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Signaling

Signaling. Econ 171. Breakfast: Beer or quiche? A Fable *. *The original Fabulists are game theorists, David Kreps and In-Koo Cho. Breakfast and the bully . A new kid moves to town. Other kids don’t know if he is tough or weak.

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Signaling

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  1. Signaling Econ 171

  2. Breakfast: Beer or quiche?A Fable * *The original Fabulists are game theorists, David Kreps and In-Koo Cho

  3. Breakfast and the bully • A new kid moves to town. Other kids don’t know if he is tough or weak. • Class bully likes to beat up weak kids, but doesn’t like to fight tough kids. • Bully gets to see what new kid eats for breakfast. • New kid can choose either beer or quiche.

  4. Preferences • Tough kids get utility of 1 from beer and 0 from quiche. • Weak kids get utility of 1 from quiche and 0 from beer. • Bully gets payoff of 1 from fighting a weak kid, -1 from fighting a tough kid, and 0 from not fighting. • New kid’s total utility is his utility from breakfast minus w if the bully fights him and he is weak and utility from breakfast plus s if he is strong and bully fights him.

  5. Nature Tough Weak New Kid New Kid Beer Quiche Beer Quiche Bully Fight Don’t Don’t Fight Bully B 1 0 1+s -1 0 0 Fight Don’t -w 1 Fight Don’t 1 0 s -1 1-w 1 0 0

  6. How many possible strategies are there for the bully? • 2 • 4 • 6 • 8

  7. What are the possible strategies for bully? Fight if quiche, Fight if beer Fight if quiche, Don’t if beer Fight if beer, Don’t if quiche Don’t if beer, Don’t if quiche

  8. What are possible strategies for New Kid • Beer if tough, Beer if weak • Beer if tough, Quiche if weak • Quiche if tough, Beer if weak • Quiche if tough, Quiche if weak

  9. Separating equilibrium? • Is there an equilibrium where Bully uses the strategy Fight if the New Kid has Quiche and Don’t if the new kid has Beer. • And the new kid has Quiche if he is weak and Beer if he is strong. • For what values of w could this be an equilibrium?

  10. Best responses? • If bully will fight quiche eaters and not beer drinkers: • weak kid will get payoff of 0 if he has beer, and 1-w if he has quiche. • So weak kid will have quiche if w<1. • Tough kid will get payoff of 1 if he has beer and s if he has quiche. • So tough kid will have quiche if s>1

  11. Suppose w<1 and s<1 • We see that if Bully fights quiche eaters and not beer drinkers, the best responses are for the new kid to have quiche if he is weak and beer if he is strong. • If this is the new kid’s strategy, it is a best response for Bully to fight quiche eaters and not beer drinkers. • So the outcome where Bully uses strategy “Fight if quiche, Don’t if beer “ and where New Kid uses strategy “Quiche if weak, Beer if tough” is a Nash equilibrium.

  12. If w>1 Then if Bully uses strategy “Fight if quiche, Don’t if beer”, what will New Kid have for breakfast if he is weak?

  13. Pooling equilibrium? • If $w>1$, is there an equilibrium in which the New Kid chooses to have beer for breakfast, whether or not he is weak. • If everybody has beer for breakfast, what will the Bully do? • Expected payoff from Fight if beer, Don’t if quiche depends on his belief about the probability that New Kid is tough or weak.

  14. Payoff to Bully • Let p be probability that new kid is tough. • If new kid always drinks beer and bully doesn’t fight beer-drinking new kids, his expected payoff is 0. • If Bully chooses strategy Fight if Beer, Don’t if Quiche, his expected payoff will be -1/2p+1/2(1-p)=1-2p. Now 1-2p<0 if p>1/2. So if p>1/2, “Don’t fight if Beer, fight if Quiche” is a best response for Bully.

  15. Pooling equilibrium • If p>1/2, there is a pooling equilibrium in which the New Kid has beer even if he is weak and prefers quiche, because that way he can conceal the fact that he is weak from the Bully. • If p>1/2, a best response for Bully is to fight the New Kid if he has quiche and not fight him if he has beer.

  16. What if p<1/2 and w>1? There won’t be a pure strategy equilibrium. There will be a mixed strategy equilibrium in which a weak New Kid plays a mixed strategy that makes the Bully willing to use a mixed strategy when encountering a beer drinker.

  17. What if s>1? • Then tough New Kid would rather fight get in a fight with the Bully than have his favorite breakfast. • It would no longer be Nash equilibrium for Bully to fight quiche eaters and not beer drinkers, because best response for tough New Kid would be to eat quiche.

  18. Signaling Game Problems

  19. Problem 1, p 348 If Buyers believe that the fraction of good cars on market is q, their Expected Value of a random car is 12000q+7000(1-q)=7,000+5,000q • 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> 10,000and 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 good • Is confirmed. We have a pooling equilibrium.

  20. There is also a separating equilibrium 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.

  21. 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.)

  22. 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.

  23. 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?

  24. Problem 5, page 350 George Bush and Saddam Hussein

  25. 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.

  26. 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.

  27. 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))

  28. 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.

  29. 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.

  30. 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.

  31. 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.

  32. 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.

  33. 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.

  34. Professor Drywall’s Lectures

  35. Another 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.

  36. 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.

  37. 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?

  38. 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?

  39. 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.

  40. 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.

  41. So there we are.

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