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.
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.