Two months ago, the Nobel committee selected two eminent economists, Thomas Schelling of the University of Maryland and Robert Aumann of Hebrew University, to receive this year's prize for their work on game theory. Aumann, however, has produced so many seminal ideas with real-world applications that I'd like to sketch one that received very little mention in the news articles. It concerns the notion of "common knowledge," which is crucial in understanding the stock market and other phenomena.
A bit of information is "common knowledge" among a group of people if all parties know it, know that the others know it, know that the others know they know it, and so on. It is much more than "mutual knowledge," which requires only that the parties know the particular bit of information, not that they be aware of the others' knowledge. As Aumann showed, one can prove a theorem that can be roughly paraphrased as follows: Two individuals cannot forever agree to disagree. As their beliefs, formed in rational response to different bits of private information, gradually become common knowledge, their beliefs change and eventually coincide.
Subterranean Information Processing
Very abstract stuff, but there is an interesting example that demonstrates how the notion might enable us to explain sudden bubbles or sudden crashes in stock markets. These changes, which sometimes seem to be precipitated by nothing at all, might be the result of "subterranean information processing."
When private information becomes common knowledge, people change their beliefs. Furthermore, as anyone who has overheard teenagers' gossip with its web of suppositions can attest, this transition to common knowledge sometimes relies on convoluted inferences about others' beliefs. The same can be said of the gradual unraveling of a political scandal such as Plamegate or congressional influence-peddling.
Sergiu Hart, an economist also at Hebrew University and one of a number of people who have built on Aumann's result, illustrates with an example relevant to the stock market. Superficially complicated, it nevertheless requires no particular background besides an ability to listen and observe, and then figure out what others really think.
Hart asks us to consider a company that must make a decision. Let's suppose it's a small telecommunications company that must decide whether to develop a new handheld device or a cell phone with a novel feature. Assume that the company is equally likely to decide on one or the other of these products, and assume further that whatever decision it makes, the product chosen has a 50 percent chance of being very successful. Thus there are 4 equally likely outcomes: Handheld+, Handheld-, Phone+, Phone- (where Handheld+ means the handheld device was chosen for development and becomes a success, Handheld- means the handheld was chosen but turns out a failure, and similarly for Phone+ and Phone-).
Let's say there are two very influential investors, Alice and Bob. They both decide that at the current stock price if the chances of success of this product development are strictly greater than 50 percent, they should (continue to) buy, and if they're 50 percent or less they should (continue to) sell.
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.