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 Paradox

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.

Unraveling Logic

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 concludes 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 has not yet been 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 either.

The same reasoning eliminates the second-to-last lecture, the third-to-last lecture, and so on, working backwards through the entire semester. Therefore, the surprise quiz is impossible.

Theory versus Practice

Yet surprise quizzes clearly do happen in practice and often surprise students. The resolution of the paradox lies in what we mean by "surprise." In game theory, when rational agents interact under common knowledge, there is no room for true "surprise." Behavior can be probabilistic or random, but this only creates uncertainty about which action will be taken from a known set of possibilities.

If it snows in August, most of us would say we are "surprised." A boring, nerdy game theorist would instead say "snow in August is a low probability event." This might seem like mere semantics, but it highlights an important difference between how game theorists think of uncertainty versus how people experience it. Formally, we would say that the quiz timing is random, unknown to students ahead of time, but cannot be a true "surprise" in the sense the teacher promised. Students know a quiz will occur, they just don't know when.

Market Corrections

One application of this reasoning appears in the stock market. Financial analysts often predict when the next market correction—a large decline in stock prices—will occur. If market participants are rational and these predictions become common knowledge, such predictions can never be correct

Suppose everyone expects 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 if everyone tries to sell before the predicted correction, their collective selling is the correction, now in March. But if investors know that many will sell in March, then they would sell in February. But if investors know that investors know that... Again, this reasoning unravels back to today. If the future correction is common knowledge, why would anyone be holding stock today?

Practical Lesson

Behaviorally, the lesson is this: rationality and common knowledge are theoretical constructs that are rarely satisfied in practice. Assuming that actual people are fully rational and abide by common knowledge is quite heroic. After all, most of us have been genuinely surprised by quizzes, snow, or market movements.

If using game theory to formulate personal strategy, the key is not to act as if everyone is fully rational with common knowledge, but to recognize that people are only "boundedly rational." They reason through a few steps of logic, but obviously not infinitely many. Success comes from staying just one step ahead of the reasoning of the competition.

Like in the p-beauty contest game, the answer that satisfies both rationality and common knowledge never wins. The person who reasons just one step ahead of the average class student wins the game. Similarly, the investor who sells just before the market correction, one step ahead of the crowd, makes the largest profit.