the unpleasant professor problem. payoff information professor can give exam on monday, wednesday or...
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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.
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.
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
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
2.51
2.50
32
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,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
The Yule Ball Tale
How many proper subgames does this game have? 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.
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
The Yule Ball Tale
Suppose Ron knows whether Victor asked
How many proper subgames does this game have? 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 says No to Victor
Ron
Valentine’s lesson:Subgame Perfection does not solve all of love’s quandries