The Surprise Quiz

Common Knowledge and Rationality

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 on the first day of class that a surprise quiz will be given at some point over the semester. "You should be ready every class, because I promise you that you will not expect it," the professor boasts. Common knowledge and rationality imply that the professor cannot keep this promise.

Imagine a student the evening before the very last class who realizes 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 expected, and not a surprise. Hence, the rational student knows that the quiz cannot be given on the very last class.

Now imagine a student preparing for the last week of class, with two lectures left. If the quiz were not yet given, it must be either in the last class or the next-to-last class. By the above reasoning, it cannot be in the last class, as everyone would then expect it. So, this leaves only the next-to-last class. But if the quiz were to be given in the next-to-last class, then the previous night the student would again know that it must be the next day, so this would not be a surprise.

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 isn't a puzzle at all, but simply an issue of definitions. In game theory, when rational agents interact, there is no room for "surprise." Instead, behavior is often 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 if we attempt to describe the behavior of actual people. After all, most of us have been "surprised" by quizzes, snow, or other events similar to the "surprise quiz paradox."

One application of this concept is the stock market. One often hears financial "analysts" predicting when the next correction--or large decline in stock prices--will occur. If market participants are even slightly rational, such wide-scale predictions can never be correct. Say that we expect a correction to begin sometime next April. A rational investor would not hold stock during this period of decline, and instead would sell just before the correction begins, in March. But then the actual correction would occur in March, not April as predicted. If investors know that rational investors would sell in March, then they would sell in February. Again, this unravels back to today. If a future correction is common knowledge, why is anyone buying now?

Behaviorally, the lesson is this: rationality and common knowledge are theoretical constructs that are rarely satisfied in practice. If using game theory to formulate personal strategy, the key is not to act as if everyone is fully rational, but to recognize that people are only "boundedly rational" and to remain just one step ahead of the competition. Like in the p-beauty contest experiment, the answer that satisfies both rationality and common knowledge never wins. The person who reasons just one step ahead of the class wins the game, just like the investor who sells just before the market correction makes the largest profit.