Final Lecture
Thoughts on subgame perfection?
``Life can only be understood backwards; but it
must be lived forwards.”
Søren Kierkegaard
Some Problems from Chapter 12
Problem 1, Chapter 12
Find a separating equilibrium for this game
Cheap-talk Signaling games
• This is cheap-talk signaling game.
• Player 1 is the sender, Player 2 is the receiver.
• Notice that the payoffs to the two players, depend only
on Player 1’s type and the action, x or y, of Player 2.
• Given Player 1’s type and the action of Player 2, the
payoffs are the same whether Player 1 says a or b.
• But what Player 1 says can matter if it affects what
Player 2 does.
• Contrast this to the beer-quiche game, where the
payoff to both players depends both on the new kid’s
type and on whether he has beer or quiche. That is a
case of costly signaling.
Equilibrium for a signaling game
• In a Bayes-Nash equilibrium for a signaling
game, we need to specify the receiver’s
beliefs.
• Then we check whether, when receiver takes
his best action based on these beliefs, the
sender takes actions that are consistent with
these beliefs.
Getting started
• Since player 1 can be one of two types and
there are two possible messages, a and b, for
player 1, there are only two possible strategies
for player 1 that result in separating equilibria.
These are
– Send a if type s and b if type t
– Send b if type s and a if type t
• Let’s see if either or both of these strategies
“works”.
Strategies and beliefs
• Let’s see if we can find beliefs for the receiver
(player 2) that make for a separating
equilibrium where player 1 sends message a if
type s and sends message b if type t.
• Recall that Player 2 sees player 1’s message,
but does not see his type.
• But if senders are using the above strategy,
then Player 2 believes that those who send a
are type s and those who send b are type t.
Best responses for 2
• Then if player 2 sees message a, he believes
that player 1 is type s and his best response
given his beliefs is to take action y.
• If player 2 sees message b, he believes that
player 1 is type t and his best response given
these beliefs is to take action x.
Problem 1, Chapter 12
Suppose that Player 2 believes that Player 1 will send
message a if Type s and b if Type t. What will Player 2
do?
Problem 1, Chapter 12
Suppose that Player 2 believes that Player 1 sends a if type s and b if Type t and
acts accordingly. What will Player 1 do if she is of type s?
Problem 1, Chapter 12
Suppose that Player 2 believes that Player 1 does a if type s and b if Type t and
acts accordingly. What will Player 1 do if she is of type t?
Best responses for 1
• If Player 1 believes that player 2 plays y when he
sees message a and x when he sees message b,
what will Player 1 do?
• Look at the payoffs. If Player 1 is a type s, then he
would rather that Player 2 play y than x. If he is
of type t, he would rather Player 2 play x than y.
• So his best response to the way player 2 responds
to messages is to send message a when he is type
s and t when he is type b.
Beliefs confirmed
• So we see that if the receiver believes that
sender will send message a if he is type s
and send message b if he is of type t then in
the resulting Nash equilibrium, the receiver’s
beliefs are confirmed by the sender’s actions.
Another separating equilibrium
• We found a Bayes’ Nash equilibrium in which
Player 1 uses the following strategy
– Send message a if type s and b if type t
And Player 2 believes that this is what Player 1 is
doing.
There is another separating equilibrium, in which
Player 1 uses the strategy
Send message b if type s and a if type t
And Player 2 believes that this is what Player 1 is
doing.
Language and equilibria
The two different separating equilibria are
equivalent to having two different language
conventions.
• In one language, saying a means “I am a type s”
and saying b means “I am a type t”
• In the other language, saying a means “I am a
type t” and saying b means “I am a type t.”
• There is an equilibrium for this game when the
receiver translates the message correctly and acts
accordingly.
Chapter 12, Problem 2
Nature determines Player 1’s type, which is either
t=-1,t= 1,t=2, or t=3, each with probability ¼.
Sender learns his type and sends one of three
possible messages, bumpy, smooth or slick.
Receiver observes message (but not type) an
chooses one of three actions: a=0, a=5, or a=10.
If sender is type t and receiver takes action a,
payoff of sender is a×t and payoff of receiver is
2a×t . (typo in textbook-last word should be
“action”, not “payoff”.)
Separating equilibriaum
• Part a) asks “Find a separating perfect Bayes
Nash equilibrium”.
Answer: There isn’t one. There couldn’t be,
since in a separating equilibrium each type
sends a different message. But there are 4
types and only 3 messages you can send.
Semi-separating PBNE
• You might be able to guess that it will be fairly
easy to separate the type t=-1 from the other
types.
• Notice that this type and only this type wants
the receiver to take action 0.
• Note also that the receiver will want to take
action 0 if and only if the sender’s type is t=-1.
Let’s try this
• Start with receiver’s beliefs. Suppose receiver
believes that senders strategy is
– Say “bumpy” if you are of type -1 and say
“smooth” or “slick” if you are of type 1, 2, or 3.
• If receiver hears “bumpy” and believes that
those who say “bumpy” are type -1, then his
best response is 0. If receiver hears
“smooth” or “slick”, his best response is 10.
Sender’s response
• Suppose that sender believes that receiver’s
strategy is
– If sender says “bumpy”, take action 0, if sender says
“smooth” or “slick”, take action 10.
• Type -1 senders want receiver to take action 0.
Other types of senders want receivers to take
action 10.
• So given receiver’s strategy, best response of
sender is
– Say “bumpy” if type=-1, say “smooth” or “slick” if
type=1,2, or 3.
Beliefs confirmed
• So we see that if receiver believes that sender
will say “bumpy” if of type -1 and otherwise
will say “smooth” or “slick”, then in the
resulting Nash equilibrium, the receiver’s
beliefs about how senders behave will be
confirmed.
Problem 3, Ch. 12
Receiver believes that
Sender is of type x, y, or z with probabilities ¼, ¼,1/2.
Sender
Type
Receiver
Action
Sender
Payoff
Receiver
Payoff
x
a
3
3
x
b
2
1
x
c
1
2
y
a
4
1
y
b
5
3
y
c
3
4
z
a
3
2
z
b
9
1
z
c
10
0
Is there a separating equilibrium?
Suppose sender announces his type by saying m1 if he is an x, m2 if
he is a y and m3 if he is a z.
What would receiver do if he knows this is what sender does?
He’d take actions
A) a if m1, c if m2, a if m3
B) a of m1, b if m2, c if m3
C) b if m1, c if m2, a if m3
D) a if m1, c if m2 or m3.
Sender
Type
Receiver
Action
Sender
Payoff
Receiver
Payoff
x
a
3
3
x
b
2
1
x
c
1
2
y
a
4
1
y
b
5
3
y
c
3
4
z
a
3
2
z
b
9
1
z
c
10
0
Is there a separating equilibrium?
If receiver believes that sender sends m1 if x, m2 if y and m3 if z and
does his best response then receiver does a if m1, c if m2 and a if
m3. If receiver does this, then a best response for the sender is to
use the following strategy.
A)
B)
C)
D)
Send m1 if x, m1 if y, m2 if z
Send m1 if x, m1 if y, m3 if z
Send m3 if x, m2 if y, m1 if z
Send m1 if x, m1 if y, m3 if z
Note: The slide displayed in
class didn’t offer a correct answer.
On this slide, A) is correct.
Sender
Type
Receiver
Action
Sender
Payoff
Receiver
Payoff
x
a
3
3
x
b
2
1
x
c
1
2
y
a
4
1
y
b
5
3
y
c
3
4
z
a
3
2
z
b
9
1
z
c
10
0
There is no separating equilibrium!
If receiver believes that sender sends m1 if x, m2 if y and m3 if z and
does his best response then receiver does a if m1, c if m2 and a if
m3. If receiver does this, then it is not a best response for the
sender to do what the receiver
believes he will do. The sender
would be better off sending message
m1 rather than m2 when he is of
type y
Sender
Type
Receiver
Action
Sender
Payoff
Receiver
Payoff
x
a
3
3
x
b
2
1
x
c
1
2
y
a
4
1
y
b
5
3
y
c
3
4
z
a
3
2
z
b
9
1
z
c
10
0
But maybe there is a semi-separating equilibrium
Candidate: Sender sends message m1 if type x, message m2 if type y
or z.
Recall that probability is ¼ that sender is of type x, ¼ that sender is
type y, ½ that sender is type z.
If receivers believe that senders
send m1 if of type x and m2 if
they are of type y or z, what is
The probability that someone who
sent message m2 is a type y?
A) 1/3
B) 1/ 4
C) 1/ 2
Sender
Type
Receiver
Action
Sender
Payoff
Receiver
Payoff
x
a
3
3
x
b
2
1
x
c
1
2
y
a
4
1
y
b
5
3
y
c
3
4
z
a
3
2
z
b
9
1
z
c
10
0
Looking for a semi-separating equilibrium
Candidate: Sender sends message m1 if type x, message m2 if type y or z.
If receivers believe that senders send m1 if of type x and m2 if
they are of type y or z, what is the best response for a receiver if
she sees a message m1?
A) a
B) b
C) c
Sender
Type
Receiver
Action
Sender
Payoff
Receiver
Payoff
x
a
3
3
x
b
2
1
x
c
1
2
y
a
4
1
y
b
5
3
y
c
3
4
z
a
3
2
z
b
9
1
z
c
10
0
Looking for a semi-separating equilibrium
Candidate: Sender sends message m1 if type x, message m2 if type y
or z.
Recall that probability is ¼ that sender is of type x, ¼ that sender is
type y, ½ that sender is type z.
If receivers believe that senders
send m1 if of type x and m2 if
they are of type y or z, what is
The probability that someone who
sent message m2 is a type y?
A) 1/3
B) 1/ 4
C) 1/ 2
Sender
Type
Receiver
Action
Sender
Payoff
Receiver
Payoff
x
a
3
3
x
b
2
1
x
c
1
2
y
a
4
1
y
b
5
3
y
c
3
4
z
a
3
2
z
b
9
1
z
c
10
0
Best response of receiver who sees m2.
If sender sends message m2, and receiver believes that he sends this
message whenever he is of types y or z, then the probability that an
m2 sender is type y is 1/3 and the probability that he is of type z is 2/3
Expected payoff to receiver from
playing a is
1/3 x 1+2/3 x 2=5/3
From playing b is
1/3 x 3 + 2/3 x1=5/3
From playing c is
1/3 x 4 + 2/3 x 0 =4/3.
Sender
Type
Receiver
Action
Sender
Payoff
Receiver
Payoff
x
a
3
3
x
b
2
1
x
c
1
2
y
a
4
1
y
So best responses for receivers who
see m2 are a or b (or any mix of these). y
b
5
3
c
3
4
z
a
3
2
z
b
9
1
z
c
10
0
• If receiver believes that sender sends message m1 if he is of type x
and m2 if of type y or z, then doing a if m1 and b if m2 is a best
response for receiver.
• If Sender knows that this is what receiver will do, then
• when sender is of type x, sender will get 3 if he says m1 and 1
if he says m2.
Sende Receiver Sender
• When sender is of type y, sender
r Type Action
Payoff
gets 4 if he says m1 and 5 if he says m2. x
a
3
• When sender is of type z, sender gets
x
b
2
3 if he says m1 and 9 if he says m2.
Receiver
Payoff
3
1
x
c
1
2
y
• So if receiver believes that senders
send m1 if of type x and m2 if of type y or z, y
Then it is in the interest of senders to do
y
exactly that.
z
• These beliefs are self-confirming. We have
z
a semi-separating equilibrium.
a
4
1
b
5
3
c
3
4
a
3
2
b
9
1
z
c
10
0
We found a semi-separating
equilibrium
Final exam
• Exam will ask questions from all chapters.
• Some problems will be easy, some will be
harder.
May all your subgames be happy..
Even if not always regular and proper.