Signaling Game Problems - PowerPoint Presentation

Slide1

Signaling Game ProblemsSlide2

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 that receiver observes. Action may depend on type.

Receiver takes an action given sender’s signal.

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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 signalsSlide4

Example: Problem 1, Chapter 11

Quality

Probability

Value to Seller

Value to Buyer

Good

Car

q10,00012,000Lemon1-q6,0007,000

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.Slide5

Nature

Good car

Lemon

Keep

Keep

Owner

Sell

Sell

Buy

Buy

Don’t buy

Don’t buy

Owner

Buyer

0

0

0

0

0

0

0

0

P-10,000

12,000-P

P-6000

7000-P

q

1-q

Extensive form of Lemons Game if price of used car is P

NatureSlide6

Is there a pooling equlibrium?

Quality

Probability

Value to Seller

Value to Buyer

Good

Car

q10,00012,000Lemon1-q6,0007,000

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.Slide7

There is also a separating equilibrium

Quality

Probability

Value to Seller

Value to Buyer

Good

Car

q10,00012,000Lemon1-q6,0007,000

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.Slide8

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. Slide9

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.Slide10

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.Slide11

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.Slide12

Best responses of Sellers

An item with warranty could sell for (almost) V.

An item with no warranty could sell for (almost) LV.

A high quality seller who offers a guarantee would get expected revenue of

hV

from a sale at price V and would have profit HV-c.

If he offered no guarantee, he could only sell his product for LV and have profit LV-c<HV-cA low quality seller would make profit LV-c by pricing at LV and offering no warranty.If he offered a guarantee, he could price at V but his profits would be LV-c. If Buyers believe that the only items without warranties are low quality:Slide13

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)Slide14

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. Cost to a low quality seller is (1-L)V>(1-H)V.

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<(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 so doesn’t increase his profit. Slide15

Problem 11.6 (Advertising)Slide16

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 to consumer of high quality is $85

Value to consumer of low quality is $30.

A customer who goes to a high quality restaurant will come back a second time.Slide17

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. Slide18

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*. Slide19

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*. Slide20

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 30-A*.

So there is a separating equilibrium.Slide21

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.Slide22

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. Slide23

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*Slide24

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 self-confirming.

When A*>15, these beliefs lead to separating equilibrium

When A*<15, they lead to pooling equilibrium.Slide25

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.Slide26

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. Slide27

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.

1/6

1/5

1/4

1/31/2Slide28

Conditional probability (Bayes’ Law)

P(H|C)=P(C and H) /P(C)

=1/6÷(1/6+1/2)=1/4.Slide29

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.Slide30

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.Slide31

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. Slide32

Problem 11.7Slide33

Finding an equilibrium

Find a semi-separating equilibrium in which the sender chooses the same action if she is of type t1 or t2 and a different action if she is type t3.Slide34

Thinking it through

If the receiver believes that the sender is type t3, the receiver will always take action b since that is strictly dominant. If receiver always takes action b with t3 sender, a t3 sender will take action z.

If the receiver believes that types t1 and t2 send the same signal which is different from z, then receiver will play a when sender does not do z.

It could not be an equilibrium for both t1 and t2 to do x, because t2’s would prefer z.Slide35

Let’s try y

Suppose that type 1 and type 2 players play y.

When would t1’s do that?

Suppose that the receiver believes that anybody who does x or z is a type 3 and that anybody who does y is a type 1 or 2.

Then a receiver will respond b to x or z and will respond a to y.

If receiver does this, types 1 and 2 will both play y and type 3 will play z.

Receiver’s beliefs are confirmed.Slide36

Problem 11.5

George Bush and Saddam HusseinSlide37

The story

Bush believes that probability Hussein has WMDs is w<3/5.

When is there a perfect Bayes-Nash equilibrium with strategies?

Hussein: If WMD, Don’t allow, if no WMD allow with probability h.

Bush: If allow and WMD, Invade. If allow and no WMD, Don’t invade, If don’t allow, invade with probability b.Slide38

Payoffs for Hussein if he has no WMDs

Payoff from not allow is 2b+8(1-b)=8-6b

Payoff from allow is 4, since if he allows Bush will not invade.

Hussein is indifferent if 4=8-6b or equivalently

b=2/3.

So he would be willing to use a mixed strategy if he thought that Bush would invade with probability 2/3 if Hussein doesn’t allow inspections.Slide39

Probability that Hussein has WMD’s if he uses mixed strategy

If Hussein does not allow inspections, what is probability that he has WMDs?

Apply Bayes’ law.

P(

WMD|no

inspect)=

P(WMD and no inspect)/P(no inspect)=

w/(w+(1-w)(1-h)) Slide40

Bush’s payoffs if Hussein refuses inspections

If Bush does not invade:

1 w/(w+(1-w)(1-h)) +9(1-(w/(w+(1-w)(1-h)))

If Bush invades:

3 w/(w+(1-w)(1-h)) +6(1-w/(w+(1-w)(1-h))

Bush will use a mixed strategy only if these two payoffs are equal.

We need to solve the equation

1 w/(w+(1-w)(1-h)) +9(1-(w/(w+(1-w)(1-h))) =3 w/(w+(1-w)(1-h)) +6(1-w/(w+(1-w)(1-h)) for h.Slide41

Solution

Solving equation on previous slide, we see that if Saddam refuses inspections, Bush is indifferent between invading and not if h=3-5w/3(1-w). (Remember we assumed w<3/5) so 0<h<1)

If Saddam has no WMD’s, he is indifferent between allowing and not allowing inspections Bush would invade with probability 4/5 if there are no inspections.Slide42

Describing equilibrium strategies

Saddam: Do not allow inspections if he has WMD. Allow inspections with probability

h=3-5w/3(1-w) if he has no WMD. (e.g. if w=1/2, h=1/3. If w=1/3, h=2/3.)

Bush: Invade if Saddam has WMD and allows inspections, Don’t invade if Saddam has no WMD and allows inspections. Invade with probability 4/5 if Saddam does not allow inspections. Slide43

Enough signals for today