Download - SN- Lecture 7
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Theory of
Lecture 7
Strategic InteractionGame Theory
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To cover formal notation in game theory
Lecture 7AimAimTo understand the definitions of:
+ Dominance
+ Best Response
+ Nash Equilibrium
+ Pareto Dominance
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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
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Pick a Number
Practical 9
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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
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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
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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
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KEEP CALM
it’sJUST
NOTATION
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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John F. Nash
Nash Equilibrium
http://www.youtube.com/watch?v=2d_dtTZQyUM
Lets check out a video
Nobel Prize 1994
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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?
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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
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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?
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Practical 10Battle of the Sexes
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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
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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
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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
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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
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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
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Questions?