Heterogenous Agents in a Model
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Effects of the First Agent's Precision
How does a change in the precision of Agent 1 alter the likelihood of a cascade? To answer this question, we performed a statistical experiment as before, with the following variations: This time, we held the precision of all agents except number 1 constant at 0.7. The precision of Agent 1, however, varied from 0.5 to 1.0. Once again, we tried to obtain the probability of an UP or a DOWN cascade occurring by the time of Agent 7's decision. Most of the time, we had to use at least 10,000 runs to achieve statistical significance for each data point.
Results of the experiment plotted with statistical error bars are below. In each case, a straight horizontal line represents the probability for a uniform society, one in which Agent 1 has a precision of 0.7, just like everyone else.
There are several interesting things to note about these graphs. First of all, from the point of view of Agent 7, it's better if the precision of Agent 1 is slightly lower than everybody else's than if it's slightly higher. This phenomenon is discussed in detail by the authors of "Fads and Fashion as Informational Cascades." In fact, as we see from the graphs, over a small range of Agent 1's precision just above everyone else's, the probability of an incorrect cascade by Agent 7 is actually higher than in a situation where precision is uniform across the agents. Also note an interesting discontinuity right at the point where the precision of Agent 1 equals that of all other agents.
On the other hand, any other change in the precision of Agent 1 improves the situation for Agent 7. Note that as the precision of Agent 1 goes to 0.5, probabilities tend to the same value as that of uniform precision - when Agent 1's precision is 0.5, the information he produces is deemed useless by other agents, and they simply ignore him. Also, when his precision goes to 1.0, the other agents recognize his authority, and the society approaches 100% likelihood of entering the correct cascade.
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