Theory of

Lecture 7

Strategic InteractionGame Theory

To cover formal notation in game theory

Lecture 7AimAimTo understand the definitions of:

+ Dominance

+ Best Response

+ Nash Equilibrium

+ Pareto Dominance

Game Theory

Put yourself in other’s shoes to try & figure out what they are going to do

Rule from past Lecture

We also know from previous lectures that Game Theory has real-world relevance

It’s outcomes relate to social phenomena

Lets do some formal stuff

Pick a Number

Practical 9

Some notation

Players i , j

Ingredients of a gameWhat formally makes something a game?

Strategies si

particular strategy for player i

Si

Set of all possible strategies for player i

all of you

Numbers Game

13

{1,2,3,...,100}

sparticular play of the game

{s1, s2, s3,..., s12}

Strategy Profile

Some notation

Payoffs

One more ingredient

ui(s1,...,si,...,s12)= ui(s)

Numbers Game

ui(s)= 50-error, if win

0, otherwiseOthers Strategy s-i

everyone’s choice except i’s

Some notation

Payoffs

One more ingredient

ui(s1,...,si,...,s12)= ui(s)

Numbers Game

ui(s)= 50-error, if win

0, otherwiseOthers Strategy s-i

everyone’s choice except i’s

For those of you who are Math-phobic

KEEP CALM

it’sJUST

NOTATION

Example

Players

Think of

5,-1 11,3 0,0

6,4 0,2 2,0

Top

Bottom

Left Cent Right

1

2

Strategies

Payoffs

Players

Does 1 has a dominated strategy?5,-1 11,3 0,0

6,4 0,2 2,0

T

B

L C R

Strategies

Payoffs

1 & 2

s1={T,B}

s2={L,C,R}

u1(T,C)=11

u2(T,C)=3

Example

Players

Does 1 has a dominated strategy?5,-1 11,3 0,0

6,4 0,2 2,0

T

B

L C R

Strategies

Payoffs

1 & 2

s1={T,B}

s2={L,C,R}

u1(T,C)=11

u2(T,C)=3

No. Player one doesn’t have one

Does 2 has a dominated strategy?

Example

Players

Does 1 has a dominated strategy?5,-1 11,3 0,0

6,4 0,2 2,0

T

B

L C R

Strategies

Payoffs

1 & 2

s1={T,B}

s2={L,C,R}

u1(T,C)=11

u2(T,C)=3

No. player one doesn’t have one

Does 2 has a dominated strategy?

Yes. C dominates R

Example

Player i’s strategy si’ is strictly dominated by player i’s strategy si if

Same definition as last time, a little more formal

Definition

ui(si, s-i) > ui(si’, s-i) for all s-i

5,-1 11,3 0,0

6,4 0,2 2,0

T

B

L C R

Best Response

What are the payoffs for player 1

ui(T,L)=5If 2: Left

5,-1 11,3

6,4 0,2

T

B

L C

5,-1 11,3 0,0

6,4 0,2 2,0

T

B

L C R

If 2: Cent

ui(B,L)=6

ui(T,C)=11 ui(B,C)=0

What are the payoffs for player 1

ui(T,L)=5If 2: Left

5,-1 11,3

6,4 0,2

T

B

L C

5,-1 11,3 0,0

6,4 0,2 2,0

T

B

L C R

If 2: Cent

ui(B,L)=6

ui(T,C)=11 ui(B,C)=0If 2 choose L, player 1 is better with B

If 2 choose C, player 1 is better with T

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Best Response

Think of a strategy that is the best you can do, given your belief about

what the other person will do

Formal Definition:

Best Response

Player i’s strategy si* is a Best Response (BR) to the strategy s-i of the other player if

ui(si*, s-i) > ui(si’, s-i) for all si’ in si

Player 1

Best Response

5,-1 11,3

6,4 0,2

T

B

L C

T is a BR to CB is a BR to L

Player 2

C is a BR to TL is a BR to B

Rule 5:

Do not play a strategy that is not a best response

What happens

5,-1 11,3

6,4 0,2

T

B

L C If 2 choose C

player 1 will best respond to C with T

The players are playing a best response to each other

In (T,C) or (B,L)

If 1 choose T

player 2 will best respond to T with C

If they reach this point, neither wants to play something different if the other stays the same

John F. Nash

Nash Equilibrium

http://www.youtube.com/watch?v=2d_dtTZQyUM

Lets check out a video

Nobel Prize 1994

Formal Definition:

Nash Equilibrium

A strategy profile (s1*, s2*,..., sN*) is a Nash equilibrium (NE) if for each i, her choice is a

best response to the other players’ choices s-i*

By far the most commonly used solution concept in game theory

Although we have seen before that in many cases people don’t play a Nash equilibrium

Then why look at a Nash equilibrium?

No regrets:

Motivation

No individual can do better by deviating (changing her behavior)Do I regret my actions? NO

Self-fulfilling beliefs:If everyone beliefs that the others are going to best respond, then everyone will play their best response to it

Nash Equilibrium

5,-1 11,3

6,4 0,2

T

B

L C

The combination of strategies (T,C) or (B,L) are part of the set

of Nash equilibria

NE={(T,C),(B,L)}

Think about the games we have played so far

Do they have more than 1 equilibrium?

Practical 10Battle of the Sexes

Multiplicity

What are the best responses for player 1?

It is not always unique

One main critique to Nash equilibrium

10,7 0,0

0,0 7,10

A

B

A B

Example: Battle of the sexes

Multiplicity

What are the best responses for player 1?

It is not always unique

One main critique to Nash equilibrium

10,7 0,0

0,0 7,10

A

B

A B

Example: Battle of the sexes

What are the best responses for player 2?A if A & B if B

Multiplicity

What are the best responses for player 1?

It is not always unique

One main critique to Nash equilibrium

10,7 0,0

0,0 7,10

A

B

A B

Example: Battle of the sexes

What are the best responses for player 2?A if A & B if B

A if A & B if B

It is not clear which one will be chosen

Pareto Dominance

(A,A) & (B,B) are Nash equilibria

Social Welfare - Efficiency

One final concept - Link to society

2,2 0,1

1,0 1,1

A

B

A B

Example: Stag Hunt

(A,A) Pareto Dominates (B,B)

Good & bad equilibrium

It is a state of allocation of resources (payoffs) in which it is impossible to make any one individually better off without making at least one individual worse off

Checklist

Best response is the best action you can choose given what others choose

Do not play a strategy that is not a BR

Nash equilibrium is a state where all players are best responding to each other

Nash equilibrium is not always unique, and there are good and bad equilibria

Questions?