Signaling Game Problems

Problem 1, p 348Quality Probability Value to Seller Value to Buyer

Good Car q 10,000 12,000

Lemon 1-q 6,000 7,000

their Expected Value of a random car is12000q+7000(1-q)=7,000+5,000q

If Buyers believe that the fraction of good cars on market is q,

• In this case, we can expect all used cars to sell for about PU=7,000+5,000q. • If q>3/5, then PU=7000+5000q> 10,000 and so owners of lemons and of good cars and of will be willing to sell at price PU. • Thus the belief that the fraction q of all used cars are goodIs confirmed. We have a 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. 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.

Problem 3, page 348

• Suppose that buyers believe that product with no warranty is low quality and that with warranty is high quality.

• High quality items work with probability H and low quality items work with probability L. Consumer values a working item at V.

• Buyers are willing to pay up to LV an item that works with probability L.

• Buyers are willing to pay up to V for any item with a money back guarantee. (If it works, their net gain is V-P

and if it fails they get their money back so their net gain is 0. Therefore they will buy if P<V.)

Equilibrium

• If the item with warranty sells for just under V and that with no warranty sells for just under LV, buyers will take either one.

• Given these consumer beliefs, V is the highest price that sellers can get for high quality with warranty and LV is the highest price for the low quality without warranty.

• Seller’s profits from high quality sales with guarantee are hV-c and profits from low quality without guaranty are LV-c.

• If seller put a guarantee on low quality items and sold them for V, his profit would be LV-c, which is no better than he does without a guarantee on these.

Equilibrium

• If buyers believe that only the good items have guarantees, the Nash equilibrium outcome confirms this belief.

• If fraction of items sold that are of high quality is r, then retailer’s average profit per unit sold

Is rHV+(1-r)LV.• Retailer can not do better with a pooling

equilbrium in which he guaranteed nothing, or in one in which he guaranteed everything.

Can you show this?

Problem 5, page 350

George Bush and Saddam Hussein

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.

Payoffs for Hussein if he has no WMDs

Payoff from not allow is 2b+8(1-b)=8-6bPayoff from allow is 4, since if he allows Bush will not invade.Hussein is indifferent if 4=8-6b or equivalentlyb=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.

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))

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.

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.

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.

Problem 5, p 350

• 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.

Equilibrium where high and medium types go to college, low does not.

• If high and medium types go to college, what is the expected profit from hiring a college grad as a manager?

• Find probability p that someone is of high type given college:

• P(H|C)=P(H and C)/P( C)=(1/6) / (1/6+1/2)=1/4• Expected profit is 1/4x14+3/4x6=8.• If you hire a college grad as clerk, expected profit

is 7. So better off to hire her as manager.

Equilibrium for workers.

• High types get paid 15 as manager have college costs of 2. So net wage is 13. That’s better than the 10 that nondegree people get as clerks.

• Medium type get paid 15 as manager have college costs 4, net wage of 11, so they prefer college and managing to no college and clerk.

• Low types would get 15 as manager with college costs of 6. Net pay of 15-6=9 is less than they would get with no college and being a clerk.

Professor Drywall’s Lectures

A 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 of $1250 per month.

• One employer believes that Drywall’s lectures are useful and requires its workers attend 10 monthly lectures by Professor Drywall and payswages 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?

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?

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.

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.

So there we are.