth
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 Step1: Find out âThe Best Responseâ given the other playersâ strategy.
So , these are Aâ s best responses given the each of Bâs STR.
Nash Equilibrium finding⢠So, How to find a Nash Equilibrium? More efficient?
ćąeast
ĺsouth
輿west
ĺnorth
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 BStep2: Find out Bâs âThe Best Responseâ given the Aâs strategy.
Suppose you are a Player B, and Player A selectâTopâ STR.
Given the situation,
(Top, south) is a best response for B.
Nash Equilibrium finding⢠So, How to find a Nash Equilibrium? More efficient?
ćąeast
ĺsouth
輿west
ĺnorth
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 Bâs âThe Best Responseâ given the Aâs strategy.
We could do same thing.
Player B
Nash Equilibrium finding⢠So, How to find a Nash Equilibrium? More efficient?
ćąeast
ĺsouth
輿west
ĺnorth
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
⢠So, How to find a Nash Equilibrium? More efficient?
[5 min - Workshop]
⢠Both player solve it individually.(2 min)
⢠Player2 explain how to solve it (process)
(2 min)⢠Player1 should do feedback to the
Player2 for Player1 could explain better (on the exam paper).
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 B
⢠So, How to find a Nash Equilibrium? More efficient?
As Player A, letâs fix Bâs STR one by one. And check it red.
We can ignore Bâs payoff now.Player A
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
⢠So, How to find a Nash Equilibrium? More efficient?
Next, as Player B,Deal Aâs STR as given.
And select a best response
We can ignore Aâs payoff now.
Player A
Player B
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
⢠So, How to find a Nash Equilibrium? More efficient?
These three are (pure STR) Nash Equilibrium.
Player A
Player B
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
⢠So, How to find a Nash Equilibrium? More efficient?
If one strategy is dominated, this strategy must not Nash EquilibriumBecause this cannot be a best response.
So you can start from eliminate dominated strategy.
As Player B, Drink is dominated.
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
⢠So, How to find a Nash Equilibrium? More efficient?
If one strategy is dominated, this strategy must not Nash EquilibriumBecause 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.
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
⢠So, How to find a Nash Equilibrium? More efficient?
If one strategy is dominated, this strategy must not Nash EquilibriumBecause 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.
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
⢠So, How to find a Nash Equilibrium? More efficient?
And finally, same as before.
Exercise NashYou and your rival could select 0~4 integer. And if only $đ â $đ = 1,
you could earn the $N where N is a integer you selected and M is your rivalâs.
How many Nash Equilibrium do we have?
0 1 2 3 4
0
1
2
3
4
Exercise NashYou and your rival could select 0~4 integer. And if only $đ â $đ = 1,
you could earn the $100 where N is a integer you selected and M is your rivalâs.
How many Nash Equilibrium do we have?
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 opponentâs strategy.Given the Bâs 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 NashYou and your rival could select 0~4 integer. And if only $đ â $đ = 1,
you could earn the $100 where N is a integer you selected and M is your rivalâs.
How many Nash Equilibrium do we have?
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.
How to analyze game and to optimize your strategy?â Find an Equilibrium!
The types of Equilibrium:
⢠Pre- Nash⢠Dominant Strategy Equilibrium
⢠Iterated Dominance Equilibrium
⢠Maxmin Strategy Equilibrium
⢠Nash⢠Nash Equilibrium
⢠Mixed Strategy â Every payoff matrix has at least one Nash!
⢠Post-Nash⢠Subgame-Perfect Nash Equilibrium (for perfect and incomplete information game)
Nash Equilibrium in Mixed StrategyWhat is an Nash Equilibrium in Mixed Strategy?
Purestrategy
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 âyou always use paperâ.You surely lose. So we need to âMix Strategies.â and find a proper probability to mix the strategies.
MixedStrategy
Paperq1
Scissorsq2
Rock1-q1-q2
Paperp1
0, 0 -1, 1 1, -1
ScissorsP2
1, -1 0, 0 -1, 1
Rock1-p1-p2
-1, 1 1, -1 0, 0
Mixed StrategyThis payoff matrix below represents the probability (%) of successfully â get a score for kicker, and keep a goal for goalkeeper, on a penalty kick.
GuardLeft
GuardRight
KickLeft
58%, 42% 95%, 5%
KickRight
93%, 7% 70%, 30%
In this case, There are no (Pure Strategy) Nash Equilibrium.
But there are MAXMIN strategy.
Mixed StrategyThis payoff matrix below represents the probability (%) of successfully â get a score for kicker, and keep a goal for goalkeeper, on a penalty kick.
GuardLeft
GuardRight
Min of kicker
KickLeft
58%, 42% 95%, 5% 58%
KickRight
93%, 7% 70%, 30% 70%
Min of keeper
7% 5%
From the MAXMIN strategy, it is better -for kicker to select âkick rightâ-for keeper to select âguard leftâ
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 yourinflexibility.
Mixed StrategySo, letâs âmixâ your strategy.To mix your strategy it is important (i) randomize, and (ii) maximize your payoff or minimize rivalâs payoff.
GuardLeft
GuardRight
KickLeft
58%, 42% 95%, 5%
KickRight
93%, 7% 70%, 30%
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
There are no (Pure Strategy) Nash Eq.
Mixed StrategyExercise for understanding:
Suppose you are kicker, calculate your âminimum payoffâ if đ = 0.5, 0.4
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
kic
ker
keeper[5 min - Workshop]
⢠Both player solve it individually.(2 min)
⢠Player1 explain how to solve it (process)(2 min)
⢠Player2 should do feedback to the Player1 for Player2 could explain better (on the exam paper).
Mixed StrategyExercise for understanding:
Suppose you are kicker, calculate your âminimum payoffâ if đ = 0.5, 0.4
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
kic
ker
keeperIf p = 0.5 how should we think?1. Fix the rivalâs strategy again.
So the keeper select left.2. Kickerâs payoff will be
58% Ă 0.5 + 93% Ă 0.5ďź75.5ďź
3. Next, change rivalâs strategy.
Mixed StrategyExercise for understanding:
Suppose you are kicker, calculate your âminimum payoffâ if đ = 0.5, 0.4
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
kic
ker
keeperIf p = 0.5 how should we think?1. Fix the rivalâs strategy again.
So the keeper select left.2. Kickerâs payoff will be
58% Ă 0.5 + 93% Ă 0.5ďź75.5ďź
3. Next, change rivalâs strategy. Say keeper â Right.95% Ă 0.5 + 70% Ă 0.5ďź82.5ďź
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
đ = 0.5 75.5% 82.5%
Mixed StrategyExercise for understanding:
Suppose you are kicker, calculate your âminimum payoffâ if đ = 0.5, 0.4
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
kic
ker
keeper If p = 0.4 how should we think?1. Fix the rivalâs strategy again.
So the keeper select left.2. Kickerâs payoff will be
58% Ă 0.4 + 93% Ă 0.6ďź79ďź
3. Next, change rivalâs strategy. Say keeper â Right.95% Ă 0.4 + 70% Ă 0.6ďź80ďź
4. So if keeper select left always, your MIX strategy will result in 79%.
Mix Strategy Left Right
đ = 0.5 75.5% 82.5%
đ = 0.4 79% 80%
Mixed StrategyIn 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
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
kic
ker
keeperMix Strategy Left Right
đ = 0.5 75.5% 82.5%
đ = 0.4 79% 80%
đ = đ =Right =Left
1. If Keeperâs strategy is Left, your payoff is:58% Ă đ + 93% Ă (1 â đ)
2. If Keeperâs strategy is Right, your payoff is:95% Ă đ + 70% Ă (1 â đ)
3. So, the best mix strategy is when58% Ă đ + 93% Ă 1 â đ = 95% Ă đ + 70% Ă 1 â đ
â đ =23
60
Mixed StrategyIn 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
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
kic
ker
keeperMix Strategy Left Right
đ = 0.5 75.5% 82.5%
đ = 0.4 79% 80%
đ =23
6079.6% 79.6%
Mixed StrategySuppose you are keeper. Find your âqâ
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
kic
ker
keeperMix Strategy đ = ?
Left
Right
[5 min - Workshop]⢠Both player solve it individually.(2 min)⢠Player2 explain how to solve it (process)
(2 min)⢠Player1 should do feedback to the Player2 for
Player1 could explain better (on the exam paper).
Mixed StrategySuppose you are keeper. Find your âqâ
Leftq
Right1-q
Leftp
58%, 42% 95%, 5%
Right1-p
93%, 7% 70%, 30%
kic
ker
keeper 1. Fix the rivalâs strategy again.So the kicker select left.
2. Keeperâs payoff will be 42% Ă đ + 5% Ă (1 â đ)
3. Next, change rivalâs strategy. Say kicker â Right.7% Ă đ + 30% Ă (1 â đ)
4. The mix probability that Kicker feel indifferent is:42% Ă đ + 5% Ă 1 â đ = 7% Ă đ + 30% Ă (1 â đ)
5. Solve for đ and we will get
Mix Strategy đ =5
12
Left 20.42%
Right 20.42%
Mixed StrategyThe result is:
Mix Strategy đ =5
12
Kicker Left 20.42%
Kicker Right 20.42%
Mix Strategy Keeper Left Keeper Right
đ =23
6079.6% 79.6%
Keeperâs payoff
Kickerâs payoff
This is zero-sum game, so kickerâs payoff + keeperâs payoff = 100%
Game Theory in dynamic(=sequential) setting and Strategic Move: Perfect but incomplete information game
[advanced]Information of Game TheoryComplete / Incomplete information
InformationPerfect
-knows full history Imperfect:
Simultaneous games
Complete:Knows the
rules, payoffTennis, soccer (amateur-level)
Rock - Paper â ScissorsTennis, soccer (pro-level)
Sealed bid auction
Incomplete Price negotiation of used cars Hiring talents
âDynamic Gameâ (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 playersâ behavior tomorrow.
⢠Remember the topic we already dealt was âStaticâ = Simultaneous game.
âDynamic Gameâ
Also called, extensive form.
Different timing of strategic choice
Multiple decision points
Notation â Game Trees / extensive form
Morning You
Wake up!Sleep
foreverWake up!
Sleepforever
(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 startto end.
Not every path is an equilibrium path.
You before sleep
Set Alarm Donât set Alarm
Notation â Game Trees / extensive form
Morning You
Wake up!Sleep
foreverWake up!
Sleepforever
(before, morning)
(10, 0) (0, 10)(-2, -5)(8, -1)
Information setYou before sleep
Set Alarm Donât set Alarm
Notation â Game Trees / extensive formWhat is a difference of these two âinformation setâ?
Morning You should select your action, Without knowing what option You Before sleep select.Say, âI forgot set alarm or not. Should I wake up nowâŚ?â
Morning You
Wake up!Sleep
foreverWake up!
Sleepforever
(10, 0) (0, 10)(-2, -5)(8, -1)
You before sleep
Set Alarm Donât set Alarm
Morning You
Wake up!Sleep
foreverWake up!
Sleepforever
(10, 0) (0, 10)(-2, -5)(8, -1)
You before sleep
Set Alarm Donât set Alarm
You know you set alarm or not.And you will decide wake up or not.
Notation â Game Trees / extensive formSo, what does it mean?Player n+1 will explain to Player n.
You
Your Rival
Rock PaperScissors
R PS R PSR PS
Notation â Game Trees / extensive formThis 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.
Notation â Game Trees / extensive formCan 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
SubgameSubgame: A smaller part of the whole game starting from any one node and continuing tothe 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
SubgameSubgame: A smaller part of the whole game starting from any one node and continuing tothe 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) (-â, -â)(-100, 1M)
PILOT
Obey Hijacker Resist Hijacker
Think Backward.
How to find an Equilibrium on Dynamic Game?
HIJACKER
Success! Give in Explode
(100, -50) (-â, -â)(-100, 1M)
PILOT
Obey Hijacker Resist Hijacker
Think Backward.Focus this subgame first,(We assume Hijacker is enough rational.)
And we will think it âBackwardâ which means,We will think the subgame first. So, Give in = (100, â50) > Explode = (ââ, ââ)
So, pilot should only compare the result of the subgameSuccess (â100, â1đ) and, Give in (100, â50)
So, PILOT should resist.
This way of thinking is called Backward induction:ruling out the actions that players would not play ifthey were actually given a chance to choose.
subgame
ReferenceNUS 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_EconomicsAmazon.com
https://www.youtube.com/watch?v=ErJNYh8ejSA
The Art of Strategy
http://ykamijo.web.fc2.com/lecture1.html