I’ve been reading Networks, Crowds, and Markets, a great textbook by David Easley and Jon Kleinberg. I’m very grateful to Cambridge University Press for surprising me with an unsolicited review copy. I’m more than halfway through its 700+ pages. Much of the material is familiar in this “interdisciplinary look at economics, sociology, computing and information science, and applied mathematics to understand networks and behavior”. But I’m delighted by much that is new to me, including a particularly elegant description of an information cascade.
The experimenter puts an urn at the front of the room with three marbles in it; she announces that there is a 50% chance that the urn contains two red marbles and one blue marble, and a 50% chance that the urn contains two blue marbles and one red marble…one by one, each student comes to the front of the room and draws a marble from the urn; he looks at the color and then places it back in the urn without showing it to the rest of the class. The student then guesses whether the urn is majority-red or majority-blue and publicly announces this guess to the class.
Let’s simulate how a set of rational students would perform in this experiment.
The first student has it easy: if he selects a blue marble, he guesses blue; if he selects a red marble, he guesses red. Either way, his guess publicly discloses the first marble’s color.
Thus the second student knows exactly the colors of the first two selected marbles. If he selects the same color as the first student, he will make the same guess. If, however, the second student selects a red marble, he has no reason to prefer one color over the other. Let’s assume that, when the odds are 50/50, an indifferent student breaks symmetry by selecting the color in his hand. That way, we guarantee that the second student discloses the color of the marble he selects.
Things get interesting with the third student’s selection. What happens if the first two students have both guessed red, but the third student selects a blue marble? Rationally, the third student will guess red, since he knows that two of the first three selected marbles were red. In fact, if the first two students select red marbles, *every* subsequent student will ignore his own selection and guess red. Of course, analogous reasoning applies if we reverse the colors.
Generalizing from this case, we can see that the sequence guesses locks in on a single color as soon as two consecutive students agree. I leave it as an exercise to the reader to determine that, if the urn is majority-red, there is a 16/21 probability that the sequence will converge to red and a 5/21 probability that it will converge to blue.
A 5/21 probability of arriving at the wrong answer may not seem so bad. But imagine if you could see the actual marbles sampled and not just the guesses (i.e., each student provides an independent signal). The law of large numbers kicks in quickly, and the probability of the sample majority color being different from the true majority converges to 0.
This example of an information cascade is unrealistically simple, but is eerily suggestive of the way many sequential decision processes work. I hope we all see it as a cautionary tale. The wisdom of the crowd breaks down when we throw away the independent signals of its participants.