Furthermore, they know whether the other one is buying or selling, and they are each privy to a different piece of private information about the company. Because of her contacts, Alice knows which product decision was made, Handheld or Phone, but not whether it will be successful or not.
Because of his position with another company, Bob stands to get the "rejects" from a failed phone project, so he knows whether or not the cell phone was chosen for development and failed. That is, Bob knows whether Phone- or not.
Let's assume that the handheld device was chosen for development. So the true situation is either Handheld+ or Handheld-. Alice therefore knows Handheld, while Bob knows that the decision is not Phone- (or else he would have received the rejects).
After the first week, Alice sells since Handheld+ and Handheld- are equally likely, and each investor sells if he or she thinks the probability of success is 50 percent or less. Bob buys since he estimates that the probability for success is 2/3. With Phone- ruled out, the remaining possibilities are Handheld+, Handheld-, and Phone+, and two out of three of them are successes.
After the second week, it is common knowledge that the true situation is not Phone- since otherwise Bob would have sold in the first period. This is not news to Alice, who continues to sell. Bob continues to buy.
After the third week, it is common knowledge that it is not Phone (neither Phone+ nor Phone-) since otherwise Alice would have bought in the second week. (If she knew it was Phone and not Phone-, it would have to be Phone+.) Thus it's either Handheld+ or Handheld-. Both Bob and Alice now think the probability of success is only 50 percent, thus both sell, and there is a mini-crash of the stock price. (Selling by both influential investors triggers a general sell-off.)
Note that at the beginning of this process both Alice and Bob know that the true situation was not Phone-, but this knowledge was mutual, not common. Alice knew that Bob knew it was not Phone-, but Bob didn't know that Alice knew this. From his position the true situation might have been Phone+, in which case Alice would know Phone but not whether the situation was Phone+ or Phone-.
The example can be varied in a number of ways: There needn't be merely three weeks before a crash, but an arbitrary number of days, weeks or months; there may be a bubble (sellers suddenly switching to become buyers) instead of a crash; there may be an arbitrarily large number of investors or investor groups; or there may be a number of issues that determine whether an investor buys or sells.
In all these cases the stock's price can move in response to no external news. Nevertheless, the subterranean information processing leading to common knowledge among the investors eventually leads to precipitous and unexpected movement in the stock's price. Analysts will express surprise at the crash (or bubble) because "nothing happened."
Professor of mathematics at Temple University, John Allen Paulos is the author of best-selling books including "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.