Assignment:
1. In this problem we will ask whether an information cascade can occur if each individual sees only the action of his immediate neighbor rather than the actions of all those who have chosen previously. Let's keep the same setup as in the Information Cascades chapter, except than when individual ichooses he observes only his own signal and the action of individual i- 1.
(a) Briefly explain why the decision problems faced by individuals 1 and 2 are un- changed by this modification to the information network.
(b) Individual 3 observes the action of individual 2, but not the action of individual 1. What can 3 infer about 2's signal from 2's action?
(c) Can 3 infer anything about 1's signal from 2's action? Explain.
(d) What should 3 do if he observes a high signal and he knows that 2 Accepted?
What if 3's signal was low and 2 Accepted?
(e) Do you think that a cascade can form in this world? Explain why or why not. A formal proof is not necessary, a brief argument is sufficient.
2. In this question we consider a variation on the model of information cascades from Chapter 16. Suppose that there is a new technology which individuals sequentially decide to adopt or reject. Let's suppose that anyone who adopts the new technology receives either a positive or a negative payoff from using the new technology. Unlike the model used in Chapter 16, these payoffs are random and have the property that the average payoff is positive if the technology is Good, and negative if the technology is Bad. Anyone who decides to reject the new technology always receives a payoff of exactly zero.
As in the model used, each person receives a private signal about the technology and observes the actions of all those who chose previously. However, unlike the model used in Chapter 16, each person is also told the payoffs received by each of those who moved previously. [One interpretation of this is that a government agency collects information about individuals' experiences and distributes it for free as a public service.]
(a) Suppose that the new technology is actually Bad. How does this new information about payoffs (the payoffs received by each of those who moved previously) affect the potential for an information cascade of choices to adopt the new technology to form and persist? [You do not need to write a proof. A brief argument is sufficient.]
(b) Suppose that the new technology is actually Good. Can an information cascade of rejections of the new technology occur? Explain briefly.
4. Let's consider the model of information cascades. Assume that the probability that the state is Good (G) is p = 1/2, and that the probability of a High signal given a Good state is q = 2/3. (The probability of a Low signal given a Bad state is also q = 2/3.) Remember that each person observes a signal and the choices (but not the signals) of all those who chose before him. Each person chooses between Accept (A) and Reject (R).
Suppose that you are the tenth person to make a choice and you have observed that everyone before you chose R. That is, we are in an R-cascade.
(a) What is the probability that this is an incorrect cascade? [The probability that the state is actually G given that we are in this R-cascade.]
(b) Now let's suppose that before you (person 10) receive your signal, you decide to ask person 9 about the signal that they observed. Let's suppose that person 9 observed a High signal, that person 9 tells you that his signal was High, and that you know that person 9 is telling the truth. After this, you receive your own signal. What decision should you make, A or R, and how does it depend on which signal you receive?
(c) Now let's consider person 11. Person 11 observes only his own signal and the choices of those who decided before him (1 to 10). Person 11 knows that you have observed both your signal and person 9's signal. Person 11 cannot observe these signals; all he knows is the choices that have been made. The first nine people have chosen R. What should person 11 do if you choose R? What should he do if you choose A? Why? Remember that person 11 observes a signal, and so his choice can depend on his signal as well as the earlier choices.