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Signaling Game Problems

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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 signals

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Example: Problem 1, Chapter 11 QualityProbabilityValue to SellerValue to Buyer Good Carq10,00012,000 Lemon1-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.

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Nature Good car Lemon Keep Owner Sell Buy Don’t buy Owner Buyer 0000 0000 0000 0000 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 Nature

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Is there a pooling equlibrium? QualityProbabilityValue to SellerValue to Buyer Good Carq10,00012,000 Lemon1-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.

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There is also a separating equilibrium QualityProbabilityValue to SellerValue to Buyer Good Carq10,00012,000 Lemon1-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.

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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.

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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.

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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.

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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.

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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

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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)

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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.

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Problem 11.6 (Advertising)

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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.

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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.

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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*.

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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*.

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Suppose 150. 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.

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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.

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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.

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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*

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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.

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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.

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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.

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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)1/6 B)1/5 C)1/4 D) 1/3 E)1/2

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Conditional probability (Bayes’ Law) P(H|C)=P(C and H) /P(C) =1/6÷(1/6+1/2)=1/4.

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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.

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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.

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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.

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Problem 11.7

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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.

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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.

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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.

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Problem 11.5 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)=8-6b 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.

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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

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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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Enough signals for today

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