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Intro to information cascades with uneven signal reliabilities

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When we were working with information cascades in class, we operated under two very large assumptions about the environment of the experiment–there is a 50-50 chance that the experiment is in a predetermined “good” or “bad” state.  In addition, we assumed that a “bad” state has the same probability of showing a low signal as a “good” state has of showing a high.  In the context of the balls-in-an-urn analogy, this is like saying a red-major urn will have the same percent of red balls as a blue-major urn will have blue balls.  What if instead we had non-symmetric states?  For example, a red-major urn might have 70% red balls, and a blue-major urn might have 80% blue balls.  How many initial rejects would lead to a cascade? We’d have a table describing the game that looks more like:

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Now, a cascade of rejects and a cascade of accepts might form under different circumstances.  Let’s consider the case where there is a rejection cascade, where S_H is the event where the current person receives a high signal and everyone prior is reliable, and S_L a low signal, with everyone prior being reliable, G for when the actual state is good, B for when it’s bad.  For a person to reliably guess a state based on his signal, we want Pr[G|S_H], Pr[L|S_L] >= 0.5.  To determine how many people is required to start this cascade, we want to calculate who the last reliable person is:

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From these calculations, we can not only get the number of reliable students(by calculating the smallest n such that the inequality on the left is violated), we also get a sense about the bounds of the experiment itself, namely, q must be >=0.5

There are still many other calculations that can be done with this, e.g., what kind of difference(or perhaps it’s a ratio) of rejects/accepts would prevent a cascade from happening, in other words, what is the “threshold” for this cascade to begin at any point, rather than just given a beginning sequence of a single vote? We can still generalize this problem one step further with a different probability of the experiment being in a “good” and “bad” state, but I’ll leave it as an exercise for the reader.

Source:

Wikipedia(http://en.wikipedia.org/wiki/Information_cascade) for the image, but otherwise from a mental itch I had to try to scratch after doing homework 6.

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