Chapter 5

1. Chapter 5 Sequential Rationality

2. Example 1: should we offer the position? A candidate to the computer science department has already written 11research papers, and the departmentwould like to decide on whether to make her a job offer based onthe quality of the papers. There are 11 committee members who are each given one paper to read in order to make a recommendation. Initially, each paper may be "good" or "bad" with equal probabilities, and the department has chosen to make an offer to a candidate if he has a majority of "good" papers. Committee member values the correctrecommendation of the committee, at $1000 to him, but valuesthe time he needs to spend on reading the paper at $400.

3. Example 1: should we offer the position? Asimple mechanism asks all the committee members, simultaneously, for their recommendations. The strategy tuple where all agents choose to read the papers and report truthfully is not an equilibrium. Consider the perspective of agent 1: assuming allagents replied (truthfully, or not), then agent 1 can alter the outcome only if the other 10 replies split evenly between 0 and 1 whichhas a probability of approximately 0.25. Therefore, by guessing, and assuming all other agents compute, he will gain 0.25 X 500 + 0.75 X 1000 = $875. 25% of the time the right decision is made (as he has a 50% chance of guessing right) and 75% of the time he gets the right decision as the others make the correct decision. However, by computing an agent gains 1000-400=$600, and so player 1 has no incentive to compute (the same for all 11 agents). .25*600 + .75*600 = $600

4. Example 1: should we offer the position? This elicitation mechanism will also fail if only agent 1 has the above cost and all other agents have zero costs (the same analysis will hold for agent 1). If however agents 2,3,….,11 are asked first for their recommendations, and agent 1 is approached only if there is a tie among the ten recommendations, then all agents will have enough incentive to invest the effort as 100% of the time you are asked you gain $600 as compared to $500 for guessing! This illustrates the power of sequential mechanisms. This motivates the careful discussion of sequential elicitation mechanisms, i.e. the construction of mechanisms that approach agents in a well designed sequence. Goal is to change the way the “game” is played so there is no temptation to shirk! This is the positive outcome of the study! We aren’t trying to teach you to be a shirker, but to devise mechanisms to de-incentivize shirking.

5. Subgame perfection ensures that the players continue to play rationally as the game progresses. • We have a problem with imperfect information, however, as we might not have subgames at all. sequential equilibrium: a pair (,µ) where  is a behavior strategy profile and µ is a system of beliefs consistent with  such that no player can gain by deviating from  at any member of the information set.

6. 5.1 Market for Lemons • Two player game in which one player has more information than the other. • Like my example of forgetting which cards had been played, but in this case, one player just knows more. • Selling cars. • ph is the reservation price (least he is willing to accept) for a high quality car. • pl is the reservation price (least he is willing to accept) for a low quality car.

7. Similarly, the buyer has similar reservation prices: • H highest price he is willing to pay for high quality car. • L highest price he is willing to pay for a low quality car. • Look at sequential game when a seller puts a car on the market. Nature reveals the quality of the car to the seller, but the buyer doesn’t know it. • Seller must decide whether to ask ph or pl. • Buyer doesn’t know price.

8. buy (p-ph, H-p) 2 not X (0,0) ph 1 (p-pl, L-p) buy 2 not pl Y (0,0) G (p-ph, H-p) Nature ph buy B 2 not M (0,0) 1 pl (p-pl, L-p) buy 2 N not (0,0)

9. Note, there are no subgames. • What should player 2 do? • If price is close to ph, could assume the car is of high value. If close to pl, could assume the car is of low value. • The seller knows this. Both would agree to split the profit in half, so gains would be equal. • BUT…Why not just charge a high price for every car! • So now the buyer doesn’t know what to do. • Could assume has equal likelihood of being high or low (uniform distribution)

10. His expected value is • ½(H-p) + ½(L-p) • Since he needs a postive expected value • ½(H-p) + ½(L-p) >0 • p  ½ (H+L) • If offered a high price, he believes he is at node X with probability ½ and at node Y with probability ½. • But if he gets such a low offer, he knows he isn’t at X or Y, but at N.

11. Consider a case by case analysis • Case 1: ½(H+L)  ph Both types of cars could really be for sale, so buyer will buy at this price. Case 2: ½(H+L) < ph No high quality car can be sold for that price, so you are at node N. Buyer insists on somewhere between pl and L, as he knows he is looking at a low quality car. Equilbrium – which is driven by consistent beliefs

13. Lets take another look. utilities = (seller, buyer) • Lots of prices could be offered – not just ph or pl. • Use line to represent infinite number of nodes (p-ph, .5(H+L)-p) Like before only assume equally likely you have Good as Bad car buy X not price, p (0,0) Good Nature (p-pl, .5(H+L)-p) Bad buy Y price, p not (0,0)

14. Case 1: ½(H+L) >ph • u1(p,s) = p-ph (if s(X) = buy) • = p-pl (if s(Y) = buy) • = 0 (if s(X) = not buy) • =0 (if s(Y) = not buy) • u2(p,s) = ½(H+L)-p if s = buy • =0 if s = not buy • Optimal stategy: • p* = ½(H+L) s* = buy if ½(H+L)  p • = not buy if ½*(H+L) <p

15. Case 2: ph > ½(H+L) • Seller knows that buyer will never buy a car at a price greater than ½(H+L) so only low-quality cars are on the market. • u1(p,s) = p-pl (if buy) • = 0 (if not buy) • u2(p,s) = L-p (if buy) • = 0 (if not buy) • p* = L • S* = buy if L > p • = not buy if L < p

16. Notice – no subgames • role of beliefs is critical.

17. Let’s look again using different values…Example of adverse selection: the market for ‘lemons’ You want to buy a used car. There are two types of cars – good cars and lemons.

18. Example of adverse selection: the market for ‘lemons’ Suppose the seller always knows what type of car they are selling. What happens in the market depends on whether buyers can also tell the type of car.

19. The market for ‘lemons’ – case 1: symmetric information (Both can tell a lemon)

20. The market for ‘lemons’: case 2 – asymmetric information Suppose the buyer cannot tell a good car from a lemon before they buy. Will you buy if the price is at least $8,000? NO! At this price, every seller will want to sell. But this means that if you buy a car it has a 60% chance of being a lemon (worth $6,000 to you) and a 40% chance of being good (worth $10,000 to the buyer). So the expected value of a car to the buyer is $7,600. So you will not pay more than $8,000 for a car with an expected value of $7,600!

21. The market for ‘lemons’: case 2 – asymmetric information Suppose the buyer cannot tell a good car from a lemon before they buy. Will you buy if the price is between $6,000 and $8,000? NO! At this price, only the sellers of ‘lemons’ will want to sell. Every car being offered is a lemon and you will not pay more than $6000. So we expect that the market will have a price of between $3,000 and $6,000 with only lemons sold

22. The market for ‘lemons’ – case 2: asymmetric information

23. Adverse selection So: the problem of adverse selection can lead to the complete collapse of the market for good cars A similar problem faces insurance companies and the market for ‘loanable funds’. Adverse selection can also lead to ‘statistical discrimination’.

24. Responses to adverse selection • Note that the problem of adverse selection harms the un-informed parties and some of the informed parties. • In the lemons example, it meant that buyers could not buy good cars. But also sellers of good cars could not get a reasonable price for their cars. • Un-informed buyers may try to overcome the information asymmetry by searching for more info • Informed sellers may try to overcome the problem by • Warranties • Signaling

25. Example: job-market model of bilateral uncertainty – uncertainty on both sides. • Workers are uncertain about what job descriptions advertised by firms really mean • Firms are uncertain about the qualifications of workers before they are interviewed. Both types of uncertainty can be resolved, but both processes are costly. Intermediaries (recruiters) can perform the job matching but only at the cost of transforming the firm’s objectives between the parties.

26. Each branch has an associated probability firms info sets Knows what fits, but can’t tell if good person good person, fits employee info sets Knows if he’s good or bad but can’t tell it he’s a fit. bad person, fits good person, doesn’t fit bad person, doesn’t fit

28. Information and market failureCan you answer these questions? • Why does a new car lose about one quarter of its value when you drive it away from the dealer? (can’t convince others it is a good car) • Why do manufacturers of products that almost never break down still offer warranties? (need to convince others the product is good) • Why is car insurance more expensive for all younger drivers? (paying more of the actual cost of converage) • Why do insurance companies make you pay the first part of any claim (the deductible)? (No temptation to let a flood water set so you get all new furniture. No temptation to be careless.)

29. Information and market failure • There are two basic types of information asymmetry that can lead to market failure • Adverse selection: one party to a deal has private information that affects the value of the deal • Moral Hazard: one party to a deal has to take an action that cannot be perfectly monitored by the other party. The action affects the value of the deal. Called a moral hazard as there may be an incentive to do something immoral (dishonest) – like shirk at your job.

30. Moral hazard can be present any time two parties come into agreement with one another. Each party in a contract may have the opportunity to gain from acting contrary to the principles laid out by the agreement. • For example, when a salesperson is paid a flat salary with no commissions for his/her sales, there is a danger that the salesperson may not try very hard to sell because the wage stays the same regardless of how much or how little the owner benefits from the salesperson's work.

31. Sometimes people do better than break even when misfortune strikes, and this possibility has greatly interested economists. • If the misfortune costs a person $1000, but insurance will pay $2000, the insured person has no incentive to avoid the misfortune and may act to bring it on. • For example, if you have full replacement costs on your house insurance, you may be happy when grape juice ruins your 10 year old carpet. Obviously, deliberately throwing grape juice to get insurance reimubursement is illegal. • This tendency of insurance to change behavior falls under the label moral hazard.

32. Adverse Selection Asymmetric information is feature of many markets - some market participants have information that the others do not have 1) The hiring process – a worker might know more about his ability than the firm does - the idea is that there are several types of workers - some are more productive than others are 2) Insurance – insurance companies do not observe individual characteristics such as driving skills 3) Project financing – entrepreneurs might have more information about projects than potential lenders 4) Used cars – sellers know more about the car’s quality than buyers Adverse selection is often a feature in these settings - it arises when an informed individual’s decisions depend on his privately held information in a way that adversely affects uninformed market participants .

33. Warranties Lets return to our car market example. Remember that buyers are willing to pay up to $10,000 for a good car but only up to $6,000 for a lemon. The problem is – which is which? Suppose that good cars never break down. However, a lemon breaks down often (that is why it is a lemon). Say lemons break down 80% of the time. Fixing a broken down car is expensive – about $5,000. Suppose now that a car seller offers you the following deal – “Buy the car for $9,000. If it breaks down, the seller will not only fix your car for you but also pay you $3,000 in cash as compensation” Should you buy the car? But, have you ever been offered that good of a warranty?

34. 5.2 Beliefs • beliefs are in important in finding solution to a game without subgames. • beliefs must be consistent with the way game is played. For example, in the Star Trek/Game Theory book gift example – you needed to know the likelihood a gift would be offered given the type of book. The game structure is key to deciding which beliefs you need to formulate. • System of beliefs: assigns probability distribution to nodes in the information set. (What node do I think I am at) • Use µ to represent that probability • Behavior strategy  for a player is the probability he will take each edge. (mixed strategy – what edge will I take) • completely mixed strategy – at every node, every choice is taken with positive probability

35. Example 5.2 • Player 1 plays a with .1 • Player 2 plays T with .1 • Player 1 plays L and L’ with .1 (4,2) .001 .1 L R .1 .9 T (0,3) .009 E .1 X B .9 L .1 (1,7) .009 F a R .9 (3,0) (2,6) O .081 .1 L’ G b .1 T .9 .9 R’ (2,4) .081 Y B .9 (3,5) .081 L’ .1 H .9 R’ (4,3) .729

36. Note, than in general, the probability you are at E is .01. • The conditional probability of p(E|X) = .1 as once you know you are at X, the probability of E is greater. • In the book example, the probability of giving a gift could be different depending on what the book is. A person might be MUCH more likely to give Star Wars as a gift than Game Theory.

37. 5.3 Bayes Consistent • A system of beliefs µ is said to be Bayes consistent with respects to a mixed behavior profile  if µ can be generated by . • In other words, you beliefs about probabilities make a behavior profile reasonable. • For example, if I1={E,F} and I2={G,H} • if µ(E|I1) = µ(G|I2) = 0 and • µ(F|I1) = µ(H|I2) = 1 • µ(X) = 0, µ(Y) = 1

38. Example 5.2 This means… • O-> Y -> H -> (4,3) is a good plan. (4,2) .1 L R .1 .9 T (0,3) E .1 X B .9 L .1 (1,7) F a R .9 (3,0) O (2,6) .1 L’ G b .1 T .9 .9 R’ (2,4) Y B .9 (3,5) L’ .1 H .9 R’ (4,3)

39. 5.4 Expected Payoff • We compute expected payoff in the regular way – multiply the payoff by the probability you get it.

40. (4,2) 1 Example 5.6 .6 N .4 (0,3) E(I2={NF}) = .2*E(N) +.8*E(F) .2 2 (1,7) .6 .8 X F .4 (2,6) The value player 1 uses for E(X) use HIS beliefs and his strategy. (3,0) 1 (2,4) G (3,5) Y H (4,3)

41. 5.5 Sequential Equilibrium • Sequential Equilbrium: a pair (,µ) where  is a strategy profile and µ is a system of beliefs consistent with  such that no player can gain by deviating from . • Note that what I believe may not be exactly the case, as I am not privy to the other person’s strategy. My strategy must be consistent with what I believe to be true.

42. Kreps-Wilson • Every sequential game with imperfect information has a sequential equilibrium. • I interpret that to mean, that given my understanding that is the “stable” thing to do. It might not be the right thing, but given what I know, I can do no better.

43. (4,2) 1 Example 5.10 L 1 N R 0 (0,3) T 1/2 2 (1,7) . B 1/2 X F (2,6) a 0 (3,0) L’ 1/6 1 R’ 5/6 (2,4) G b 1 (3,5) Y H (4,3) Can show equilibrium by seeing if can gain with other strategy.

44. Warranties A GOOD SELLER CAN MAKE SUCH AN OFFER Don’t buy (0,0) Breakdown under warranty (zero chance) Buyer Offer deal Good seller (-$7,000, $?) Buy Don’toffer deal No breakdown (100%) (0,0) (1000,$?)

45. Warranties But the buyer can infer this – the seller of a lemon would not offer the deal. Don’t buy (0,0) Buyer Offer deal Breakdown (80%) Lemon seller (-$2,000, $?) Buy Don’toffer deal No breakdown (20%) (0,0) ($6,000, $?)

46. Warranties Don’t buy (0,0) Buyer Offer deal Breakdown (80%) Lemon seller Expected payoff for lemon seller if buyer accepts offer is -$400. So if the lemon seller will not offer deal! Buy Don’toffer deal No breakdown (20%) (0,0)

47. Warranties • So the warranty ‘works’ • Only the good sellers will offer the warranty • Buyers can buy the car with the warranty, sure that they are buying a good car (and will never need to use the warranty) • But it is not worth while for the sellers of ‘lemons’ to offer the warranty – their cars break down and the warranty costs more than the increased price that they receive for their cars

48. Signaling • The warranty is an example of a ‘signal’ that the ‘good’ seller can send to the buyer. • In our example here the signal had no cost to the ‘good’ seller, but was expensive for the bad seller. This is not generally the case. • “This is a Good Car” sign is ineffective: every type of seller will use it, and it will provide no new info • For a signal to ‘work’ it requires three features • It must be less costly to the ‘good’ type than to the ‘bad’ type. • Even given the cost, it must be better for the ‘good’ type to distinguish themselves than be mistaken for a ‘bad’ type. • The cost to the ‘bad’ type must be high enough so that they do not want to pretend to be a ‘good’ type

49. Other examples: the early career rat race • Suppose there are ordinary and talented workers • Your boss can observe the quality of your work but not how difficult you found the task • If everyone spends the same time, the talented workers will be recognised and gain promotion • So the ordinary workers work harder to try and ‘appear’ to be talented • So to distinguish themselves, the talented workers also have to work hard

50. Other examples: the early career rat race • What will be the outcome? • Could get a separating equilibrium. This is where the signal ‘works’. The talented workers work too hard but are recognised. The ordinary workers just give up. • Or could get a pooling equilibrium. In this situation, the ordinary workers work hard and talented workers work normally. The boss interprets ‘ordinary’ performance as a sure sign of lack of talent. But the boss cannot infer anything from exceptional work – because everyone is doing it!