0 0 0 1 1 0 10 10 Swerve Hang Tough Swerve Hang Tough Player 2 Pllayer 1 Does either player have a dominant strategy A Yes B No ID: 562503
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Slide1
Clicker Question-A Chicken Game
0, 0 0, 1 1, 0 -10, -10
Swerve
Hang Tough
Swerve
Hang Tough
Player 2
Pllayer
1
Does either player have a dominant strategy?
A) Yes
B) NoSlide2
Nash Equilibrium in Chicken Game?
0, 0 0, 1 1, 0 -10, -10
Swerve
Hang Tough
Swerve
Hang Tough
Player 2
Pllayer
1
How many Nash
equilibria
does this game have?
None
Exactly one
At least twoSlide3
Monte Hall Problem
Let’s Draw a Game Tree…
Problem 6, chapter 2Slide4
Information sets in Monte Hall game
In last move, contestant knows which door he chose and which Monte opened. The door that Monte opened is neither the one that contestant opened nor the one with the good prize.Six things contestant might see.
1/ 2, 1/ 3, 2/ 1, 2/ 3, 3/ 1, 3/ 2. Slide5
Finding information sets
If contestant sees 1/ 2, then the node might be 1/ 1 /2 or it might be 3/ 1/ 2. Why couldn’t it be 2/ 1/ 2?So one information set is {1/1/2,3/1/2}.What are the others?Slide6Slide7
HW Problem 2.11
Perfect Recall?Slide8
Problem 8
Figure PR2.8
Harrington: Games, Strategies, and Decision Making, First EditionCopyright © 2009 by Worth PublishersSlide9
Nash EquilibriumSlide10
Clicker Question 1:
6,0
3,24,12,32,32,81,12,05,2
Player 1
Strategy A1
Strategy B1Strategy C1
Strategy A2 Stategy B2 Strategy C2
Is the outcome where Player 1 plays B1 and Player 2 plays C2 a Nash equilibrium?
YesNoSlide11
Clicker Question 2:
6,0
3,24,12,32,32,81,12,05,2
Player 1
Strategy A1
Strategy B1Strategy C1
Strategy A2 Stategy B2 Strategy C2
Is the outcome where Player 1 plays A1 and Player 2 plays B2 a Nash equilibrium?
YesNoSlide12
Definition
A strategy profile is a Nash Equilibrium if each player’s strategy maximizes his payoff given the strategies used by the other players.Slide13
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.Slide14
Prisoners’ Dilemma Game
10, 10 0, 11 11, 0 1, 1Cooperate
Defect
Cooperate
Defect
PLAyER 1
Player 2Slide15
Battle of Sexes
2,1
0,0 0,0 1,2BobAliceMovie A
Movie B
Movie A
Movie B
BRA
(A)=ABRA(B)=B
BRB(A)=ABRB(B)=BSlide16
Best Responses and Nash
Equilibria for this game?BR2(a)=z BR1(w)=bBR2(b)={w,x,z} BR1(x)=bBR
2(c)=y BR1(y)=b
BR2(d)={y,z} BR
1(z)={a,d} Slide17
Find Nash equilibria
for these gamesChickenPure coordination (Driving Game)Slide18
How many Nash equilibria
(in pure strategies)?
There might be just one.There might be more than one.There might not be any.Slide19
Rock, Paper Scissors,
Where is Nash equilibrium?Slide20
When is Nash equilibrium “the right answer”?
Players are “rational”. Each player’s strategy maximizes his payoff, given his beliefs about the strategies used by the other players.
Each player’s beliefs about the other players’ strategies are correct.When is 2) a reasonable assumption?Slide21
3-Hunter Stag Hunt
2,2,2
0,1,01,0,01,1,0Hunter 3 does StagHunter 3 does HareHunter 2
Stag
Hare
Stag
Hare
Hunter 1
0,0,1
0,1,1
1,0,11,1,1
Hunter 2
Stag
Hare
Stag
Hare
Find the Nash
equilibria
Hunter 1Slide22
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.)Slide23
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.Slide24
Clicker Question 3:
6,0
3,24,12,32,32,81,12,05,2
Player 1
Strategy A1
Strategy B1Strategy C1
Strategy A2 Stategy B2 Strategy C2
Does this game have more than one Nash equilibrium?
YesNoSlide25
Clicker Question:
What are the Nash
Equilibria for this game?A) Player 1 plays a and Player 2 plays z.Player 1 plays d and Player 2 plays z.Player 1 plays b and Player 2 plays y.Both outcomes A) and B) are Nash equilibria.There are no Nash
equilibria
4,1