Signaling Game Problems
Signaling game.
• Two players– a sender and receiver.
• Sender knows his type. Receiver does not. It is
not necessarily in the sender’s interest to tell
the truth about his type.
• Sender chooses an action (signal) that receiver
observes. Action may depend on type.
• Receiver takes an action given sender’s signal.
In Equilibrium
• Receiver has beliefs about actions (signals)
that will be taken by each type of sender.
• Receiver chooses best action given senders’
signals and his beliefs about how signals are
related to types.
• Senders choose signals that maximize their
payoffs given response of receivers.
• Beliefs of receivers about senders’ types are
confirmed.
Pooling and Separating
• Pooling equilibrium—All types of senders send
the same signal.
• Separating equilibrium—Each type of sender
sends a different signal.
• Semi-separating equilibrium—Some, but not
all types send same signals
• In equilibrium beliefs of receivers about
meaning of signals are confirmed by
outcomes.
An Education Fable
• Imagine that the labor force consists of two
types of workers: Able and Middling with
equal proportions of each.
• Employers are not able to tell which type they
are when they hire them.
• A worker is worth $1500 a month to his boss if
he is Able and $1000 a month if he is
Middling.
• Average worker is worth
• $ ½ 1500 + ½ 1000=$1250 per month.
Competitive labor market
• The labor market is competitive and since
employers can’t tell the Able from the
Middling, all laborers are paid a wage equal
to the productivity of an average worker:
$1250 per month.
Enter Professor Drywall
Drywall claims
• My 10-lecture course raises worker
productivity by 20%!
• One employer believes that Drywall’s lectures
are useful and requires its workers attend 10
monthly lectures by Professor Drywall and pays
wages of $100 per month above the average
wage.
– Middling workers find Drywall’s lectures
excruciatingly dull. Each lecture is as bad as losing
$20.
– Able workers find them only a little dull. To them,
each lecture is as bad as losing $5.
• Which laborers stay with the firm?
• What happens to the average productivity of
laborers?
The outcome
• Middling workers quit. Raise of $100 per
month isn’t enough to compensate them for
listening to Drywall. (They’d need $200.)
• Able workers stay. They don’t like the
lectures, but $100 per month is enough to
compensate for Drywall. (Drywall costs them
$50.
• Labor force is now all Able workers. Average
productivity rises from $1250 to $1500.
Other firms see what happened
• Professor Drywall shows the results of his
lectures for productivity at the first firm.
• Firms decide to pay wages of about $1500 for
people who have taken Drywall’s course.
• Now who will take Drywall’s course?
• What will be the average productivity of
workers who take his course?
Do we have an equilibrium now?
A Drywall Premium?
• Suppose those who take Drywall’s course get
$1500 wage and those who don’t get $1250.
• For Middling workers, a $250 wage premium is
enough to make them take the course.
• Able workers will also take the course.
• Average productivity of those who take the
course is now $ ½ 1000 + ½ 1500=1250.
• Firm’s belief that those who take Drywall’s course
are all Able is not confirmed.
• This is not an equilibrium.
Professor Drywall responds
• Professor Drywall is not discouraged.
• He claims that the problem is that people
have not heard enough lectures to learn his
material.
• Firms believe him and Drywall now makes his
course last for 30 hours a month.
• Firms pay almost $1500 wages for those who
take his course and $1000 for those who do
not.
A Separating Equilibrium
• Able workers will prefer attending lectures
and getting a wage of $1500, since to them
the cost of attending the lectures is
$5x30=$150 per month.
• Middling workers will prefer not attending
lectures since they can get $1000 if they don’t
attend. Their cost of attending the lectures
would be $20x30=$600, leaving them with a
net of $900.
• Firms’ beliefs are confirmed. Those who
attend Drywall’s lectures are Able and those
who don’t are Middling
But there is also a pooling equilibrium
• Suppose that employers believe that Professor
Drywall is a crackpot and that those who
attended his lectures are no more productive
than average.
• They would not pay a premium for those who
take Drywall’s course.
• Nobody would take the course.
• Employers’ beliefs are consistent with
observation.
So there we are.
Example: Problem 1, Chapter 11
•
•
•
•
Quality
Probability
Value to Seller
Value to Buyer
Good Car
q
10,000
12,000
Lemon
1-q
6,000
7,000
A used car owner wants to sell his car.
The fraction q of used cars are good and 1-q are “lemons”
Only the current owner (seller) knows if his car is good or a lemon.
There are many buyers whose values are as above.
• Sender is seller. Receivers are buyers.
• Types of senders—good car owners, lemon owners
• Possible actions taken by types—sell your used car or keep it.
Extensive form of Lemons Game if price of used car is P
Nature
q
1-q
Good car
Lemon
Owner
Owner
Nature
Keep
Keep
Sell
Sell
0
0
0
0
Buyer
Buy
P-10,000
12,000-P
Don’t buy
0
0
Buy
Don’t buy
P-6000
7000-P
0
0
Is there a pooling equlibrium?
Quality
Probability
Value to Seller
Value to Buyer
Good Car
q
10,000
12,000
Lemon
1-q
6,000
7,000
• In a pooling equilibrium, both types of owners would sell
their car.
• Suppose buyers believe that all used car owners are selling.
• A buyer gets a random draw of lemon or good car which is worth
P=12,000q+7,000(1-q)=7,000+5,000q.
• Owners of good cars will sell their cars only if P≥10,000.
• So there can be a pooling equilibrium only if 7,000+5,000q≥10,000
• This implies q≥3/5.
• So if q≥3/5, there is a pooling equilibrium at a price of about
7,000+5q. If q<3/5. there is no pooling equilibrium.
There is also a separating equilibrium
Quality
Probability
Value to Seller
Value to Buyer
Good Car
q
10,000
12,000
Lemon
1-q
6,000
7,000
Suppose that buyers all believe that the only used cars on the
market are lemons. Then they all believe that a used car is only
worth $7000. The price will not be higher than $7000.
At this price, nobody would sell his good car, since good used
cars are worth $10,000 to their current owners.
Buyer’s beliefs are confirmed by experience. This is a
separating equilibrium. Good used car owners act differently
from lemon owners.
Pooling and separating
• Notice that in this case, if q>3/5, there is both a
separating and a pooling equilibrium.
• In the pooling equilibrium, everybody believes
that all used cars come to market and used cars
sell for more than $10,000.
• In separating equilibrium, everybody believes
that only lemons come to market. Used cars sell
for $7000.
• In either case, beliefs are confirmed by
experience.
Signaling Equilibrium as
Self-confirming Beliefs
• Receiver has beliefs about probability distribution
of types and how each type will act.
• Receiver chooses a strategy that is a best
response, given these beliefs and actions that the
sender takes.
• Each sender-type strategy is a best response,
given the way receiver reacts.
• Receiver’s beliefs about how each type will act
are “confirmed” outcome.
Problem 11.3 (Product Quality)
• A product can be of either high or of low quality. Some
retailers have high quality products, some have low
quality.
• Production cost is c for either type.
• High quality items work with probability H and low
quality items work with probability L. Consumer
values a working item at V.
• Value of a product to consumer is V if it works, 0 if it
doesn’t.
Separating equilibrium with warranty
• Find a separating equilibrium where high
quality firm offers warranty, low quality firm
offers no warranty and high quality firm
charges a higher price than low quality firm.
Buyers’ beliefs and behavior
• Suppose that buyer believes that items
without warranty are of low quality and items
with warranty are of high quality.
• With these beliefs,
– Buyer would pay up to LV for item without
warranty.
– Buyer would pay up to V for item with warranty.
Best response of High quality seller
If Buyers believe that the only items without
warranties are low quality:
• An item with warranty could sell for (almost) V. An item
with no warranty could sell for (almost) LV.
• When a unit with warranty fails, seller has to give
money back. For a high quality seller, expected cost of
paying off warranties is (1-H) (V+x) where x is the cost
per unit of handling the repayment.
• Expected profit per item sold with warranty is
V-(1-H)(V+x)-c=HV-(1-H)x-c.
• If sold without warranty, profit must be LV-c.
• Selling with warranty gives higher profit if
HV-(1-H)x>LV.
Best response of Low Quality seller
• Low quality seller could sell at price V with
warranty, at price LV with no warranty.
• Profit if it sells with warranty is
V-(1-L)(V+x)-c=L(V+x)-c=LV-c-x(1-L).
• Profit per unit if it sells without warranty is
LV-c
• Selling without warranty is best response for
Low quality seller.
An equilibrium
• So if buyers believe that only the good items
have guarantees:
– the best response of high quality sellers is to price
at V and offer a warranty
– The best response of low quality sellers is to price
at LV and offer no warranty.
• This is an equilibrium. The buyers’ beliefs
that only high quality sellers have warranties is
confirmed by the way sellers act in response to
these beliefs. (Self-confirming beliefs)
Why is does signaling “work” here?
• It is cheaper for the high quality seller to offer a
warranty than for a low quality seller.
• Cost to high quality seller is (1-H)(V+x). Cost to a
low quality seller is (1-L)(V+x)>(1-H)(V+x).
• So if having a warranty raises price you can
charge from LV to V:
– a warranty increases high quality seller’s revenue by
(1-L)V and his costs by (1-H)(V+x)<(1-L)V and so
increases his profit.
– a warranty increases low quality seller’s revenue by
(1-L)V and his costs by (1-L)(V+x) so reduces his profit.
Problem 11.6 (Advertising)
The setup
• Nature determines a restaurant’s quality, high
or low with probability ½ either way.
• Production cost is $35 per meal for either
type.
• Price of a meal is fixed at $50 in either type.
– Value of high quality to consumer is $85
– Value of low quality to consumer is $30.
• A customer who goes to a high quality
restaurant will come back a second time.
Find a Separating equilibrium
• A restaurant can choose an amount A to
spend on advertising. Customers observe A.
• Find a separating equilibrium in which high
quality restaurants spend A on advertising and
low quality restaurants do not advertise.
Beliefs and Behavior
• Suppose that customers believe that low
quality restaurants spend less than A* on
advertising and high quality restaurants spend
at least A*.
• With these beliefs, they will not go to a
restaurant that spends less than A* and will go
to one that spends A*.
What will restaurants do?
• If consumers have these beliefs, low quality
restaurants will have profits 50-35-A*=15-A*
if they spend A* on advertising and 0 if it
spends less than A*.
• For high quality restaurants, a consumer who
comes once will come twice. So it will have
profits 100-70-A*=30-A* if it spends A* on
advertising and 0 if it spends less than A*.
Suppose 15<A*<30
• Then if low quality restaurant spends A*, its
profits will be 15-A*<0, so it doesn’t pay a low
quality restaurant to spend A* (or any money
at all) on advertising).
• If high quality restaurant spends A*, its profits
will be 30-A*>0. If it spends less its profit is 0,
if it spends more its profits are less than 30A*.
• So there is a separating equilibrium.
What makes for separation?
• Because customers once attracted to a high
quality business will return, advertising is
more valuable to a high quality than to a low
quality business and thus works as a signal in
a separating equilibrium.
Other Customer Beliefs lead to Pooling
equilibrium
• Suppose that for some number A*<15,
consumers believe that a restaurant that
spends A* on average is equally likely to be
good or bad, while any restaurant that spends
less than A* is sure to be bad.
Response of Restaurants
• All restaurants would find it profitable to
advertise at level A*<15.
• If they spent less they would get 0 profits.
• So all would advertise at level A*.
• Average payoff to customer from going to a
restaurant restaurant that advertises at A*
would be ½x85+ ½x35-50=10, so customer
would go to any restaurant that advertises at
A*
Self-confirming beliefs
• Note that there many different beliefs would
be self-confirming.
– In fact, for any A*<30, the belief that restaurants
that spend less than A* are low quality is selfconfirming.
– When A*>15, these beliefs lead to separating
equilibrium
– When A*<15, they lead to pooling equilibrium.
Problem 11.5
• Students are of 3 types, High, medium, and low.
Cost of getting a college degree to a student is 2 if
high, 4 if medium, and 6 if low.
• 1/6 of students are of high type, ½ of medium
type, 1/3 are of low type.
• Salaries for managers are 15, and 10 for clerks.
• An employer has one clerk’s job to fill and one
manager’s job to fill. Employer’s profits (net of
wages) are 7 from hiring anyone as a clerk,
4 from hiring a low type as a manager, 6 from hiring
a medium type as manager, 14 from hiring a high
type as manager.
Problem 11. 5 The College Signaling Game
Probability low=1/3, Probability moderate=1/2, Probability high=1/6
Find a PBNE where students of low intellect do not go to college and
those of moderate and high intellect do.
Recall that the probability that an applicant is of low
intellect is 1/3, probability of moderate intellect is
1/2 and probability of high intellect is 1/6.
If the moderate and high intellect types go to
college and the low intellect types do not, what
proportion of those who go to college are of high
intellect.
A)
B)
C)
D)
E)
1/6
1/5
1/4
1/3
1/2
Conditional probability (Bayes’ Law)
• P(H|C)=P(C and H) /P(C)
=1/6÷(1/6+1/2)=1/4.
Beliefs and actions of Employer
• Suppose employer believes that applicants of low intellect
do not go to college and those of high and medium intellect
do go.
• Then if applicant has not gone to college, employer’s payoff
is 4 for manager, 7 for clerk.
• If applicant has gone to college, then employer believes he is
of medium intellect with probability ¾ and high with
probability 1/4
– Expected payoff from making him manager is 3/4x6+1/4x14=8
– Expected payoff from making him clerk is 7.
• With these beliefs, employer will make college graduates
managers, and non college applicants clerks.
Is this an equilibrium?
• Will low intellect types choose not to go to
college? Yes-they get payoff of 10 from no college
and clerk and 9 from college and manager.
• Will medium intellect types choose to go to
college? Yes-they get payoff of 11 from college
and manager and 10 from college and clerk.
• Will high intellect types choose college? Yes, they
get payoff of 13 from college and manager and 10
from no college and clerk.
Equilibrium
• We see that when employer believes that low
intellect types don’t go to college and all
others do, then it is in the interest of low
intellect types not to go to college and of high
and medium intellect types to go to college.
• So these employer beliefs are self-confirming.
Gazelle Signals:
“I’m fast, Don’t chase me.”
This behavior is called “stotting”
Peacock signals,
“I’m healthy, Mate with Me”
Enough signals for today