Taking Turns in the Dark:(Subgame perfection with incomplete information )

Econ 171

Subgame Perfection with Imperfect Information

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

Look back at kidnapper game

What is a subtree of a game?

• It is a non-terminal node, together with all of the nodes that could be reached from this node.

• A Proper Subtree is a subtree that is not the entire game.

How many subtrees does this game tree have? A) 1 B) 2 C) 3 D) 4 E) 5

How many proper subtrees does the kidnapper game have?

A) 1B) 2C) 3D) 4E) 5

What is a regular subtree of a game?

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

• A proper, regular subtree is a regular subtree that is not the entire game tree.

How many regular subtrees does this game tree have?A) 1 B) 2 C) 3 D) 4 E) 5

How many regular, proper,subtrees does this game tree have?A) 1 B) 2 C) 3 D) 4 E) 5

What is a subgame of a game?

• A subgame is 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)

How many proper subgames does this Game have?A) 1 B) 2 C) 3 D) 4 E) 5

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 strategy profile for an entire game induces a substrategy profile for each of its subgames. This substrategy profile specifies what each player will do at each of the player’s information sets in the subgame.

Subgame perfection

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

• That is, the actions taken in the proper subgame are a Nash equilibrium for the game that consists of just that subgame.

Find Subgame Perfect Nash equilibria

The subgame when Guy Kidnaps

What are the substrategies for Guy? for Vivica?

What are the Nash equilibria in this subgame?

4, 1 2, 2

5, 2 1, 4

Kill

Release

Pay Ransom Don’t Pay Ransom

Subgame in Normal Form

Guy

Vivica

What are the Nash equilibrium payoffs in this subgame?

What are the Nash equilibria in this subgame?

Find Subgame Perfect Nash equilibria

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

Is every Nash equilibrium for the previous game between Alice and Bob subgame

perfect?

A) YesB) No

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

How many proper subgames does this game have?A) 1 B) 2 C) 3 D) 4 E) 5

Bob

BobGo to Movies

How would you play in this game if you were Bob?

A) Go shoot poolB) Go to movie AC) Go to movie B

How would you play in this game if you were Alice?

A) Go to AB) Go to B

Finding subgame perfect strategy profiles

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

Find Nash equilibria for the proper subgame.Look at the truncated game with equilibriumpayoffs from subgame.

Bob

BobGo to Movies

Finding subgame perfect strategy profiles

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

Proper subgame has two N.E. Both go to A, Both go to B.We need to look at two possibilities. We may find more than one S P N E.

Bob

Truncating the tree with both go to B in the subgame

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

If both go to B if Bob goes to the movies, then Bob will go to the movies rather than play pool. The profile, Bob goes to the movies and goes to B; Alice goes to B is a SPNE

Bob

BobGo to Movies

Truncating the tree with both go to A in subgame

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

If Alice’s strategy is Go to A, then Bob’s best response is Go shoot pool and Go to Movie A if he goes to the movies. This is a SPNE as well.

Bob

BobGo to Movies

Complete Information: Alice chooses first. Find SPNE

Alice

Bpb Bob

Shoot poolShoot pool

Movie A

Movie A Movie A

Movie B

Movie BMovie B32

11

12.5 0

2.5

23

00

The Yule Ball Tale

How many strategies are possible for Hermione?A) 2 B) 3 C) 4 D) 6 E) 8

The Yule Ball Tale

How many strategies are possible for Ron ?A) 2 B) 3 C) 4 D) 6 E) 8

Dating Dilemma: Strategic Form

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,NAsk 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,NAsk 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

The Yule Ball Tale

How many proper subgames are there? A) 0 B) 2 C) 3 D) 6 E) 8

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.

The Yule Ball Tale

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

Payoffs listed in order Victor, Ron, Hermoine

Suppose Ron knows whether Victor asked

How many subgames are there now?A) 2 B) 3 C) 4 D) 6 E) 8

Ron

Suppose Ron knows whether Victor asked

Ron

What are N.E. in subgame where Victor Asks

If Victor asks, then in remaining game, there are two things Hermoine can do, say Yes or No to Victor.There are two things, Ron can do. Ask Hermoine or Don’t ask her. What are the N.E. in this subgame?

Strategic form if Victor asks:

Ask Hermoine Don’t ask Hermoine

Yes to Victor 6,3 (Victor 8) 5,6 (Victor 7)

No to Victor 8, 8 (Victor 1) 3,5 (Victor 2) Hermoine

Ron

We have two Nash equilibria for the subgame between Hermoine and Ron starting at the node where Victor asks Hermoine.

In one of them, Hermoine says Yes to Victor and Ron doesn’t ask.In the other, Hermoine says No to Victor and Ron asks.

A SPNE in which Hermoine says Yes to Victor

Ron

A SPNE where Hermoine would say No to Victor

Ron

Valentine’s lesson:Subgame Perfection does not solve all of love’s quandries

The Unpleasant Professor Problem

Payoff information

• Professor can give exam on Monday, Wednesday or Friday.

• Students will study the night before exam if they know there will be an exam next day.

• Professor prefers to have nobody prepared when exam is offered.

• He also prefers earlier exam to later.

Solve for SPNE by backward induction

• Without drawing full tree, let’s try a shortcut.• If he doesn’t give exam on Monday, then he

must either give it on Wednesday or on Friday.• If he doesn’t give it on Wednesday, students

will know exam is Friday and will all study. • That is the worse for professor than giving it

on Wednesday.• So he will not give exam on Friday.

Working back…

• So if he doesn’t give exam on Monday, he will give it on Wednesday.

• Therefore if he doesn’t give exam on Monday, students will study on Tuesday.

• If students will study on Tuesday if exam is not on Monday, professor would rather give exam on Monday.

• Only subgame perfect Nash equilibrium has exam on Monday, students study on Sunday.

Problem 8.16

Nick and Rachel divide 4 candy bars. They take turns choosing. Nick goes first. What should Nick choose first?

Preferences are:For Nick For RachelSnickers Milky WayMilky Way Kit KatKit Kat Baby RuthBaby Ruth Snickers

Hint: No matter what happens, Nick will get two bars. Rachel will never choose Snickers.