Monte Hall Problem

Let’s Draw a Game Tree…

Problem 6, chapter 2

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.

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?

HW Problem 2.11

Perfect Recall?

Nash Equilibrium

Clicker Question 1:

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 2:

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

Definition

A strategy profile is a Nash Equilibrium if each player’s strategy maximizes his payoff given the strategies used by the other players.

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?

The Gingrich-Santorum game?

Textbook example. 3 candidates. Winner takes all. Strategy positive effort +1 for self or negative effort -2 for smeared candidate.Starting point. Candidate 1 has initial score 2, candidates 2 and 3 have initial scores 0. Nash equilibria. All positive. Candidates 2 and 3 both smear 1.Other Nash equilibria?

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 3:

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

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