Check your (Mis)understanding?Number 3.5 page 79

Answer Key claims that: • For player 1 a strictly dominates c• For player 2, y strictly dominates w and x.These claims are correct.The key claims that • if player 1 knows that player 2 will not use w or x, then a dominates b for player 1.This claim is not true!

Moral of the Story?

• Question authority.• Even expensive, well-written textbooks make

mistakes.• So do instructors.

Nash Equilibrium

The real John Nash Hollywood’s Version

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) YesB) No

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?

A) NoneB) Exactly oneC) At least two

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

6,0 3,2 4,12,3 2,3 2,81,1 2,0 5,2Player 1

Strategy A1

Strategy B1

Strategy C1

Strategy A2 Stategy B2 Strategy C2

Is the outcome where Player 1 plays B1 and Player 2 plays C2 a Nash equilibrium?A) YesB) No

Clicker Question

6,0 3,2 4,12,3 2,3 2,81,1 2,0 5,2Player 1

Strategy A1

Strategy B1

Strategy C1

Strategy A2 Stategy B2 Strategy C2

Is the outcome where Player 1 plays A1 and Player 2 plays B2 a Nash equilibrium?A) YesB) 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

10, 10 0, 11

11, 0 1, 1

Cooperate Defect

Cooperate

Defect

PLAyER 1

Player 2

Battle of Sexes

2,1 0,0

0,0 1,2

Bob

Alice

Movie A

Movie B

Movie A

Movie B

BRA(A)=ABRA(B)=B

BRB(A)=ABRB(B)=B

Best Responses and Nash Equilibria for this game?

BR2(a)=z BR1(w)=bBR2(b)={w,x,z} BR1(x)=bBR2(c)=y BR1(y)=bBR2(d)={y,z} 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

2,2,2 0,1,0

1,0,0 1,1,0

Hunter 3 does Stag Hunter 3 does Hare

Hunter 2

Stag Hare

Stag

Hare

Hunter 1

0,0,1 0,1,1

1,0,1 1,1,1

Hunter 2

Stag

Hare

Stag Hare

Find the Nash equilibria

Hunter 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

6,0 3,2 4,12,3 2,3 2,81,1 2,0 5,2Player 1

Strategy A1

Strategy B1

Strategy C1

Strategy A2 Stategy B2 Strategy C2

Does this game have more than one Nash equilibrium?A) YesB) No

Game of previous slidereduced 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?

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

4,1

A coordination game

• You choose one of three parties to go to Party X, Party Y, Party ZYou 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. Aftereach 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

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?