EC 330 / ECO2007 Signaling and screening information D&S chapter 9 1. Some areas of economics in which asymmetries in information (that is, some game players having private information that others lack) are crucial for explanation and prediction include: - the structures of contracts - the organization of companies - markets for labour - government regulation of business 2. Direct communication of information – which game theorists call „cheap talk‟ – isn‟t always strategically useless. When interests of players are aligned, as in an assurance game, cheap talk can be used to select focal points. If this is so then sending a signal constitutes an action in the game and must be so represented. In the assurance game between Harry and Sally in a previous lecture, for example, we would include a first round in which Harry (it could have been Sally instead) can choose among the following pure strategies: - H1: Signal “I‟m going to Starbuck‟s”, then go to Starbuck‟s - H2: Signal “I‟m going to Local Latte”, then go to Local Latte - H3: Signal “I‟m going to Starbuck‟s”, then go to Local Latte - H4: Signal “I‟m going to Local Latte”, then go to Starbuck‟s - H5: Send no signal, then go to Starbuck‟s - H6: Send no signal, then go to Local Latte. Each of these must appear as a branch from an information set assigned to Harry. Sally will choose among the following pure strategies: S1: If Harry signals “I‟m going to Starbuck‟s”, go to Starbuck‟s. S2: If Harry signals “I‟m going to Local Latte”, go to Local Latte S3: If Harry signals “I‟m going to Starbuck‟s”, go to Local Latte. S4: If Harry signals “I‟m going to Local Latte,” go to Starbuck‟s. S5: If Harry sends no signal, go to Starbuck‟s S6: If Harry sends no signal, go to Local Latte. In the assurance game, the following are all NE: (H1, S1)*, (H2, S2), (H3, S3), (H4, S4)*, (H5, S5)*, (H6, S6), (H1, S4)*, (H1, S5)*, (H4, S1)*, (H4, S5)*, (H2, S3), (H2, S6), (H5, S1)*, (H5, S4)*, (H6, S2), (H6, S3) Those marked with * are efficient. The fact that (e.g.) (H1, S4) is an efficient NE tells us that the signals are conventions – that is, cheap talk. Finally, there is an infinite number of mixed-strategy NE, all of which are inefficient, and in some – those that randomize to yield equal proportions of destinations in both directions – no signaling occurs at all. (These are called „babbling equilibria‟.) 3. Since no player in a zero-sum game should ever give away private information, as signaling games zero-sum games have only babbling equilibria. 4. The most interesting cases are those in which interests are partly but not perfectly aligned. Here we can get complex mixtures of signaling and screening in equilibria. 5. Consider, for example, a broker recommending stock purchases to a client. Suppose that she knows whether the stock is good or bad and the client doesn‟t. She says the stock is good. Should the client believe the signal or not? This will depend on the balance to the broker of retaining your future business, and unloading bad stocks from her firm‟s inventory. The client must estimate her utility function. If the two utility functions are perfectly aligned, the client should always act as if the signal is true; if the two utility functions are perfectly zero-sum, the client should pay no attention to any signals. These are the extreme cases; in between lie all the intermediate ones where the client should pay some attention to signals but weight their value according to a probability function based on the utility estimates. 6. The most important kinds of signals in games are not cheap talk signals, but those which have positive costs to the signaler. If the cost in question has a direct relationship to the private information the signal concerns, then the signal is evidence (but not necessarily conclusive evidence) for its truth. 7. We‟ll work a numerical example. Suppose that there are two types of college graduates, able (A) and challenged (C). Potential employers are willing to pay $150,000 to A‟s and $100,000 to C‟s. A‟s can pass tough courses with less effort than C‟s. Suppose A‟s regard the cost of a tough course as worth $6,000 per year of salary, while C‟s regard the cost as $9,000 per year. How can the employer use this knowledge about relative costs to extract signals from student transcripts? 8. The employer seeks a number n such that anyone who has passed n or more tough courses should be offered an A salary and anyone who has passed less than n should be offered a C salary. If the employers can find n then no C types should take tough courses – since the cost of them will be wasted – and should major in Communications. A types should aim to take exactly n tough courses. 9. The condition to make sure that no C‟s are incentivized to pass as A‟s is 100,000 ≥ 150,000 – 9000n, or 9n ≥ 50, or n ≥ 5.56. 10. The condition that n is not so high so as to discourage even A types is 150,000 – 6,000n ≥ 100,000, or 50 ≥ 6n, or 8.33 ≥ n. 11. These two conditions together are called incentive- compatibility constraints because they make actions (taking tough courses to send a signal) compatible with incentives for players to do so. 12. When incentive-compatibility constraints exist such that players can signal their type if and only if their private information is true, we say that a separating equilibrium exists. 13. Note that in our example the A types bear the full cost ($36,000) of the screening device (since C types take no costly tough courses). They bear this cost to prevent C types from confusing the market. (So C types inflict a negative externality on A types.) 14. Whether a situation of this kind has a separating equilibrium depends on the proportions of the types in the population relative to the cost of the negative externality. Suppose no one paid the negative externality so employers paid salaries that reflected the probability that a randomly drawn person is an A or a C. Imagine proportions are as follows: 20% of people are A‟s and 80% are C‟s. Then employers will pay 0.2 $150,000 + 0.8 $100,000 = $110,000. When they pay the negative externality cost, A‟s make $114,000 ($150,000 - $36,000), so at these proportions the separating equilibrium holds. But now suppose proportions are instead as follows: 50% of people are A‟s and 50% are C‟s. Then the common salary will be $125,000 and it will not be worth the cost to A‟s to take tough courses. In this case, the only available equilibrium, in which no signaling is possible and all types are treated alike, is called a pooling equilibrium. 15. In our example, there will not likely be a pooling equilibrium because employers are competing with one another for good workers. Thus each employer will have an incentive to offer $132,000 to a person who‟s taken just one tough course. This is incentive-compatible because a tough course costs just $6,000 to an A, but her salary increment is $7,000. The employer here makes a profit of $150,000 - $132,000 = $18,000. But this now causes the pooling equilibrium to collapse. The employer who offers the increment will attract only A types. This lowers the supply of A‟s on the market, so that the average salary given pooling begins to fall from $125,000. At some point it will fall to a point where it‟s worthwhile for C‟s to take one tough course. At this point the employer trying to attract A‟s must raise her requirement to two courses and raise the salary offer by a minimum of $6,000 (so C‟s won‟t find two courses worthwhile.) This adjustment process continues until the market is back at the original separating equilibrium. 16. Note that it‟s arbitrary that we picked the employer as the party whose screening action triggers the collapse of the pooling equilibrium. It could as well have been an A student who sends a signal by taking one tough course and then offering her services for $132,000. At the beginning of the process where the pooling equilibrium is in place, no C can match this signal. 17. Our restriction in the example to 2 types is just for simplicity. We could have any number of types, with a resulting whole hierarchy of signals represented by educations of varying degrees of difficulty. An additional potential complication arises from the utility functions of the universities. On the one hand, they‟re in the business of selling signaling devices to both sides of the market. This gives them an incentive not to make tough courses any easier. On the other hand, they can grab a quick profit by allowing standards to relax, and thus selling signaling devices at a discount to C types before the employers have learned that the information has been degraded. If all universities do this, we can get a situation in which employers continuously search for the new equilibrium, which universities keep moving. If pursued unchecked, this behavior will destroy the market for education. The problem is that the quality of the information market is a commons good, which the universities face together in the form of an n-person PD. (In reality, there isn‟t just one „market for education‟. There‟s a market for engineering education, another market for medical education, another market for business education, etc.. At any given time in a given country, some of these markets will be closer to separating equilibria and others will be closer to pooling equilibria.) 18. In a market with a signaling equilibrium, those who do not pay to send the signal will be assumed to have the bad information. Thus people in such markets cannot opt out of the game. 19. Now we consider contracting in which (as there usually is) there‟s private information on at least one side. 20. Suppose I‟m hiring a manager for a project. The project will earn $600,000 if it succeeds. Its probability of success is 60% if the manager puts in routine effort, but rises to 80% if the manager puts in high effort. 21. Imagine the manager demands $100,000 for a routine effort and $150,000 for a high effort. Since 20% of $600,000 is $120,000, I should pay the manager the extra $50,000 – if I can be sure the manager really will put in the extra effort in return. But how can I be sure of this? How would I know if the manager put in only routine effort? After all, there‟s a 20% chance of failure even with a high effort – so the proof isn‟t in the pudding. 22. Still, since high effort improves the probability of success, success provides some information about the probability of effort having been high. Suppose I offer the manager a base salary of s plus a bonus of b to be paid if the project succeeds. Then the manager‟s expected earnings are s + .6b for a routine effort and s + .8b for a high effort. His expected extra earnings from the better effort are (s + .8b) – (s + .6b) or (.8 - .6)b = .2b. For the extra effort to be worth his while it must be that .2b ≥ $50,000, or b ≥ $250,000. Once again, this is the incentive-compatibility constraint in this instance. 23. In addition, the manager must make at least $150,000 to work for me at all. Thus his participation constraint is given by s + .8b ≥ $150,000. 24. I want to maximize my profit by paying the manager just enough to induce the high effort. I want the smallest s that satisfies the participation constraint, s = $150,000 - .8b. But b must be at least $250,000, so s can be no more than $150,000 - .8 $250,000 = $150,000 - $200,000 = - $50,000. I must pay the manager a negative salary. This could mean either that he puts up an equity stake of $50,000 as a capital investor, or that he is fined $50,000 if the project fails. Institutional rules and practices may rule this out, in which case I must overfill the participation constraint – thus paying a cost for the asymmetry of information. The smallest non-negative salary I can provide is 0 + .8 $250,000 = $200,000. Is this extra $50,000 worth it for me? By paying it I get an expected profit of .8 $600,000 - $200,000 = $280,000. Had I offered only the salary sufficient for the routine effort I would have made .6 $600,000 - $100,000 = $260,000. So, yes. But had the project promised to make only $400,000 if successful, the answer would have been no. 25. Situations in which there is efficiency loss due to people‟s incentives not to consider the other party‟s utility in a game are called moral hazard problems. It is the reason why, for example, insurance schemes include deductibles – they incentivize the insured party to take some responsibility for guarding against loss. 26. We now learn how to model, in general, games of asymmetric information between two players. 27. Suppose there is an attacker and a defender. The Defender may be tough or weak, and the Attacker doesn‟t know which at the time when she must choose between invading or not invading. Suppose A estimates the probability of D‟s being tough is .25, and D knows this is A‟s estimate. If A invades and D is tough there is a major war and the outcome is (-10, -10). If A invades and D is weak, the outcome is (5, -5). Suppose now that before A decides, D has a move in which he can send a signal. Can D send a signal that will change the NE of the game? The intuitive answer might seem to be `no‟. After all, weak parties have more incentive to try to prevent attack than strong ones; so a signal designed to prevent attack might seem to be necessarily self-defeating. This can be true in many cases – but not necessarily, so long as the cost of the signal stands in the right proportional relations to the defender‟s payoffs. 28. Consider the following extensive-form game: 1. Nature D is W: .75 D is T: .25 2. 3. D D NS S NS S 4. 5. A A 6. 7. A A NI I NI I NI I NI I (0, 0) (-5, 5) (-6, 0) (-11, 5) (0, 0) (-10, -10) (-6, 0) (-16, -10) 29. D‟s toughness or weakness is not up to it (at least in the short run), so we have parametric uncertainty here. Thus we must bring Nature (`Player 0‟) into the game. Suppose that Nature makes D tough with probability .25 and weak with probability .75. D knows what Nature has done. It chooses between sending and not sending a signal. A observes the signals but not Nature‟s move. This is reflected in the distribution of the information sets. 30. How do we locate NE here? Zermelo‟s algorithm can‟t literally be applied because the multiple-membered information sets, so SPE is an inappropriate equilibrium concept. But it has an analogue, called sequential equilibrium. We‟ll digress from the game at hand to explain this. 31. Consider the three-player game below known as „Selten's horse‟ (for its inventor, Nobel Prize winner Reinhard Selten, and because of the shape of its tree): 32. One of the NE of this game is (L, r2, l3). This is because if Player I plays L, then Player II playing r2 has no incentive to change strategies because her only node of action, 12, is off the path of play. But this NE seems to be purely technical; it makes little sense as a solution. This reveals itself in the fact that if the game beginning at node 14 could be treated as a subgame, (L, r2, l3) would not be an SPE. Whenever she does get a move, Player II should play l2. But if Player II is playing l2 then Player I should switch to R. In that case Player III should switch to r3, sending Player II back to r2. And here's a new, „sensible‟, NE: (R, r2, r3). I and II in effect play „keepaway‟ from III. 33. This NE is „sensible‟ in just the same way that a SPE outcome in a perfect-information game is more sensible than other non-SPE NE. However, we can't select it by applying Zermelo's algorithm. Because nodes 13 and 14 fall inside a common information set, Selten's Horse has only one subgame (namely, the whole game). We need a „cousin‟ concept to SPE that we can apply in cases of imperfect information, and we need a new solution procedure to replace Zermelo's algorithm for such games. 34. Notice what Player III in Selten's Horse is wondering about as he selects his strategy. "Given that I get a move," he asks himself, "was my action node reached from node 11 or from node 12?" What, in other words, are the conditional probabilities that III is at node 13 or 14 given that he has a move? Now, if conditional probabilities are what III wonders about, then what Players I and II must make conjectures about when they select their strategies are III's beliefs about these conditional probabilities. In that case, I must conjecture about II's beliefs about III's beliefs, and III's beliefs about II's beliefs and so on. The relevant beliefs here are not merely strategic, as before, since they are not just about what players will do given a set of payoffs and game structures, but about what they think makes sense given some understanding or other of conditional probability. 35. We don‟t want to impose any stronger rationality assumptions than necessary. Bayes’s rule is the minimal true generalization about conditional probability that an agent could know if it knows any such generalizations at all. Bayes's rule tells us how to compute the probability of an event F given information E (written „pr(F/E)‟): pr(F/E) = [pr(E/F) × pr(F)] / pr(E). We will assume that players do not hold beliefs inconsistent with this equality. 36. We may now define a sequential equilibrium . A SE has two parts: (1) a strategy profile § for each player, as in all games, and (2) a system of beliefs μ for each player. μ assigns to each information set h a probability distribution over the nodes x in h, with the interpretation that these are the beliefs of player i(h) about where in his information set he is, given that information set h has been reached. Then a sequential equilibrium is a profile of strategies § and a system of beliefs μ consistent with Bayes's rule such that starting from every information set h in the tree player i(h) plays optimally from then on, given that what he believes to have transpired previously is given by μ( h) and what will transpire at subsequent moves is given by §. 37. We now demonstrate the concept by application to Selten's Horse. Consider again the uninteresting NE (L, r2, l3). Suppose that Player III assigns pr(1) to her belief that if she gets a move she is at node 13. Then Player II, given a consistent μ(II), must believe that III will play l3, in which case her only SE strategy is l2. So although (L, r2, l3) is a NE, it is not a SE. This is of course what we want. 38. The use of the consistency requirement in this example is somewhat trivial, so consider now a second case: 39. Suppose that I plays L, II plays l2 and III plays l3. Suppose also that μ(II) assigns pr(.3) to node 16. In that case, l2 is not a SE strategy for II, since l2 returns an expected payoff of .3(4) + .7(2) = 2.6, while r2 brings an expected payoff of 3.1. Notice that if we fiddle the strategy profile for player III while leaving everything else fixed, l2 could become a SE strategy for II. If §(III) yielded a play of l3 with pr(.5) and r3 with pr(.5), then if II plays r2 his expected payoff would now be 2.2, so (L, l2, l3) would be a SE. Now imagine setting μ(III) back as it was, but change μ(II) so that II thinks the conditional probability of being at node 16 is greater than .5; in that case, l2 is again not a SE strategy. 40. Thus ends the digression. We now apply informal SE reasoning to the Defender / Attacker game. The question we want to answer is whether the game has a separating equilibrium in which D signals and A‟s behavior is influenced by the signal. 41. Suppose D chooses S if her type is T and chooses NS if her type is W. Suppose A chooses I if D chooses NS and NI if D chooses S. Is this an SE? 42. If A finds himself in the S information set, he infers from his conjecture about D‟s strategy that he is at node 7. In that case NI is his best reply. At the NS information set, A will infer that he is at node 4, where I is his best reply. Now consider a T-type D. If she chooses S A will infer that he is at node 7 and choose NI. This yields D her preferred payoff of the two outcomes descending from node 7. If D chooses NS, A will infer that he‟s at node 4 when he‟s really at node 5. He‟ll then choose I, yielding D a worse payoff than the other outcome descending from node 5. I leave it to you to show by parallel reasoning that a W-type D will prefer to choose NS. Thus we have a separating SE. 43. Suppose we alter the game so that the cost of signaling falls to 4, less than the cost to a type-W D of giving in without a fight. Here‟s our new tree in that case: 1. Nature D is W: .75 D is T: .25 2. 3. D D NS S NS S 4. 5. A A 6. 7. A A NI I NI I NI I NI I (0, 0) (-5, 5) (-4, 0) (-9, 5) (0, 0) (-10, -10) (-4, 0) (-14, -10) 44. A still plays as before. But now the type-W D gets a payoff of -5 if she chooses NS and A replies with I, whereas she could get -4 if she chose S. Thus the separating equilibrium is not a SE, because D has an incentive to lie, so A should ignore her signal. We have a pooling equilibrium where all types of D choose S. We can also construct a pooling equilibrium in which all types of D choose NS. If T-type D‟s do this, then A‟s will choose by assuming that observation of NS indicates that D is W with pr(.75). In this case if A chooses I he gets an expected payoff of (-10 .25) + (5 .75) = 1.25. If A chooses NI his expected payoff is 0. Thus A will always choose I. This is best for D whether she‟s T-type or W- type: T-type gets -14 from choosing S and -10 from choosing NS, while W-type gets -9 from choosing S and - 5 from choosing NS. Thus we have a pooling SE. 45. Finally, we can have semiseparating equilibria, in which there‟s mixing by one player and by one type of the other player. In our game, this can arise when one type of D gets the same payoff from S and NS when A mixes. For example, suppose all T-type D‟s choose S while W-type D‟s mix. In that case observing S gives A some, but not perfect, information. We‟ll pass over the details of this somewhat esoteric case. Just be aware that it‟s possible. 46. Examples of real-life separating equilibria: - Venture capitalists will be willing to lend larger amounts to entrepreneurs who are willing to risk larger stakes of their own. - Sellers of high-quality goods offer warranties that sellers of low-quality knock-offs can‟t afford. - Female birds prefer males with gaudy or exaggerated feathers (think of male peacocks) because only healthier males can afford to carry the extra weight and/or be more conspicuous to predators.
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