2. To cover formal notation in game theory
Lecture 7AimAim
To understand the definitions of:
+ Dominance
+ Best Response
+ Nash Equilibrium
+ Pareto Dominance
3. 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
5. Some notation
Players i , j
Ingredients of a game
What 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}
s
particular play of the game
{s1, s2, s3,..., s12}
Strategy Profile
6. Some notation
Payoffs
One more ingredient
ui(s1,...,si,...,s12)= ui(s)
Numbers Game
ui(s)= 50-error, if win
0, otherwise
Others
Strategy s-i
everyone’s choice except i’s
7. Some notation
Payoffs
One more ingredient
ui(s1,...,si,...,s12)= ui(s)
Numbers Game
ui(s)= 50-error, if win
0, otherwise
Others
Strategy s-i
everyone’s choice except i’s
For those of you who are Math-phobic
10. 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
11. 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
12. 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
13. 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
14. 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
15. 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
If 2 choose L, player 1 is better with B
If 2 choose C, player 1 is better with T
<
>
Best Response
16. 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
17. Player 1
Best Response
5,-1 11,3
6,4 0,2
T
B
L C
T is a BR to C
B is a BR to L
Player 2
C is a BR to T
L is a BR to B
Rule 5:
Do not play a strategy that is not a
best response
18. 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
19. John F. Nash
Nash Equilibrium
http://www.youtube.com/watch?v=2d_dtTZQyUM
Lets check out a video
Nobel Prize 1994
20. 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?
21. 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
22. 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?
24. 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
25. 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
26. 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
27. 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
28. 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
