Conference 6Intermediate Microeconomics - Fall 201808/10/2018Problem 1: Akerlof’s lemonsRead The Economist‘s article on the lemon dilemna, developped by Akerlof in 1970 to illustrateadverse selection. We are going to consider a simple set up in which such adverse selection oc-curs. Assume there are two types of cars on the used car market: good quality, well-maintainedcars and “lemons”, which are poor quality, very damaged cars. A buyer is ready to pay $20000for a good car and $10000 for a lemon. A seller on this market that sells both good cars andlemons is ready to sell a good car for $17000 and a lemon for $8000. The seller knows whattype of car he is selling, but the buyer has no way of observing a car’s type. The seller will notaccept a price below his willingness to sell, while once the buyer has offered a price, he has tobuy the car.1. Describe this situation as a strategic game: who are the players? What are their actionsand payoffs? Write the game’s matrix.The players are the buyer and the seller. The buyer can either buy a car for $20000 (H) orfor $10000 (L). The seller can either sell a good quality car (G) or a lemon (L).H LG (3, 0) (0, 0)L (12, −10) (2, 0)2. What is the Nash equilibrium in pure strategies? How does this equilibrium translate intoadverse selection?At equilibrium the seller offers a lemon and the buyer pays $10000. Only bad cars are soldon this market, and tis is because of the information gap between the seller and buyer oncar qualityProblem 2: The Bystander Dilemma. Someone is screaming in the night, and is beingmurdered. Should you call the police? There are n people living in the neighborhood (includingyou) and they all heard the scream. Calling the police would save the victim’s life, and thatwould make everybody happy. Suppose that gives everyone a payoff equal to v. Calling thepolice, however, requires some effort, it costs each person who calls c with 0 < c < v. If nobodycalls, everyone gets a payoff of 0. Hence we have v > v − c > 0, that is every individual prefersthat someone else calls to calling themself, but prefers calling and saving the victim to no onecalling.11. We are going to show that there is no symmetric pure strategy Nash equilibrium. Asymmetric pure strategy Nash equilibrium is a situation in which everybody makes thesame choice (call or not call), and has no incentives to change her mind after learning thechoice of others.a. Draw the game’s matrix if n = 2 and find the Nash equilibriaC NC (v − c, v − c) (v − c, v)N (v, v − c) (0, 0)The Nash equilibria are (C, N) and (N, C).b. Now assume n > 2. Show that there a no symmetric pure Nash equilibria by showingsomeone always deviates in a symmetric situation.There are only two possible symmetric Nash equilibria in pure strategy, either everybodycalls the police or nobody does. If everybody calls the police, everybody is getting v − c,and provided that others maintain their choice, every single individual would benefitby not calling the police so as to avoid paying the cost c. If nobody else calls the police,every single individual would be better off calling the police and getting v − c insteadof 0. Therefore there is no symmetric pure strategy Nash equilibrium.2. Find the symmetric mixed strategy equilibria:a. if n = 2Assume player 1 decides not to call the police with probability p. Then player 2 isindifferent between calling and not calling the police ifp(v − c) + (1 − p)(v − c) = p × 0 + (1 − p)vv − c = v − vpp∗=cvNote that p∗is always between 0 and 1.b. if n > 2In a symmetric mixed strategy equilibrium, all agents use the same probability p of notcalling the police. Because we have ruled out symmetric pure strategy equilibria, weknow that 0 < p < 1. If an agent randomizes between the two actions calling and notcalling, she must be indifferent between these two actions (for otherwise she would justchoose the one that she prefers with probability 1). Taking as given that every otherplayer remains silent with probability p, the payoff of an agent who decides not to callpolice is v(1 − pn−1), that is she receives the payoff v if at least another person callswhich occurs with probability 1 − pn−1. If on the other hand she decides to call, herpayoff is v − c. The indifference condition for this agent then implies that we shouldhave p =cv1n−1. Note that limn→∞p = 1, that is the probability that any particularindividual calls the police goes to 0 as the density of the neighborhood increases.Note: the probability that someone else calls the police is 1 − pn−1, which decomposesas follow: 1 is the sum of all possible events’ probabilities. pn−1is the probability that2no one among the n − 1 other players calls the police. So the probability that at leastone other player calls the police is equal to 1 minus the probability that no one callsthe police.3. How do you expect the probability of someone calling the police to change with the densityof population n?We could expect that the probability that at least someone calls increases with the density:anyone in particular may be less likely to call the police, but after all there are more peoplewitnessing the crime. But we can calculate that this probability is actually equal to 1− cnn−1,and since c < 1 this probability decreases with n. But then at least we can be reassuredthat this probability remains bounded away from 0 as it converges to 1 − c, and if c is smallthe person is likely to get some help.4. Are there asymmetric pure strategy equilibria? (that is everyone is either calling or notcalling, and no one wants to change their mind.)Consider a strategy profile in which at least two people are calling the police. Then each ofthe callers would be better off if she could change her decision while others don’t. Therefore,in any pure strategy Nash equilibrium there is at most one person calling the police in thegroup. It is easy to see that no silent witness has any incentive to change her mind and callthe police since the victim will be saved anyway and she can save v. The single caller hasno incentive to change her mind either as the victim would then die a horrible death, theprospect of which is well worth spending some time making a phone call.Comments: People seem less likely to offer help in case of emergency if there are other peoplearound, a phenomenon called the bystander dilemma by social psychologists. A famous exampleis the case of Kitty Genovese