Signaling Game Problems
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Transcript of Signaling Game Problems
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Signaling Game Problems
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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.
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There is also a separating equilibriumQuality 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.
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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.)
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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.
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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?
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Problem 5, page 350
George Bush and Saddam Hussein
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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.
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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.
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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))
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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.
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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.
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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.
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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.
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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.
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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.
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Professor Drywall’s Lectures
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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.
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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.
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• 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?
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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?
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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.
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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.
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So there we are.