1. The Unpleasant Professor Problem

2. 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.

3. 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.

4. 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.

5. 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 Rachel
Snickers Milky Way
Milky Way Kit Kat
Kit Kat Baby Ruth
Baby Ruth Snickers
Hint: No matter what happens,
Nick will get two bars. Rachel will never choose Snickers.

6. Taking Turns in the Dark: (Subgame perfection with incomplete information )
Econ 171

7. Subgame Perfection with Imperfect Information
How can the notion of subgame perfection help us if there is incomplete information? Look back at kidnapper game

9. 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.

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

11. How many proper subtrees does the kidnapper game have?1 2 3 4 5

12. 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.

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

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

15. 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)

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

17. 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.

18. 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.

19. Alice and Bob Play in the Dark
How many proper subgames does this game have? 1 2 3
More than 3
Go to A
Go to B
Alice
Alice
Go to B
Go to A
Go to A
Go to B 2 3 1 3 2

20. Alice and Bob Play in the Dark
How many subgame perfect Nash equilibria does this game have? 1 2 3 4
Go to A
Go to B
Alice
Alice
Go to B
Go to A
Go to A
Go to B 2 3 1 3 2

21. Alice, Bob, and the outside option
Go to Movies
Bob
Go shoot pool
Go to A
Go to B
2.5 1
Alice
Alice
Go to B
Go to A
Go to A
Go to B 2 3 1 3 2
How many proper subgames does this game have?
A) 1 B) 2 C) 3 D) 4 E) 5

22. How would you play in this game if you were Bob?
Go shoot pool
Go to movie A
Go to movie B

23. How would you play in this game if you were Alice?
Go to A
Go to B

24. Finding subgame perfect strategy profiles
Bob
Go to Movies
Bob
Go shoot pool
Go to A
Go to B
2.5 1
Alice
Alice
Go to B
Go to A
Go to A
Go to B 2 3 1 3 2
Find Nash equilibria for the proper subgame.
Look at the truncated game with equilibrium
payoffs from subgame.

25. Finding subgame perfect strategy profiles
Bob
Go to A
Go to B
2.5 1
Alice
Alice
Go to B
Go to A
Go to A
Go to B 2 3 1 3 2
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.

26. Truncating the tree with both go to B in the subgame
Bob
Go to Movies
Bob
Go shoot pool
Go to A
Go to B
2.5 1
Alice
Alice
Go to B
Go to A
Go to A
Go to B 2 3 1 3 2
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

27. Truncating the tree with both go to A in subgame
Bob
Go to Movies
Bob
Go shoot pool
Go to A
Go to B
2.5 1
Alice
Alice
Go to B
Go to A
Go to A
Go to B 2 3 1 3 2
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.

28. Complete Information: Alice chooses first. Find SPNE
Movie A
Movie B
Bpb
Bob
Shoot pool
Shoot pool
2.5 1
2.5
Movie A
Movie A
Movie B
Movie B 3 2 3 2 1

29. The Yule Ball Tale How many strategies are possible for Hermione?
A) 2 B) 3 C) 4 D) 6 E) 8

30. The Yule Ball Tale How many strategies are possible for Ron ?
A) 2 B) 3 C) 4 D) 6 E) 8

31. Dating Dilemma: Strategic Form
Victor Asks
Hermione
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
1,8*,8*
3,2,4
Don’t
7*,6*,5*
2,5,3
2,5*,3
Ron
Victor Doesn’t Ask
Hermione
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*
Don’t
5,4,1
Ron

32. The Yule Ball Tale How many proper subgames does this game have?
A) 0 B) 2 C) 3 D) 6 E) 8

33. 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 which Hermione always says “yes” to Ron if she hasn’t accepted a date from Victor.

34. Victor Asks
Hermione’s strategy
Yes to Victor
No to Victor
Ask
8,3,6
1,8*,8*
Don’t Ask
7*,6*,5*
2,5,3
Ron’s Strategy
Victor Doesn’t Ask
Hermione’s strategy
Yes to Victor
No to Victor
Ask
4,7*,7*
4*,7*,7*
Don’t Ask
5,4,1*
Ron’s Strategy
Payoffs listed in order Victor, Ron, Hermoine

35. The Yule Ball Tale

36. Suppose Ron knows whether Victor asked
How many proper subgames does this game have?
A) 2 B) 3 C) 4 D) 6 E) 8

37. Suppose Ron knows whether Victor asked

38. 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?

39. Strategic form if Victor asks:
Ron
Ask Hermoine
Don’t ask Hermoine
Yes to Victor
6, (Victor 8)
5,6 (Victor 7)
No to Victor
8, 8 (Victor 1)
3,5 (Victor 2)
Hermoine
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.

40. A SPNE in which Hermoine says Yes to Victor
Ron

41. A SPNE where Hermoine says No to Victor
Ron

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