Networks: Fall 2013 Homework 8David Easley and´Eva Tardos Due at 11:15am, Wednesday, November 27, 2013As noted on the course home page, homework solutions must be sub mi t te d by upload to course’sBlackboard site. The file you upload must be in PDF format. It is fine to write the homeworkin another format such as Word; from Word, you can save the file ou t as PDF for uploading.You can choose type ”pdf” when you save the file, or print it and choose ”Adobe pdf” as yourprinter. (Changing the file extension from doc, or docx to pdf does not change the forma t , onlymakes the file unreadable)Blackboard will stop accepting homework uploads after the posted due date. We cannot ac-cept late homework except for University -a p p r oved excuses (which include illness, a familyemergency, or travel as part of a University sports team or other University activity).Reading: The questions below are prim ar i l y based on the material in Chapters 7, 19 and 22of the book.(1) (10 points)In this pr o b l em we are going to examine how various events might a↵ect the price andquantity of used cars traded. Suppose that th er e are two types of used cars: good ones and b adones. Sellers of used cars know the type of car that they own. Buyers do no t know which typeof car any particular seller has. Buyers do know that there are good and bad used cars, andthey know that of the 100 people who own used cars and are interested in selling their car, 50have good cars and 50 have b a d cars. Let’s suppose that there are 200 possible buyers of usedcars. [ We assume t h at there are more buyers than sellers to make the analysis straightforward.]A seller who has a good used car values it at $10,000 and a seller who has a bad used car valuesit at $4 , 0 0 0 . A seller is willi n g to sell his car for any price g r ea t er than or equal to his valuefor the car; no s el l er is willing to sell his car for a price less than his value for the car. Buyersvalues for good and ba d used cars are $13,000 and $8,000, respectively. As in class we willassume that buyers each want at most one used car and they are willing to pay their expectedvalue for a used car.We are also going to assume that every year th e numbers of used cars available to be sol dare as described above. That is, every year there are both a new 100 people looking to sell usedcars, 200 people looking to b u y used cars and the values remain unchanged from year-to-year,unless specified otherwise in the questions below. [Any c ar s that are unsold in some year areremoved from the market so that they do not e↵ect the equilibrium in subsequent years.](a) Find all of the equil i b r i a in a typical year in the market for used cars. For each equilib-rium provide the price of used cars and the number of used cars of each type traded.(b) Now suppose that at the beginning of some year there is a negative news story aboutthese cars that makes both good and bad used cars less attract i ve to buyers. The e↵ect of t h i s1news story is to lower the values that buyers place on any type of used car this year by $1,000.[It has no e↵ect on the values that sellers place on their cars.] Find all of the equilibria in themarket for used cars this year.(c) At the end of the year there is a n ews story noting that the price of used cars fe ll thisyear by much more than $1,000 as a result of the story in part (b). This story argues thatbuyers must have overreacted to the previous story as prices should n o t have fallen by morethan $1,000; buyer values fell by $1,000, so why should the price of used cars fall by more than$1,000? Explain why this subsequent story may be wrong. If the story has the facts correct(that the price of used cars fell by more than $1 , 00 0 ), but h a s reached the wrong conclusionwhat do you k n ow about the equilibrium that must have been i n place prior to the story?(d) At the beginning of the next year values of used cars to buyers and sellers are as inpart (b). That i s, all buyer valu e s have been permanently reduced by $1,000 as a result of t h enews story in part (b). Now at the beginning of this year the government introdu ces a “cashfor clunkers” program which results in one-half of the bad used cars being removed from themarket. This program h a s no e↵ect o n any buyer or seller values for good or bad used cars,and it has no e↵ect on the number of good cars or on the number of buyers. All it does is toreduce the number of bad used cars from 50 to 25. Find all equilibria in the market for usedcars after this new program is introduced.(2) (10 points)In this problem we will consider the rel at i o n sh i p between Nas h equilibria and evolutionarilystable strategies.(a) First, consider the game in Figure 1:Player APlayer BX YX 2, 2 4, 1Y 1, 4 3, 3Figure 1: The payo↵ matrix for Question (1a).Find all of the pure strategy Nash equil i b ri a and al l of the evolutionarily stable str at eg i esfor this game.(b) Next, consider the game in Figure 2:Player APlayer BX YX 2, 2 3, 1Y 1, 3 3, 3Figure 2: The payo↵ matrix for Question (1b).2Find all of the pure strategy Nash equil i b ri a and al l of the evolutionarily stable str at eg i esfor this game.(c) There should only be one di↵erence in your answers to parts (a) and (b) of this question.Provide a b ri ef explanation of why the answers di↵er.(3) (10 points)A s ocial networking site is studying how new applications (apps) spread from one country toanother. In particular, they’re t r y i n g to und er st a n d the foll owing phenomenon that takes placebetween two neighboring countries, which we’ll call X and Y . The site has 10 million usersin each country, and to make the question easy to exp r ess, we’ll assume that the friendship samong these users have the following simple structure.• Each u ser i n X has 50 friends: 40 of these friends ar e a l so i n X,and10ofthemareinY .• Each user in Y has 30 friends: 20 of these friends are also in Y , and 10 of them are in X.The site’s admin i st r at o r s have observed the general principle that a user will tend to startusing a new app on her own (without any additional ma r keting from the sit …