The Random Dresser (Clicker Question) Wilbur has – 3 left shoes, all of different colors – 5 right shoes, all of different colors – 4 right gloves, all.

The Random Dresser(Clicker Question)

• Wilbur has– 3 left shoes, all of different colors– 5 right shoes, all of different colors– 4 right gloves, all of different colors – 2 left gloves of different colors.

• How many different ways can Wilbur dress himself?

A) 14 B) 28 C) 48 D) 120 E) 240

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.

Information sets

• {1/1/2, 3/1/2}• {1/2/3, 2/2/3}• {1/3/2, 3/3/2}• {2/1/3, 1/1/3}• {2/2/1, 3/2/1}• {2/3/1, 3/3/1}

To stay or to switch: that is the question.

• If your initial choice was right, then you win if you stay and lose if you switch.

• If your initial choice was wrong, you lose if you stay and you win if you switch.

• What is the probability that your initial choice was right?

• What is probability you win if you stay?• What is probability you win if you switch?

Dominant strategies

Clicker Question

1 , 3 5, 3

2, 4 7, 2

Player 2Strategy A Strategy B

Player 1

Strategy A

Strategy B

A ) Strategy A strictly dominates Strategy B for both Players.

B) Strategy B strictly dominates A for Player 1. Strategy A weakly dominates B for Player 2.

C) Strategy B strictly dominates A for Player 1. Strategy A strictly dominates B for Player 2.

D) Strategy B strictly dominates Strategy A for both players.

E) No strategy in this game is strictly dominated

Strict and Weak Dominance

• Strategy A strictly dominates strategy B for a player if that player gets a higher payoff from doing A than from doing B no matter what the other player(s) do.

• Strategy A weakly dominates strategy B for a player gives at least as high a payoff no matter what the other player(s) do and for some actions of the others gives a higher payoff.

Clicker Question

2,2 0, 3

3, 0 1,1

Player 2Strategy A Strategy B

Player 1

Strategy A

Strategy B

A ) Strategy A strictly dominates Strategy B for both Players.

B) Strategy B strictly dominates A for Player 1. Strategy A weakly dominates B for Player 2.

C) Strategy B strictly dominates A for Player 1. Strategy A strictly dominates B for Player 2.

D) Strategy B strictly dominates Strategy A for both players.

E) No strategy in this game is strictly dominated

Game Theory Doctrine (A tautology)

• A rational player who understands the payoffs of a game and who tries to maximize his own payoff will

A) never use a strictly dominated strategy. B) will always use a strictly dominant strategy if

one exists.

Dominant strategies?

1 0 , 10 0, 11

1 1, 0 1, 1

Strategy A

Strategy A Strategy B

Strategy B

Player 1

Player 2

Does either strategy strictly dominate the other for Player 1? Does either strategy strictly dominate the other for Player 2? What is the predicted outcome? What are games like this called?

How about this one?

1 0 , 10 0, 10

1 0, 0 1, 1

Strategy AStrategy A Strategy B

Strategy BPlayer 1

Player 2

Does either strategy strictly dominate the other for Player 1? Does either strategy weakly dominate the other for Player 1? How about player 2?

Clicker Question

1 0 , 10 0, 10

1 0, 0 1, 1

Strategy AStrategy A Strategy B

Strategy BPlayer 1

Player 2

If I were playing this game just once with a stranger whom I would never meet again, I would:A) Play Strategy AB) Play Strategy B

Rousseau’s Stag Hunt

2 , 2 0 , 1 1 , 0 1 , 1

StagStag Hare

HarePlayer 1

Player 2

Are any strategies weakly dominated?

Are any strategies strictly dominated?How would you play?

Clicker Question

2 , 2 0 , 1 1 , 0 1 , 1

StagStag Hare

HarePlayer 1

Player 2

If you were playing Rosseau’s stag hunt with astranger, whom you will never meet again, which strategy would you play? A) Stag B) Hare

Gaming Pigs(Iterated dominance)

Are there dominated strategies for Big Pig?

How about Little Pig?

How would you “solve” this game?

What went on in the pigpen

The Entry Game

Challenger

Stay out

0 Challenger’s payoff1 Incumbent’s payoff

Challenge

Incumbent

Give in Fight

10

-1 -1

Challenger’s payoffIncumbent’s payoff

Strategic Form of Entry Game

0,1 0,1

1,0 -1,-1

Give in Fight

Stay out

Enter

Challenger

Incumbent

Dominance in Entry Game?

• No dominant strategy for Challenger. Which is better depends on what incumbent will do.

• Give-in is weakly dominant for Incumbent.

• If Challenger believes that Incumbent is rational, Challenger believes that Incumbent will give in.

• In this case, predicted outcome is Challenger enters and incumbent gives in.

What if incumbent could precommit?

• Could the incumbent make a credible threat to fight if challenger enters. If he could, he could get challenger to stay out.

• On blackboard we will draw a game that allows incumbent the choice of committing to to be badly punished if he gives in.

• Lets do this so that the “solution” is that the incumbent chooses to make this commitment and the challenger stays out.

• Tools for understanding this “solution” will arrivelater in this course.

Kidnapping with imperfect information

Strategic Form

Dominated strategies?

• Guy doesn’t have any dominated strategies• But for Vivica, Don’t Pay dominates Pay.• What does iterated elimination of dominated

strategies tell us?• If Guy knows that Vivica is rational, he knows

she won’t pay ransom.• If Guy knows that Vivica won’t pay ransom, he

is better off not kidnapping.

Kidnapping with Perfect Information

Kidnapping with complete information

Pay Ransom Don’t Pay Ransom

Kidnap-- Kill if R, Kill if NR 4,1 2,2

Kidnap—Release if R, Kill if NR 5,3 2,2

Kidnap— Kill if R, Release if NR 4,1 1,4

Kidnap—Release if R, Release if NR 5,3 1,4

Don’t Kidnap– Kill, Kill 3,5 3,5

Don’t Kidnap—Release, Kill 3,5 3,5

Don’t kidnap--Kill, Release 3,5 3,5

Don’t kidnap—Release, release 3,5 3,5

Vivica

Guy

Are any strategies strictly dominated for either player?

Dominated strategies?

• Neither strategy dominates for Vivica• For Guy, Kidnap—Release if Ransom, Kill if No

ransom weakly dominates all other strategies that start with Kidnap.

• So if Vivica believe that Guy is rational, then she believes that if Guy Kidnaps, he will kill if no ransom and release if ransom.

• So Vivica would pay ransom• So Guy would Kidnap and release after receiving

ransom.

Does Player 1 have a dominated strategy? Hint: Compare b and d.

Does Player 2 have a dominated strategy? Hint: Compare y and z.

Iterated Elimination of Dominated Strategies-Stage 1

If each knows the other won’t play a dominated strategy, we have a smaller game.

The game After first round of Eliminaton

Reduced Game after one iteration.

This is the game if each knows that the other is rational and each knows that the other knows that the other is rational.

Are there any dominated strategies?

Reduced Game after 2 rounds of iterated elimination of strictly dominated strategies. (Note that x couldn’t have been eliminated in the first round.)

Reduced Game after 3 rounds of iterated elimination.a is eliminated. This couldn’t have been done in earlier rounds.

Are there any strictly dominated strategies in this game?

We have eliminated 12 of 16 strategies, but to get any further,We’re going to need more tools.

Iterated elimination and Common Knowledge

Strategy a dominates c for Player 1. Strategy y dominates w and x for Player 2. Rational players won’t use these strategies. If each knows other is rational, then Player 2 know s that 1 won’t play c and 1 knows that 2 won’t play w or x.If both are rational and believe other is rational, Player 1 knows that 2 won’t play x or y, so Player 1 can eliminate b. Player 2 knows that Player 1 won’t play c, so Player 2 can eliminate y. If Player 1 knows that Player 2 knows that Player 1 is rational, thenPlayer 1 knows Player 2 will Play z. What will Player 1 do?

…And Steer Clear of Dominated Strategies

See you on Thursday

Clicker Question

A) No strategies are strictly dominated for Player 1. Strategy w is dominated for Player 2

B) No strategies are strictly dominated for either player.

C) Strategy c is strictly dominated for Player 1. Strategies w and x are strictly dominated for Player 2.

D) Strategy d is strictly dominated for Player 1. Strategies x and y are Strictly dominated for Player 2.

E) No strategies are strictly dominated for Player 1. Strategies x and y are Strictly dominated for Player 2.