Clicker Game:
Choose one of the numbers below. You will get 1 point
if your number is the closest number to 3/4 of the
average of the numbers chosen by all class members,
otherwise you will get 0 points…
•
•
•
•
•
A) 6
B) 5
C) 4
D) 3
E) 2
Iterated Deletion of Dominated
Strategies
• Would a smart player choose 6?
• If you believe that everybody in the class is
smart, would you ever choose 5?
• If you believe that everybody in the class is
smart and believes everybody else is smart,
what would you conclude?
• What if you take this another step?
Nash Equilibrium
The real John Nash
Hollywood’s Version
Clicker Question-A Chicken Game
Player 2
Swerve
Swerve
Hang Tough
0, 0
0, 1
1, 0
-10, -10
Pllayer 1
Hang Tough
Does either player have a dominant strategy?
A) Yes
B) No
Nash Equilibrium in Chicken Game?
Player 2
Swerve
Swerve
Hang Tough
0, 0
0, 1
1, 0
-10, -10
Pllayer 1
Hang Tough
How many Nash equilibria does
this game have?
A) None
B) Exactly one
C) More than one
Definition
A strategy profile is a Nash Equilibrium if each
player’s strategy maximizes his payoff given the
strategies used by the other players.
Clicker Question
Strategy A2 Stategy B2 Strategy C2
Strategy A1
Strategy B1
Player 1
Strategy C1
6,0
2,3
1,1
3,2
2,3
2,0
4,1
2,8
5,2
Is the outcome where Player 1 plays B1 and Player
2 plays C2 a Nash equilibrium?
A) Yes
B) No
Clicker Question
Strategy A2 Stategy B2 Strategy C2
Strategy A1
Strategy B1
Player 1
Strategy C1
6,0
2,3
1,1
3,2
2,3
2,0
4,1
2,8
5,2
Is the outcome where Player 1 plays A1 and Player
2 plays B2 a Nash equilibrium?
A) Yes
B) No
Best response mapping
Best response for a player is a mapping from
actions by the others to the action (or actions)
that maximizes the player’s payoffs given the
actions of the others.
In Nash equilibrium, every player is doing the
best response to what the other players are
doing.
Prisoners’ Dilemma Game
Player 2
Cooperate
P
L
A Cooperate
y
E
R
1
Defect
Defect
10, 10
0, 11
11, 0
1, 1
Battle of Sexes
Bob
Movie A
Movie A
Alice
Movie B
BRA(A)=A
BRA(B)=B
Movie B
2,1
0,0
0,0
1,2
BRB(A)=A
BRB(B)=B
Best Responses and Nash Equilibria for this game?
BR2(a)=z
BR2(b)={w,x,z}
BR2(c)=y
BR2(d)={y,z}
BR1(w)=b
BR1(x)=b
BR1(y)=b
BR1(z)={a,d}
Find Nash equilibria for these games
• Chicken
• Pure coordination (Driving Game)
How many Nash equilibria
(in pure strategies)?
There might be just one.
There might be more than one.
There might not be any.
Rock, Paper Scissors,
Where is Nash equilibrium?
When is Nash equilibrium
“the right answer”?
1. Players are “rational”. Each player’s strategy
maximizes his payoff, given his beliefs about
the strategies used by the other players.
2. Each player’s beliefs about the other players’
strategies are correct.
When is 2) a reasonable assumption?
3-Hunter Stag Hunt
Hunter 3 does Hare
Hunter 3 does Stag
Hunter 2
Hunter 2
Stag
2,2,2
0,1,0
Stag
Hunter 1
Hare
Stag
Hare
1,0,0 1,1,0
Find the Nash equilibria
Stag
Hunter 1
Hare
Hare
0,0,1 0,1,1
1,0,1 1,1,1
Weakly dominated strategies?
• Nobody will use a strictly dominated strategy
in Nash equilibrium.
• If there is a strictly dominant strategy for all
players, it is a Nash equilibrium. (example
Prisoners’ Dilemma.)
• Nash equilibrium does not exclude possibility
of using a weakly dominated strategy. (A
voting example with unanimous preferences.)
Nash and domination
Every Nash equilibrium survives the iterated
elimination of strictly dominated strategies.
Not every outcome that satisfies the iterated
elimination of strictly dominated strategies is a
Nash equilibrium.
Clicker Question
Strategy A2 Stategy B2 Strategy C2
Strategy A1
Strategy B1
Player 1
Strategy C1
6,0
2,3
1,1
3,2
2,3
2,0
4,1
2,8
5,2
Does this game have more than one Nash
equilibrium?
A) Yes
B) No
Game of previous slide
reduced by IDSDS
Strategy B2 Strategy C2
Strategy A1 3,2
4,1
Strategy C1 2,0
5,2
Find the Nash equiibria for reduced game. These must be
Nash equilibria for the original game.
Note that strategy profiles C1,B2 and A1,C2 are not
Eliminated by IDSDS, but are not Nash equillibria.
Clicker Question:
What are the Nash Equilibria for this game?
4,1
A) Player 1 plays a and Player 2 plays z.
B) Player 1 plays d and Player 2 plays z.
C) Player 1 plays b and Player 2 plays y.
D) Both outcomes A) and B) are Nash equilibria.
E) There are no Nash equilibria
A coordination game
• You choose one of three parties to go to
Party X, Party Y, Party Z
You like big parties and your payoff will be the
number of people who attend the same party
that you do.
We will play this repeatedly with clickers. After
each round, you will see how many people
chose each option. Then you play again.
Which Party do you choose?
• A) Party Y
• B) Party X
• C) Party Z
Remember to
Your
On
Fictional play version
• You do the best response given the average of
previous responses.
• Will this converge?
• If it converges, it converges to Nash
equilibrium. Why?