Taking Turns in the Dark(Subgame perfection with

imperfect information)

Econ 171

Homework from Chapter 8

Galileo and the Papal inquisition.

Strategic FormThree Players.—What are the strategies?

Confess Before Torture. Confess if tortured

Confess before torture . Do not confess if tortured

Do not confess before torture. Confess if tortured

Do not confess before torture. Do not confess if tortured

Torture

Do not torture

Galileo’s Strategy

Payoffs if Pope Refers the Case to the Inquisition

Inquisitor’sStrategy

Confess Before Torture and Confess if tortured

Confess before torture but do not confess if tortured

Do not confess before torture, confess if tortured

Do not confess before torture, do not confess if tortured

Torture

Do not torture

Galileo’s Strategy

Payoffs if Pope Does not refer the Case to the Inquisition

Inquisitor’sStrategy

Some Nash equilibria

• Pope refers, Galileo will confess before torture and will confess if tortured, Inquisitor will torture if Galileo doesn’t confess beforehand.

• Pope refers, Galileo confess before torture, would not confess if tortured, Inquisitor will torture.

• Pope doesn’t refer, Galileo will not confess before torture, wouldn’t would confess if tortured, Inquisitor would torture if G doesn’t confess.

More Nash equilibria

• Pope doesn’t refer the case. Galileo would not confess either before or after torture. Inquisitor would torture.

• Pope doesn’t refer the case. Galileo would not confess either before or after torture. Inquisitor would not torture.

Describe strategies in the subgame perfect equilibrium.

What does pope do? What does Galileo do at each of his decision nodes? What does the inquisitor do?

Piquant facts for fans of the waterboard.

• Galileo would rather confess before being tortured than be tortured.

• But if he is tortured, he would rather not confess. • Pope would like Galileo to confess without being

tortured. • Pope would also be happy if Galileo is tortured and

confesses.• But Pope would rather not refer the case if Galileo

would be tortured and not confess.• So Galileo is not brought before the Inquisition.

Describe the subgame perfect N.E.

• In subgame perfect Nash equilibrium, JS strategy is as follows:

“Do not kidnap. If you kidnap, then kill if ransom paid, kill if no ransom is paid, and release if bounty is offered.”JM’s strategy is “Offer a Bounty.”Note that what would happen out of equilibrium determines actions taken in equilibrium.

When does a lawmaker want a moderate law?

• Pick numbers so that a gentle law is enforced but not obeyed, and a severe law is neither enforced nor obeyed, but a moderate law is enforced and obeyed.

Subgame Perfection with Imperfect Information

Can the notion of subgame perfection help us if there is incomplete information?

What is a regular subtree of a game?

• It is a tree starting from one of the nodes of the game such that the subtree contains an entire information set if it contains at least one node from that set.

• A subgame is defined to be a regular subtree together with the associated payoffs.

• A proper subgame of a game is a subgame that does not contain the entire game. (by analogy to a proper subset of a set)

Subgame perfection

• In a game with imperfect information, a strategy profile is a subgame perfect Nash equilibrium if for every subgame of the game, its substrategy profile is a Nash equilibrium.

What is a substrategy profile?

• A strategy profile for a game specifies what a player will do at every information set in the game and specifies the payoffs at the end of the game.

• A substrategy profile of the original strategy profile specifies what will happen at every information set in the subgame.

Alice and Bob Play in the Dark

Bob

Go to A Go to B

Go to A

Alice Alice

Go to B Go to A Go to B

23 0

011

32

How many proper subgames does this game have?

A) 0B) 1C) 2D) 3E) More than 3

Alice and Bob Play in the Dark

Bob

Go to A Go to B

Go to A

Alice Alice

Go to B Go to A Go to B

23 0

011

32

How many subgame perfect Nash equilibria does this game have?

A) 0B) 1C) 2D) 3E) 4

Alice, Bob, and the outside option

Go to A Go to B

Go to A

Alice Alice

Go to B Go to A Go to B

23 0

011

32

2.5 1

Go shoot pool

What are the subgame perfect equilibria in this game?

Bob

BobGo to Movies

The Yule Ball Story

Page 283, in your text.

How many proper subgames (subgames not equal to the whole game) does this game have?A) 0B) 1C) 2D) 3E) More than 3

Dating Dilemma

Ron

Hermione

Victor Asks

Y,Y,Y Y,Y,N Y,N,Y Y,N,N N,Y,Y N,Y,N N,N,Y N,N,N

Ask 8,3,6 8,3,6 8,3,6 8,3,6 1,8*,8* 1,8*,8* 3,2,4 3,2,4

Don’t 7*,6*,5* 7*,6*,5* 7*,6*,5* 7*,6*,5* 2,5,3 2,5,3 2,5*,3 2,5*,3

Hermione

Victor Doesn’t Ask

Y,Y,Y Y,Y,N Y,N,Y Y,N,N N,Y,Y N,Y,N N,N,Y N,N,N

Ask 4,7*,7* 6,1,2 4,7*,7* 6,1,2 *4,7*,7* 6,1,2 *4,7*,7* 6,1,2

Don’t 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1Ron

Simplifying the Game

If Hermione ever reaches either of the two nodes where Ron gets to ask her, she would say Yes. So a subgame perfect equilibrium must be a Nash equilbrium for the simpler game in whichHermione always says “yes” to Ron if she hasn’t accepted a date from Victor.

Yes to Victor No to Victor

Ask 8,3,6 1,8*,8*

Don’t Ask 7*,6*,5* 2,5,3

Victor Asks

Hermione’s strategy

Ron’s Strategy

Yes to Victor No to Victor

Ask 4,7*,7* 4*,7*,7*

Don’t Ask 5,4,1* 5,4,1*

Hermione’s strategy

Victor Doesn’t Ask

Ron’s Strategy