In his paper ” On the convergence of informational cascades” (http://www.sciencedirect.com/science/article/B6WJ3-45P12S4-N/2/8a374f4e23f6ea6d04deb1891875e7c1) , In Ho Lee discusses the formation of informational cascades when the action space is continuous.

In the basic model of Bikhchandani, Hirshleifer and Welch, each person makes a binary “Accept/Reject” action after considering his own signal and the actions of his predecessors. It turns out that the binary action space is a necessary ingredient for an informational cascade to take place. This is because each person’s action may not be a reflection of his private signal as he may have acted based on the actions of those before him. As a result, information, in the form of valuable private signals, is lost each time a person chooses to act against his own signal.

This loss of meaningful information can be avoided if we allow a continuous action space. Each individual observes the actions of those before him and chooses an action that has a value between that and his own. In other words, he modifies the predecessors’ actions slightly according to his own judgement. His action thus encapsulates some information about his private signal. Thus, the next person can make a better decision, preventing a cascade.

A real-life example of a continuous action space is found in the carnival games in which participants are asked to guess the number of marbles contained in a large glass jar, which is anywhere from 0 to 200. Participants make their guesses sequentially and write them down on a white board, or something to that effect. Each person makes a private guess and then updates his guess by finding the mean of the sum of all the previous guesses and his own. As a result, he fine-tunes the most recent guess according to his own estimate. Assuming most people make educated guesses, the guesses will tend to the actual number when the number of guesses is large. This works because, intuitively, for every one guess that is less than the actual number, there is another one more than the actual number by the same amount. Finding the mean just cancels them out in the same way we eliminate random error in experiments. An informational cascade never forms in this senario because the continuous reporting preserves information about each person’s private signal.