econ 1 game theory

7/29/2019 Econ 1 Game Theory

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An introduction to gametheory

Today: The fundamentals of

game theory, including Nashequilibrium


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Today Introduction to game theory

We can look at market situations with two

players (typically firms)

Although we will look at situations whereeach player can make only one of two

decisions, theory easily extends to three ormore decisions


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Who is this?


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John Nash, the person portrayed

in A Beautiful Mind


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John Nash One of the early

researchers in

game theory

His work resultedin a form of

equilibriumnamed after him


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Three elements in every game Players

Two or more for most games that are

interesting

Strategies available to each player

Payoffs

Based on your decision(s) and thedecision(s) of other(s)


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Game theory: Payoff matrix A payoff

matrix

shows thepayout toeachplayer,

given thedecision ofeachplayer

Action C Action D

Action A 10, 2 8, 3

Action B 12, 4 10, 1

Person 1

Person 2


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How do we interpret this box? The first number in

each box determines

the payout forPerson 1

The second numberdetermines the

payout for Person 2

ActionC ActionD

ActionA

10, 2 8, 3

ActionB

12, 4 10, 1

Person1

Person 2


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How do we interpret this box? Example

IfPerson 1

chooses Action Aand Person 2chooses Action D,then Person 1

receives a payoutof8 and Person 2receives a payoutof3

ActionC ActionD

ActionA

10, 2 8, 3

ActionB

12, 4 10, 1

Person1

Person 2


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Back to a Core Principle:

Equilibrium The type of equilibrium we are looking

for here is called Nash equilibrium

Nash equilibrium: Any combination ofstrategies in which each players strategy ishis or her best choice, given the otherplayers choices (F/B p. 322)

Exactly one person deviating from a NEstrategy would result in the same payoutor lower payout for that person


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How do we find Nash

equilibrium (NE)? Step 1: Pretend you are one of the players

Step 2: Assume that your opponent picks aparticular action

Step 3: Determine your best strategy (strategies),given your opponents action Underline any best choice in the payoff matrix

Step 4: Repeat Steps 2 & 3 for any other opponent

strategies Step 5: Repeat Steps 1 through 4 for the other

player

Step 6: Any entry with all numbers underlined is NE


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Steps 1 and 2 Assume that

you are

Person 1 Given that

Person 2chooses

Action C,what isPerson 1sbest choice?

Action

C

Action D

Action

A

10, 2 8, 3

Action B 12, 4 10, 1

Person1

Person 2


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Step 3: Underline

best payout,given thechoice of the

other player

ChooseAction B,

since12 > 10underline 12

Action

C

Action D

Action

A

10, 2 8, 3

Action B 12, 4 10, 1

Person1

Person 2


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Step 4 Now

assume

that Person2 chooses

Action D

Here,

10 > 8Choose andunderline10

Action C Action

D

Action A 10, 2 8, 3

Action B 12, 4 10, 1

Person1

Person 2


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Step 5 Now,

assume youare Person 2

IfPerson 1chooses A 3 > 2

underline 3

IfPerson 1chooses B 4 > 1

underline 4

Action C Action D

Action

A

10, 2 8, 3

Action

B

12, 4 10, 1

Person1

Person 2


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Step 6

Whichbox(es) haveunderlines

under bothnumbers?

Person 1chooses B

and Person2 chooses C

This is theonly NE

Action C Action D

Action A 10, 2 8, 3

Action B 12, 4 10, 1

Person1

Person 2


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Double check our NE

What ifPerson 1deviates

from NE? Could

choose Aand get 10

Person 1spayout islower bydeviating

Action C Action D

Action A 10, 2 8, 3

Action B 12, 4 10, 1

Person1

Person 2


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Double check our NE

What ifPerson 2deviates

from NE? Could

choose Dand get 1

Person 2spayout islower bydeviating

Action C Action D

Action A 10, 2 8, 3

Action B 12, 4 10, 1

Person1

Person 2


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Dominant strategy

A strategy isdominant if thatchoice is

definitely madeno matter whatthe otherperson chooses

Example:Person 1 has adominantstrategy ofchoosing B

Action C Action D

Action A 10, 2 8, 3

Action B 12, 4 10, 1

Person1

Person 2


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New example

Suppose inthis examplethat two

people aresimultaneouslygoing todecide on thisgame

Yes No

Yes 20, 20 5, 10

No 10, 5 10, 10

Person1

Person 2


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New example

We will gothrough the

same steps todetermine NE

Yes No

Yes 20, 20 5, 10

No 10, 5 10, 10

Person1

Person 2


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Two NE possible

(Yes, Yes) and(No, No) areboth NE

Although (Yes,Yes) is the moreefficientoutcome, wehave no way topredict whichoutcome willactually occur

Yes No

Yes 20, 20 5, 10

No 10, 5 10, 10

Person1

Person 2


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Two NE possible

When there are multiple NE that arepossible, economic theory tells us little

about which outcome occurs withcertainty


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Two NE possible

Additional information or actions mayhelp to determine outcome

If people could act sequentially instead ofsimultaneously, we could see that 20, 20would occur in equilibrium


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Sequential decisions

Suppose that decisions can be madesequentially

We can work backwards to determinehow people will behave

We will examine the last decision first and

then work toward the first decision To do this, we will use a decision tree


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Decision tree in a sequentialgame: Person 1 chooses first

A

B

C

Person 1choosesyes

Person 1chooses

no

Person 2choosesyes

Person 2choosesyes

Person 2chooses no

Person 2chooses no

20, 20

5, 10

10, 5

10, 10


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Given point B,Person 2 will

choose yes(20 > 10)

Given point C,Person 2 willchoose no(10 > 5)

A

B

C

Person 1choosesyes

Person 1chooses

no

Person 2choosesyes

Person 2choosesyes

Person 2chooses no

Person 2chooses no

20, 20

5, 10

10, 5

10, 10


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IfPerson 1 isrational, she willignore potentialchoices thatPerson 2 will notmake

Example: Person

2 will not chooseyes after Person 1chooses no

A

B

C

Person 1choosesyes

Person 1chooses

no

Person 2choosesyes

Person 2choosesyes

Person 2chooses no

Person 2chooses no

20, 20

5, 10

10, 5

10, 10


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IfPerson 1 knowsthat Person 2 isrational, then shewill choose yes,since 20 > 10

Person 2 makes adecision from point

B, and he willchoose yes also

Payout: (20, 20)

A

B

C

Person 1choosesyes

Person 1chooses

no

Person 2choosesyes

Person 2choosesyes

Person 2chooses no

Person 2chooses no

20, 20

5, 10

10, 5

10, 10


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Summary

Game theory

Simultaneous decisions NE

Sequential decisions Some NE may notoccur if people are rational


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Can you think of ways game theorycan be used in these games?

econ 1 game theory

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