The Surprise Quiz

The assumptions of common knowledge and rationality can have some rather "surprising" implications. Consider the example of a surprise quiz.

The surprise quiz paradox concerns a teacher who announces during the first lecture that a surprise quiz will be given at some point over the semester. Imagine a student the evening before the last lecture realizing that the quiz has not yet been given. The rational student knows that the quiz must be the next day. But then, the quiz is not a surprise. Hence, the rational student knows the test is not on the last day. If the quiz was to be given the next-to-last lecture, then the previous night the student would again know that it must be the next day, since, given the previous reasoning, we know that it can't be on the last day. The same reasoning eliminates the second-to-last lecture, the third-to-last lecture, and so on. Therefore, the surprise quiz is not possible.

Logicians, mathematicians, and philosophers have labored for over fifty years to solve this "puzzle." From our perspective, it is simply an issue of definitions. In game theory, when rational agents interact, there is no room for "surprise." Instead, things are probabilistic, or random. If it snows in August, some of us might say we are "surprised" while the boring, stodgy, nerdy game theorist would say "snow in August is a low probability event." In other words, snow in August is unlikely, but not surprising, much like the timing of the quiz is random and unknown to the students ahead of time, but cannot be a surprise.

A silly distinction? Probably. However, assuming perfect rationality and common knowledge is quite heroic as well. After all, most of us have been "surprised" by quizzes, inspections, or other events similar to the "surprise quiz paradox."

One application of this concept is the stock market. During the technology boom, the stock prices of almost every "e-company" ascended drastically. Most analysts both suggested that this was a price "bubble" destined to correct and simultaneously invested heavily in the technology sector. If the market is driven by fully-rational agents, this should not be possible. Say that we know that the correction will begin sometime next April. The fully-informed, rational person would sell just before the correction begins, say in March. But then the "real" correction would be in March, leading rational investors to sell in February. Again, this unravels back to today - if we know a correction is coming, why is anyone buying now?

The lesson is this: assuming rationality and common knowledge in the marketplace leads to missed opportunities. The key is not to act as if everyone is fully rational, but instead to be just