Unformatted text preview: GAME THEORY Econ 414 Lecture 2225 Incomplete Information Games Instructor: Nuno Limão slides_414_22_25_2011 1 OUTLINE Motivation From incomplete to imperfect information Application: Firm Entry and expansion Application: Trigger Strategies Application: Private information in Auctions Signaling games: motivation and structure Bayesian updating of beliefs Perfect Bayes‐Nash equilibrium Application: management trainee Application: a market for lemons slides_414_22_25_2011 2 MOTIVATION Definition: Games of incomplete information, strategic interactions where the game is not common knowledge Examples of incomplete information: Hiring employees: The firm doesn’t know whether employee is good, bad, lazy Competition: Firm’s don’t know if the competitor’s costs are high or low Buying a used car: is it low value (a lemon) or not? Selling car or health insurance to a person (are they high risk or low?) Auctions: What is the value that other bidders place on object up for auction? More generally: any time any player does not know the other’s “type” and thus payoffs with certainty Can we solve any game of incomplete information with the tools learned so far? slides_414_22_25_2011 3 Can we solve any game of incomplete information with the tools learned so far? Yes! Consider a case where the incomplete information is only regarding the payoffs. Then if a player is rational and has a dominant strategy then he/she does not need to know anything about the other players’ preferences or payoffs to decide on his optimal strategy e.g. player 1 chooses mean in PD, as does 2 if he has symmetric payoffs. Recall from lecture 5 that dominant strategy equilibrium require only rationality of each player. Game of incomplete information from player 1’s perspective Player 2 Mean Nice
Player Mean $2, ? $4, ?
1 Nice $1, ? $3, ? slides_414_22_25_2011 4 Can we solve games of incomplete information when there is no DSE? No. Try the above when 1’s payoffs for (Mean, nice) are $2,? (a conditional altruist?) Thus with the tools learned so far we can only solve a very limited set of interactions Notice this is quite different from imperfect information where the game structure (payoffs, etc) are common knowledge but players do not know what decision the other has taken The key insight from John Harsany (Nobel, 1994) that we will learn is how to transform a game of incomplete information (which we can’t solve) into one of imperfect information (which we can) slides_414_22_25_2011 5 FROM INCOMPLETE TO IMPERFECT INFORMATION Background In 1938, Nazi Germany annexed Austria. It was believed that Hitler was considering similar action against Czechoslovakia. To prevent another major war, British Prime Minister Chamberlain traveled to Munich to negotiate with Hitler. On September 29, 1939, Hitler and Chamberlain signed the Munich agreement, which gave Germany part of Czechoslovakia in exchange for Hitler’s promise that Germany would not go any further.The rest is history… What went wrong? With World War I in recent memory, Chamberlain feared a repeat of such horror. Furthermore, Chamberlain did not know Hitler’s true intentions. It is a strategic interaction where Hitler had some private information about his true intention that Chamberlain did not. slides_414_22_25_2011 6 Modeling Munich agreement from Chamberlain’s perspective: [Hitler’s payoffs unknown] Backward induction is not feasible if Chamberlain does not know Hitler’s payoffs slides_414_22_25_2011 7 But chamberlain could conjecture that either Hitler is (a) Amicable: preferred outcome is to receive concessions but not go to war . Or (b) Belligerent: high payoffs from going to war (a dominant strategy with higher payoffs if also receives concessions) Game could be solved under each of these alternatives, as shown below: Concessions if Hitler amicable Stand firm if Hitler is belligerent But this is not useful since Chamberlain does not know Hitler’s type slides_414_22_25_2011 8 Harsany’s insight: transform game of incomplete into imperfect information by Adding an initial “player”, Nature, which determines what type a player is, e.g. it determines if Hitler is amicable or belligerent Nature’s move is observed by Hitler but not Chamberlain (imperfect information) Chamberlain knows the probability of each possible type, e.g. p=0.6 of amicable. “Imperfect information version” of The Munich agreement game slides_414_22_25_2011 9 Using backward induction to solve Hitler’s optimal strategy has 1 action at each info node: (No war, war, war, war) Chamberlain’s optimal strategy: the one that maximizes expected value Concession: 0.6*3+0.4*1=2.2 Stand firm: 0.6*2+0.4*2= 2 Proposed solution and predicted outcome: Chamberlain chooses concessions and Hitler does not go to war if he is amicable and war otherwise slides_414_22_25_2011 10 BAYES‐NASH EQUILIBRIUM Definition: A Bayesian game is a modification of a standard game in which Nature initially moves by endowing players with private information 3 Steps to construct a Bayesian game 1: Identify the relevant dimension of the game along which a player does not have information. This is typically something about the other player’s payoffs, strategies or beliefs. We refer to this as the player’s type and the collection of feasible types as the type space [e.g. Hitler had two types: amicable and belligerent] 2: Explicitly model “Nature” and the initial determination of each player’s type as well as the probability assigned to each type (not necessarily a number, could be a general distribution). These probabilities are exogenous, reflect beliefs of players and are common knowledge. 3: Define the strategy sets, which must specify decision rule for every possible information node. We conceive of these as being chosen before Nature makes its choice of types so the strategies must include decisions that correspond to types even if they are not chosen by nature ex‐post. [e.g. Hitler knows that he is belligerent but his strategy must specify strategy for amicable, otherwise Chamberlain would not know how to act] slides_414_22_25_2011 11 Definition: A BayesNash equilibrium of a game is a solution concept for a Bayesian game characterized by a strategy profile that prescribes optimal behavior for every type of player, given the other players’ strategies and given the beliefs about the other players’ types slides_414_22_25_2011 12 APPLICATION: FIRM ENTRY AND EXPANSION Setup for entry and expansion game Two competing firms make simultaneous decisions Firm 1 is a potential entrant and decides whether or not to enter Firm 2 is the incumbent and decides whether or not to expand capacity. 2’s expansion costs are either high or low. Firm 1 does not observe firm 2’s costs but knows that they are low with probability 2/3 and high with probability 1/3 Payoffs under each cost type for 2 are given by 2’s expansion costs are low (2/3) Expand Not Expand
1 Enter ‐1,2 1,1
1 Not Enter 0,4 0,3 2’s expansion costs are high (1/3)
Expand
Not Expand
Enter ‐1,‐1
1,1
Not Enter 0,0
0,3 slides_414_22_25_2011 13 Exercise What is the extensive form representation of this game? What is the optimal strategy for each of the firm 2 types? What is the expected value of entering and the value of not entering for firm 1? What is the Bayes‐Nash equilibrium of this game? slides_414_22_25_2011 14 Solution Game tree Expand
Enter
Low Expansion Cost (2/3) 1 Nature 1 Not Expand Expand 2
Not Enter Enter High Expansion Cost (1/3) 2 Not Expand Expand 2 Not Expand Expand Not Enter 2 Not Expand 1,2 1,1 0,4 0,3 ‐1,‐1 1,1 0,0 0,3 2’s decision: If we look at each of the payoff matrix we see there is a dominant strategy for each type. If 2’s expansion costs are low then expand is dominant (2 > 1 and 4 > 3) If 2’s expansion costs are high then not expand is dominant (1 > ‐1 and 3 > 0) slides_414_22_25_2011 15 Player 1 expected value from Entering: (2/3) x (‐1) + (1/3) x 1 = ‐1/3. Not entering = 0 So the Bayes‐Nash equilibrium in this game is 2 expand if its costs are low and does not expand if its expansion costs are high 1 does not enter slides_414_22_25_2011 16 APPLICATION: TRIGGER STRATEGIES Setup Two players, Earp Wyatt and stranger are facing off and must simultaneously decide whether to draw or wait Earp’s skill is common knowledge but Earp can only guess if stranger is good (a gunslinger) with probability 0.75 or not (a cowpoke) with probability .25 Payoffs Key difference for the player that has private information relative to Munich game: stranger does not observe the action of the other before deciding Entry game: stranger does not have a dominant strategy for all types if he is a gunslinger then draw is a dominant strategy for stranger (3>1, 4>2) but if stranger is a cowpoke then he wants to match Earp slides_414_22_25_2011 17 Implication of not having a dominant strategy for one of the players: optimal strategy depends on his beliefs about what the other player (Earp) will do the probabilities must be common knowledge so stranger can form those beliefs Extensive form slides_414_22_25_2011 18 4 possible strategies for stranger: {___if Gunslinger, ___ if Cowpoke} {Draw, Draw} {Draw, Wait} {Wait, Wait} {Wait, Draw} Recall Bayes Nash equilibrium definition: a strategy for each player type that maximizes their payoff given the probability measures of the other’s type and the other player’s strategy. In practice Earp chooses strategy that maximizes expected payoff given stranger’s strategy Gunslinger Stranger chooses action that maximizes his payoff if he is a gunslinger given what he believes Earp will do Cowpoke Stranger chooses action that maximizes his payoff if he is a cowpoke, which in this case is the same action as he believes Earp will choose If stranger is gunslinger then draw is dominant so we can eliminate {Wait, Wait} and {Wait, Draw} slides_414_22_25_2011 19 Remaining candidate strategy pairs for a Bayes‐Nash equilibrium after eliminating strictly dominated strategies for stranger A: Stranger {Draw, wait} and Earp draws B: Stranger {Draw, wait} and Earp waits C: Stranger {Draw, draw} and Earp draws D: Stranger {Draw, draw} and Earp waits Candidate strategies for a Bayes‐Nash equilibrium that we can rule out: A, D A: Stranger {Draw, wait} and Earp draws: if stranger believes that Earp will draw then he would also want to draw if he is a cowpoke, not wait (2>1) D: Stranger {Draw, draw} and Earp waits: if stranger believes that Earp will wait then he would also want to wait if he is a cowpoke, not draw (4>3) Exercise: Is candidate B a Bayes‐Nash equilibrium? Is candidate C a Bayes‐Nash equilibrium? slides_414_22_25_2011 20 Solution for B: Yes, the following is a BNE: Stranger {Draw, wait} and Earp waits VE(draw) = (.75 x 2) + (.25 x 4) = 1.5 + 1 = 2.5 VE(wait) = (.75 x 1) + (.25 x 8) = .75 + 2 = 2.75 > 2.5 Stranger: If he believes Earp will wait then he is better drawing if he is a gunslinger (4>2) and waiting if he is a cowpoke (4>3) Solution for C: Yes, the following is a BNE: Stranger {Draw, draw} and Earp draws VE(draw) = (.75 x 2) + (.25 x 5) = 1.5 + 1.25 = 2.75 VE(wait) = (.75 x 1) + (.25 x 6) = .75 + 1.5 = 2.25 < 2.75 Stranger: If he believes Earp will draw then he is better drawing if he is a gunslinger (3>1) or a a cowpoke (2>1) slides_414_22_25_2011 21 APPLICATION: PRIVATE INFORMATION IN AUCTIONS Examples of private information in auctions While bidding in an auction it is common to be uncertain about the true value of an object and/or its valuation by other bidders. Two broad types we consider Independent and private values across bidders: e.g. private collectors may know exactly how much they value a painting but not how much others do. Firms may have known valuation for a given government license but it is not known by others (depending on their costs) Unknown but common value across bidders, e.g. art dealers bidding on an item expect their profit from reselling is the same but are uncertain of how high it is Firms may share valuation of an oil field but are uncertain about the level of reserves or future value of oil slides_414_22_25_2011 22 General Setup of first price sealed bid auction bidders simultaneously submit their written bids. The highest bid wins the item and pays a price equal to her bid. If there is more than one highest bidder, then a winner is chosen at random. Setup for 1st price auction with independent and private values The bidders value of the item is unrelated across bidders. Bids can come only in increments of 10 Two bidders: Bill and Charlene Each only knows that the other is either a high valuation type: v=100, with probability 0.4 low valuation type: v=50 with probability 0.6 slides_414_22_25_2011 23 Strategy: a bid for each player type, bti, where t=low or high and i=Bill or Charlene Consider a candidate symmetric strategy: bti = 40 if t=low and bti =60 if t=high Expected payoff for Bill of bli = 40 if he is low: 0.6*0.5*(50‐40)+0.4*0=3 Expected payoff for Bill of bhi = 60 if he is high: 0.6* (100‐60)+0.4*0.5*(100‐60)=32 Does Bill have an incentive to set bli <40? No, since his probability of getting object is zero and so expected payoff =0<3 =50? No, since his expected payoff is then 0<3 >50? No, since then if he wins he pays more than he values Does Bill have an incentive to set bhi <40? No, since his probability of getting object is zero and so expected payoff =0<3 >70? No, since at 70 probability of winning is 1 and so higher bids costly =40? NO 0.6* 0.5(100‐40)+0.4*0=18<32 =50? NO 0.6* (100‐50)+0.4*0= 30<32 =70? NO 0.6* (100‐50)+0.4*(100‐70)=30<32 Given symmetry of the game bli = 40 and bhi =60 is a BNE for both Bill and Charlene slides_414_22_25_2011 24 Questions: Why do players bid below their valuation , i.e. “shade their bid.”? How would we find the equilibrium bidding behavior with many bidders and what effect does this “competition” have? How would we find the equilibrium if there are many types, rather than just two? Setup for 1st price auction w/ n independent private values and continuum of types n>1 bidders bidder i’s valuation can range anywhere from [0,1] Types are uniformly distributed in [0,1], i.e. the probability that each player assigns to another’s valuation is Pr(vi ≤v)=v if 0≤v≤1 Nature reveals type to each bidder privately Bidders simultaneously choose bids bi to maximize their expected payoff given their type and beliefs about others Expected payoff: π i(bi, vi)=Pr(bi > bj for all j ≠i ) x (vi ‐ bi) slides_414_22_25_2011 25 Solution Note that the probability of a tie id 0 (a continuous distribution of typed) Pr(bi > bj for all j ≠i ) = Pr(bi > b1)* Pr(bi > b2) * Pr(bi > b3)*… [1] = Pr(bi > dv1)* Pr(bi > dv 2) * Pr(bi > dv 3)*… [2] = Pr(bi/d> v1)* Pr(bi /d> v 2) * Pr(bi /d> v3)*… [3] = (bi/d)n1 [4] [1] due to independence of values assumption [2] by conjecturing that others optimally shade their bids below valuation by some similar amount d<1 [3] re‐arranging [4] Uniform distribution assumption FOC for optimal bid: ∂π i(bi, vi)/ ∂bi = 0 ∂ [(bi/d)n1 x (vi ‐ bi)]/ ∂bi =0 ∂ [(bi/d)n1 x (vi ‐ bi)]/ ∂bi= ‐(bi/d)n1+ [(n‐1)/d](vi ‐ bi) (bi/d)n2 slides_414_22_25_2011 26 Evaluate at symmetric equilibrium (bi = dvi) and determine what d must be for FOC to hold ∂ [(bi/d)n1 x (vi ‐ bi)]/ ∂bi  (bi = dvi) = ‐(vi)n1+ [(n‐1)/d](vi ‐ dvi) (vi)n2 = ‐(vi)n1+ [(n‐1)/d](1 ‐ d) (vi)n1 So: ‐(vi)n1+ [(n‐1)/d](1 ‐ d) (vi)n1=0 iff [(n‐1)/d](1 ‐ d) =1 d=(n‐1)/n BNE of the auction game is bi = vi(n1)/n Note that s‐>1 as n increases so more competition leads to less shading since the probability of losing is higher when it is more likely that there is another type that is somewhere between your bid and your valuation. slides_414_22_25_2011 27 Baseline setup to illustrate common value auctions and winner’s curse Suppose first that Texaco is the only firm bidding for an oil field lease. The true (unknown) valuation of the oil field is v an integer between 10 and 100. Texaco gets estimate s, which is either v – 2 w/ probability ½ and v + 2 otherwise. Equal probability that s is under or overestimate by a similar amount implies that, the best guess for Texaco for the expected value of the field is simply s. So, this is the maximum it will be willing to pay for the lease Expected payoff if Texaco pays anything less than s is positive since with probability ½ the value is s+2 and with equal probability it is s‐2 so the expected value is s and the expected payoff is simply that minus the price p≤s slides_414_22_25_2011 28 Setup when the lease is sold via a first price sealed bid auction Now, suppose both Texaco and Exxon are bidding for the lease. Again, the true valuation of the oil field is v an integer between 10 and 100. Each gets an estimate s, either v – 2 w/ probability ½ or v + 2 with probability ½. Each firm can bid in increments of 1 from 0 to 100 If Texaco gets an underestimate, then Exxon gets an overestimate, & vice versa Objective: Show that expected payoff if Texaco pays less than s but not “sufficiently” less, is not positive because High signal leads it to win the auction but the signal was above the true value Low signal: lose the auction (so get 0) So a BNE equilibrium requires firms to bid “sufficiently” below their signal slides_414_22_25_2011 29 Suppose at a Bayes Nash equilibrium if a firm receives a bid s, then it bids s – 1. Suppose Texaco receives an estimate of st=20. Then it bids 19. Given s=20 then either the true value is v=s‐2=18 or v=s+2=22 If v= 18, then Texaco received an overestimate so Exxon gets sE=16, bids 15 & Texaco wins bid Texaco payoff is 18‐19=‐1<0 If v= 20, then Texaco received an underestimate so Exxon gets sE=24 and bids 23 and wins Texaco payoff is 0 (Exxon’s is negative) So expected payoff for Texaco of this strategy is negative and is thus not a BNE since could always not bid or bid something else to get 0 slides_414_22_25_2011 30 Broader points about common value auction: the company that gets a higher signal typically wins and so if it were to bid its signal, or something not sufficiently below, it would make a loss, which is known as the winner’s curse. Avoiding the winner’s curse: treat your signal as if it were the highest (since that is actually the case if you do win the auction). In the example above that would mean that you would never bid more than s‐2 and you would be ensured at least a zero expected payoff Optimal bidding strategy: involves shading your signal by enough As shown in appendix the bid for a case with n bidders and a uniform distribution of signals between 0 and 1 (as in experiment) is bi=[(n+2)/2n]*[(n‐1)/n]si; e.g. bid si/2 if n=2 slides_414_22_25_2011 31 Common value auctions and the winner’s curse: basics to understand experiment The true (unknown) object value is v (equal to average of signals in experiment) Each bidder obtains a signal about v, that will determine his type. E.g. high type receives a signal v+e and low type v‐e , where e is the “noise”. After privately observing one’s signal the player chooses how much to bid Login: ngl11 Prediction for bid/signal ratio =[(n+2)/2n]*[(n‐1)/n] = 0.5 if n=2 =0.562 if n=4 Observations Were all bids below signal? Did high signal types win? How close was average bid/signal in treatment 1 = ½? Was average bid in treatment 2 lower? slides_414_22_25_2011 32 SIGNALING GAMES: MOTIVATION AND STRUCTURE Motivation for signaling games: a player may have private information about his type, e.g. hardworking or lazy worker interviewing for a job high insurance risk or low insurance risk person applying for insurance high quality (cost, etc)or low quality (cost, etc) firm May want to signal your type to others (or hide it). Basic Structure of Signaling Games Two type of players: a sender and a receiver Three stages At the start, Nature chooses the sender’s type and reveals it to sender but not receiver (so there is private information) The sender chooses an action after he learns his type The receiver observes the sender’s action and uses it to update his beliefs about the sender’s type and then chooses an action of his own. slides_414_22_25_2011 33 BAYESIAN UPDATING OF BELIEFS Most standard assumption used is that players use new information to update beliefs by using Bayes's rule. Definition Bayes’s rule: Prob(y x)= Prob(x,y)/Prob(x’), where Prob(x’): probability of event x’ occurring Prob(y x) is the probability of event y’ occurring given that x’ occurred Prob(x,y) is the joint probability of both events Highlighting the belief “update component” Reverse role of x’ and y’ and re‐arrange to get Prob(x y)= Prob(x and y)/Prob(y’) Prob(x and y)= Prob(y’) Prob(x y) So Prob(y x) Prob(y) *[Prob(x y) /Prob(x)] = Updated (or Posterior) belief = Prior belief * “update component” slides_414_22_25_2011 34 Example: online chatting with someone from a college. Trying to determine if Social Sciences major without asking information Social sciences (y’) 600
900
1500 Male
Female (x’)
total Total
1900
2100
4000 At first only know fraction of SS in college so prior probability is: pr(y’)=1500/4000=37.5 Find out that "chattee" is female and know there are 900 female ss students and 2100 females so now can use this information to calculate pr(y’x’)=900/2100=42.9 What if you only know the fraction of females in social sciences in college population and fraction fraction of females but not the numbers? use Bayes rule pr(y’x’)= pr(y’,x’)/ pr(x’)=(900/4000)/(2100/4000) = 42.9 or to see the updating pr(y’x’)= pr(y’)* (Prob(x y)/Prob(x))= =(1500/4000)*[(900/1500)/(2100/4000)] slides_414_22_25_2011 35 How informative is signal In the case above we can see why the signal is informative : the probability that a student is a female given they are in ss is different (in this case higher) than the probability that they are a female (given they are in this college) , i.e. from their fraction in the college population . If those probabilities were the same then no new information and so the posterior would equal the prior. In some special cases signal is very informative, e.g. if all females were in social sciences then pr(y’x’)= pr(x’y’)=1. slides_414_22_25_2011 36 Another example: The Monte Hall problem Suppose you're on Let’s Make a Deal, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host Monte Hall, who knows what's behind the doors, opens another door, say No. 2, which he knows has a goat. He then says to you, "Do you want to switch and pick door No. 3 or do you want to stay with Door No. 1?" Is it to your advantage to switch your choice? Try it: http://math.ucsd.edu/~crypto/Monty/monty.html slides_414_22_25_2011 37 Use Bayes’ rule framework y’ = the car is behind Door 1 x = Monte opens Door 2 Prob(y x) = the probability that the car is behind Door 1 given that Monte opens Door 2 Prob(x, y) = the probability Monte opens Door 2 and the car is behind Door 1 Prob(x) = the probability Monte opens Door 2 So: Prob(y x) = Prob(x, y) / Prob(x) slides_414_22_25_2011 38 Tree below tracks possible outcomes Calculate probabilities Prob(x, y) = 1/6 (=(1/3)*(1/2)) Prob(x) = 1/6 + 1/3 = ½ Prob(y x) = Prob(x, y) / Prob(x) = (1/6) / (1/2) = 1/3 The car is behind either Door 1 or Door 3. We have shown that the probability it is behind Door 1 is 1/3 and so the probability it is behind Door 3 is 2/3 so SWITCH to 3 slides_414_22_25_2011 39 Intuition: no new information on door you chose so prior=posterior since before the probability was 1/3 that it was behind door 1. But since there are two left the host can always show you at least one of them and this will not contain any information about whether there is something behind your door. But if that probability has not changed and it must be either behind that or door 3 the probability that it is behind 3 is 2/3. See this using Bayes rule: Prob(y x) = Prob(y) *( Prob(x y) /Prob(x)) and showing that update component is equal to 1. Prob(x y)/Prob(x) = Prob(x and y)/(Prob(y’) Prob(x))=(1/6)/((1/3)(1/2)) = 1 . slides_414_22_25_2011 40 PERFECT BAYES‐NASH EQUILIBRIUM Basic Structure of Signaling Games (recall) Two type of players: a sender and a receiver Three stages: (i) Nature chooses sender’s type and reveals it to sender but not receiver; (ii) The sender chooses an action after he learns his type; (iii) The receiver observes the sender’s action and uses it to update his beliefs about the sender’s type and then chooses an action of his own Definition: Perfect Bayes Nash equilibrium For each of the sender’s possible types, the sender’s strategy prescribes an action that maximizes the sender’s expected payoff, given his beliefs about how the receiver will respond (aka as sequential rationality). The receiver uses Bayes rule (whenever possible) to update his beliefs about the sender’s type when he observes the sender’s action (aka consistent beliefs). For each action of the sender, the receiver’s strategy prescribes an action that maximizes the receiver’s expected payoff, given the receiver’s updated beliefs about the sender’s type. slides_414_22_25_2011 41 Definition: a separating strategy is one that assigns a distinct action to each type of player. Hence, the receiver can separate out each player’s type from her observed play. Definition: A pooling strategy is one where all sender types pool together in choosing the same action, regardless of the sender’s actual type. Definition: A semi‐separating strategy: is one where some types choose distinct actions and others pool. Figure: types of strategy (a) Separating, (b) Pooling, (c) semi Separating slides_414_22_25_2011 42 APPLICATION: MANAGEMENT TRAINEE Setup A person accepts a position as a management trainee. On the basis of close observation during the probationary period, the manager will decide whether to hire the trainee permanently. Suppose there are two types of workers, lazy and industrious and Only the trainee knows her type. The manager wants to hire a trainee only if the trainee is industrious. The manager’s payoff from hiring an industrious worker is 100, hiring a lazy worker is 25, and not hiring is 60. Net benefit to trainee = gross benefit –cost where Gross Benefit 130 if hired 70 if fired. Cost varies with hours worked during training and type (costlier if lazy) slides_414_22_25_2011 43 The management trainee extensive form game slides_414_22_25_2011 44 Candidate for separating PBNE of management trainee game Trainee strategy: If lazy, then work 40 hours. If industrious, then work 80 hours. Manager strategy: If the trainee worked 40 or 60 hours, then do not hire her. If the trainee worked 80 hours, then hire her. Manager’s beliefs: If the trainee worked 40 hours, then assign a probability of 1 to her being lazy. If the trainee worked 60 hours, then assign a probability of 0.6 to her being lazy and 0.4 to her being industrious. If the trainee worked 80 hours, then assign a probability of 1 to her being industrious. Verify three conditions: optimality of trainee’s strategy, manager’s strategy and consistency of beliefs Trainee’s strategy is optimal, given the manager’s strategy and beliefs. slides_414_22_25_2011 45 Manager’s beliefs are consistent. if the trainee worked 40 hours, the manager’s beliefs assign a probability of 1 to her being lazy, which is indeed consistent, since only a lazy trainee works 40 hours. If the trainee worked 80 hours then the manager assigns probability of 1 to industrious, which is consistent with optimal behavior of worker since only industrious work 80 hrs No trainee chooses 60 hours so probabilities .4/.6 are as consistent as any other. Manager’s strategy is optimal If the trainee worked for 40 hours, then the manager believes that she is lazy for sure. Then, it is optimal not to hire her (60>25). If the trainee worked for 80 hours, then the manager believes that she is industrious for sure. Then, it is optimal to hire her (100>60). If the trainee worked for 60 hours, then expected payoff given the probability is 0.6*25+0.4*100=55<60 Exercise (CYU 11.1) : is there a separating strategy with industrious choosing 60 hours? slides_414_22_25_2011 46 The management trainee game: a pooling equilibrium Trainee’s Strategy: Work 40 hours whether lazy or industrious. Manager’s Strategy: Do not hire her (regardless of how hard she worked). Manager’s Beliefs: Assign a probability of 0.75 to the trainee being lazy and 0.25 to the trainee being industrious. Verifying that it is a PBNE Optimality for trainee given beliefs: If not hired independently of how hard she worked then better off choosing minimum # hours, 40 Consistent beliefs of manager: When he observes 40 hours worked this is consistent with the optimal strategy of both types of workers. Thus 40 hours worked reveal no new information about worker type and so the posterior beliefs, (.75 lazy) should be identical to prior, as they are. Sequential rationality: given manager’s beliefs it is optimal not to hire if 60 ≥ 0.7525+0.25*100=43.75 thus, anticipating this, it is also optimal for trainees to pool by working the minimum amount of hours slides_414_22_25_2011 47 APPLICATION: A MARKET FOR LEMONS Motivation: In many markets there is asymmetric information with the seller having private information about the true quality of object. In some cases this can leave us with a market for “lemons” only, i.e. where high quality goods are never sold slides_414_22_25_2011 48 Setup: used car market A car for sale can be low, moderate, or high quality, which is known only to seller Buyer knows only that the chance of high is 20%, moderate is 50%, and low is 30%. The seller decides whether to put the car up for sale and if so, then at what price. If the car is for sale, then the buyer decides whether or not to buy it. Payoff for seller: price received for car if the car is sold, or value to seller if it is not Payoff for buyer: Value of the car minus price paid or zero if no transaction. Preliminary point: If there was perfect information then it is Pareto optimal to sell any quality since buyer is willing to pay up to value, which is always higher than the value of seller holding on to car. slides_414_22_25_2011 49 Exercise: Is there a PBNE where higher quality cars sell for more? Hint: Consider the following Seller strategy: p_h> p_m> p_l for each quality type, e.g. 22k>17k>10.5k Buyer’s (posterior beliefs): Pr(ip_i)=1 , e.g. if asked 22k then believe it is high quality Buyer’s strategy: buy if p_h≤24k ; p_m≤18k; p_l≤12k slides_414_22_25_2011 50 Solution: No separating PBNE in this game. To see this examine the 3 requirements Buyer’s beliefs are consistent given the seller’s strategy Buyer’s strategy is optimal given the seller’s strategy and consistent beliefs since payoff if high is 24‐22>0, if medium is 18‐17>0 and if low 12‐10.5>0 But seller’s strategy is not optimal (for medium and low types) since if he anticipates that buyer will take a high price as a perfectly credible signal for quality then he will choose a high price independently of his type slides_414_22_25_2011 51 Exercise: Is there a pooling equilibrium price where all types of quality are present? Hint: Consider the following Seller’s Strategy: Price at p_bar whether the car is of low, moderate, or high quality. Buyer’s Strategy: If P≤ p_bar, then buy otherwise don’t buy. Buyer’s Beliefs: For any price, the buyer believes that the chance of high is 20%, moderate is 50%, and low is 30% (so posterior is the same as prior) slides_414_22_25_2011 52 Solution: No. To see this verify the three conditions for a PBNE Beliefs are consistent since if price is the same independent of quality there is no new information in that signal so posterior is equal to prior Buyer’s strategy: it is optimal to buy iff 0.2*(24‐ p_bar) + 0.5* (18‐ p_bar) +0.3*(12‐ p_bar) ≥0 p_bar≤ 17,400 Seller’s strategy is optimal iff p_bar exceeds the value for each type of seller, e.g. p_bar≥ 20k if high quality. Thus if the car is high quality the seller would require a p_bar that exceeds what the buyer is willing to pay. Thus there is no PBNE with a common price for ALL quality types slides_414_22_25_2011 53 Existence of a market for lemons: Consider the following pooling strategy and beliefs Seller’s strategy: If the car is of low or moderate quality, then price at p_bar. If the car is of high quality, then do not put the car up for sale. Buyer’s strategy: P≤ p_bar, then buy otherwise don’t buy. Buyer’s belief: If P ≤ p_bar the car is of low quality with probability 0.375 and of moderate quality with probability 0.625. If P> p_bar, then the car is of low quality with probability 1. Show the pooling strategy above is a PBNE for some p_bar Seller’s strategy is optimal if p_bar<20k (otherwise would want to sell high quality) p_bar≥15k (get at least the value of a medium and more than low quality) slides_414_22_25_2011 54 Show the pooling strategy above is a PBNE for some p_bar (ctd) Buyer beliefs are consistent if the posterior beliefs have been correctly updated given the seller’s strategy Probability that car is high quality is 0 so it must be either low or moderate, i.e. Pr(low’)=1‐Pr(medium’) Fraction of low to moderate still the same (.3/.5) so Pr(low’)/ Pr(moderate’)=0.3/0.5 Pr(low’)0.5 = (1‐Pr(low’)0.3 Pr(low’)=.3/.8 [Alternatively apply Bayes’ rule: Prob(y x)=Prob(y)*[Prob(x y) /Prob(x)] where y= low quality, x: either low or moderate quality] Given these beliefs the buyer’s strategy is optimal iff 0.625*(18,000‐p_bar)+0.375*(12,000‐p_bar)≥0 p_bar≤15,750 Therefore the strategies and beliefs form a PBNE if 15,000≤p_bar≤15,750 A market exists but only for lemons because of asymmetric information! slides_414_22_25_2011 55 ...
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