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 Theoryfrom 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, 𝐵𝛼) (𝐴1, 𝐵𝛽)


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 𝑨𝟏 𝒇𝒐𝒓 𝑨, 𝑎𝑛𝑑𝑩𝜶 𝒇𝒐𝒓 𝑩

Example – The Price Wars

Price ↑ Price ↓

Price ↑ + +,+ + − −,++ +

Price ↓ ++ +,− − −, −

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 players’ 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 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)

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 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.


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 – 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)


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• Nash Equilibrium

• 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



Step2: Fix B’s strategyas Price ↑


Price ↑ Price ↓

Price ↑ + +,+ + − −,++ +

Price ↓ ++ +,− − −, −

Store A

Store B




Step3: Compare only A’sPayoff. And decide whichone would A choose.

A would choose Price↓If A knows B choose ↑


Price ↑ Price ↓

Price ↑ + +,+ + − −,++ +

Price ↓ ++ +,− − −, −

Store A

Store B




Step3: Compare only A’sPay off. And decide whichone would A choose.

Step4: Move to B’s next strategy. Price ↓


Price ↑ Price ↓

Price ↑ + +,+ + − −,++ +

Price ↓ ++ +,− − −, −

Store A

Store B




A would choose Price↓If A knows B choose ↓


Price ↑ Price ↓

Price ↑ + +,+ + − −,++ +

Price ↓ ++ +,− − −, −

Store A

Store B




A would choose Price↓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


Step6: Change Player.We are Store B!


Price ↑ Price ↓

Price ↑ + +,+ + − −,++ +

Price ↓ ++ +,− − −, −

Store A

Store B


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




Step 9: Fix A’s strategy as Price ↓


Price ↑ Price ↓

Price ↑ + +,+ + − −,++ +

Price ↓ ++ +,− − −, −

Store A

Store B




Step 9: Fix A’s strategy as Price ↓Step 10: Compare only B’s Payoff.B would prefer Price↓

So, B will choose Price↓ regardless A is ↑ or ↓This is called “Dominant Strategy”


Price ↑ Price ↓

Price ↑ + +,+ + − −,++ +

Price ↓ ++ +,− − −, −

Store A

Store B


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 (+ +,+ +), 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 “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=1Also, 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=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 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 Playerwhich 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• 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”



• 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.

• 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 Prisoners’ Dilemma and they are better off!!!

Government regulation



• Pre- Nash• Dominant Strategy Equilibrium

• Iterated Dominance Equilibrium← What’s NEXT



• Mixed Strategy


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!

We have only 3 options: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


Step1: We are Player A!

We have only 3 options: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


Step1: We are Player A!Step2: Fix your rival’s strategy. Say, Left.

- Top and Middle areworse 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


Step1: We are Player A!Step2: Fix rival’s strategy.Step3: change rival’s STR.Say, Center.

- Top and Bottom areworse 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


Step1: We are Player A!Step2: Fix rival’s strategy.Step3: change rival’s STR.Say, Right.

- Top and Bottom areworse 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


Step1: We are Player A!Step2: Fix rival’s strategy.Step3: change rival’s STR.Step4: Find a dominant orDominated 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


Step1: We are Player A!Step2: Fix rival’s strategy.Step3: change rival’s STR.Step4: Find a dominant orDominated 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 / Bottomare Dominated/DominantSTR.

So Move to Player B.

As Player B,We should focus only onPlayer 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!

We have only 3 options: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 thanLeft 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


Step6: We are Player B!Step7: Fix Rival’s strategy.Step8: Change Rival’s STR.Say, Bottom.

-Right is better thanLeft 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


Step6: We are Player B!Step7: Fix Rival’s strategy.Step8: Change Rival’s STR.Step9: Find a dominant orDominated 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


Step6: We are Player B!Step7: Fix Rival’s strategy.Step8: Change Rival’s STR.Step9: Find a dominant orDominated 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


Step6: We are Player B!Step7: Fix Rival’s strategy.Step8: Change Rival’s STR.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 dominanceequilibrium

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 butMiddle STR.

So, as Player B, you could choose Proper one.

Comparing 東east = 3南south = 2And 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 comparingnorth, 1, 1, 1

And others. Especially south.south, 2, 2, 5

And we know north is dominated by south.





• Maxmin Strategy Equilibrium ← Let’s talk about it!


• Mixed Strategy


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 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 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:


• Player1 explain how to solve it (process)(2 min)



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

② MAXMIN Strategy:


② MAXMIN Strategy:As the Graph Shows,For A: MIN(A1)=8MIN(A2)= –10MAX(MIN(A1), MIN(A2))→ A1

For B:MIN(B1)=4MIN(B2)=10MAX(MIN(B1), MIN(B2))→B2

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







• Mixed Strategy


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.






• Nash• Nash Equilibrium ← So finally, “Beautiful Mind”

• Mixed Strategy


“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”

• Nash Equilibrium• You are already given the strategy 𝐴𝑖 , 𝐵𝑗 , 𝐶𝑘 , … . .

• 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 BHere 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.

Nash Equilibrium finding• 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 BStep2: 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

• So, How to find a Nash Equilibrium? More efficient?

[5 min - Workshop]


• 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


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 EquilibriumBecause 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 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.


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 NashYou 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


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.


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.







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


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 “you always use paper”.You surely lose. So we need to “Mix Strategies.” 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 – 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 – 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 “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 yourinflexibility.

Mixed StrategySo, let’s “mix” your strategy.To mix your strategy it is important (i) randomize, and (ii) maximize your payoff or minimize rival’s 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 “minimum payoff” if 𝑝 = 0.5, 0.4

Leftq

Right1-q

Leftp

58%, 42% 95%, 5%

Right1-p

93%, 7% 70%, 30%

kic

ker

keeper[5 min - Workshop]


• Player1 explain how to solve it (process)(2 min)




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 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.



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 rival’s strategy again.


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%



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 rival’s strategy again.


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%.

Mix Strategy Left Right

𝑃 = 0.5 75.5% 82.5%

𝑃 = 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

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 when58% × 𝑝 + 93% × 1 − 𝑝 = 95% × 𝑝 + 70% × 1 − 𝑝

↔ 𝑝 =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

𝑃 = 0.5 75.5% 82.5%

𝑃 = 0.4 79% 80%

𝑝 =23

6079.6% 79.6%

Mixed StrategySuppose you are keeper. Find your “q”

Leftq

Right1-q

Leftp

58%, 42% 95%, 5%

Right1-p

93%, 7% 70%, 30%

kic

ker

keeperMix Strategy 𝑞 = ?

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 “q”

Leftq

Right1-q

Leftp

58%, 42% 95%, 5%

Right1-p

93%, 7% 70%, 30%

kic

ker

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 StrategyThe result is:

Mix Strategy 𝑞 =5

12

Kicker Left 20.42%

Kicker Right 20.42%

Mix Strategy Keeper Left Keeper Right

𝑝 =23

6079.6% 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 TheoryComplete / Incomplete information

InformationPerfect

-knows full history Imperfect:

Simultaneous games

Complete:Knows the

rules, payoffTennis, soccer (amateur-level)


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

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 Don’t set Alarm

Notation – 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


Notation – Game Trees / extensive formWhat 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

foreverWake up!

Sleepforever

(10, 0) (0, 10)(-2, -5)(8, -1)

You before sleep


Morning You

Wake up!Sleep

foreverWake up!

Sleepforever

(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.

Notation – 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 – 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 – 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) (-∞, -∞)(-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 subgameSuccess (−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 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

https://courses.edx.org/courses/course-v1:MITx+14.100x+3T2016/course/

https://en.wikiquote.org/wiki/Greg_Mankiw#Ch._1._Ten_Principles_of_Economics

https://www.youtube.com/watch?v=ErJNYh8ejSA

http://ykamijo.web.fc2.com/lecture1.html

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Economy & Finance