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.
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