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)
B)
C)
D)
E)
1
2
3
4
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
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?
Subgame in Normal Form
Vivica
Pay Ransom
Don’t Pay Ransom
4,
1
2,
2
5,
2
1,
4
Kill
Guy
Release
What are the Nash equilibria in this subgame?
What are the Nash equilibrium payoffs in this subgame?
Find Subgame Perfect Nash equilibria
Alice and Bob Play in the Dark
How many proper
subgames does this
game have?
Bob
Go to A
Go to B
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
A)
B)
C)
D)
E)
0
1
2
3
More than 3
Is every Nash equilibrium for the
previous game between Alice and Bob
subgame perfect?
A) Yes
B) No
Alice and Bob Play in the Dark
How many subgame
perfect Nash
equilibria does this
game have?
Bob
Go to A
Go to B
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
A)
B)
C)
D)
E)
0
1
2
3
4
Alice, Bob, and the outside option
Bob
Go to Movies
Bob
Go to A
Go shoot pool
Go to B
Alice
Go to A
2
3
2.5
1
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
How many proper subgames does this game have?
A) 1 B) 2 C) 3 D) 4 E) 5
How would you play in this game if you
were Bob?
A) Go shoot pool
B) Go to movie A
C) Go to movie B
How would you play in this game if you
were Alice?
A) Go to A
B) Go to B
Finding subgame perfect strategy
profiles
Bob
Go to Movies
Bob
Go to A
Go shoot pool
Go to B
Alice
Go to A
2
3
2.5
1
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
Find Nash equilibria for the proper subgame.
Look at the truncated game with equilibrium
payoffs from subgame.
Finding subgame perfect strategy
profiles
Bob
Go to A
Go to B
Alice
Go to A
2
3
2.5
1
Alice
Go to B
0
0
Go to A
1
1
Go to B
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.
Truncating the tree with both go to B
in the
subgame
Bob
Go to Movies
Bob
Go to A
Go shoot pool
Go to B
Alice
Go to A
2
3
2.5
1
Alice
Go to B
0
0
Go to A
1
1
Go to B
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
Truncating the tree with both go to A
in subgame
Bob
Go to Movies
Bob
Go to A
Go shoot pool
Go to B
Alice
Go to A
2
3
2.5
1
Alice
Go to B
0
0
Go to A
1
1
Go to B
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.
Complete Information:
Alice chooses first. Find SPNE
Alice
Movie A
Movie B
Bob
Bpb
Shoot pool
1
2.5
Movie A
3
2
Shoot pool
Movie A
Movie B
Movie B
1
1
0
0
2
3
0
2.5
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
Victor Asks
Hermione
Ron
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
N,Y,N
N,N,Y
N,N,N
Victor Doesn’t Ask
Hermione
Ron
Y,Y,Y
Y,Y,N
Y,N,Y
Y,N,N
N,Y,Y
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
5,4,1
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 which
Hermione always says “yes” to Ron if she hasn’t
accepted a date from Victor.
The Yule Ball Tale
Victor Asks
Hermione’s strategy
Ron’s
Strategy
Yes to Victor
No to Victor
Ask
8,3,6
1,8*,8*
Don’t Ask
7*,6*,5*
2,5,3
Victor Doesn’t Ask
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*
Payoffs listed in order Victor, Ron, Hermoine
Suppose Ron knows whether Victor asked
Ron
How many subgames
are there now?
A) 2 B) 3 C) 4 D) 6 E) 8
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:
Ron
Hermoine
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)
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
Snickers
Milky Way
Kit Kat
Baby Ruth
For Rachel
Milky Way
Kit Kat
Baby Ruth
Snickers
Hint: No matter what happens,
Nick will get two bars. Rachel will never choose Snickers.