Game Theory: Sleeping Or Studying the Night Before an Exam
A question that faces students in college far too often is whether, the night before a big exam, one should pull an all-nighter or “give up” and go to bed. While some students may believe that staying up all night to study will allow them to do better on that exam, studies show that doing more studying instead of sleeping might not be helping as much as some may think. This article (https://upchieve.org/blog/2021/3/8/is-it-better-to-sleep-or-study) posted on Upchieve by J.E. Choi describes how research has shown that memory is not quite the video reel that people think it is, that one can play back at one’s leisure. Memory is more like having a pile of Legos in one’s brain, full of little bits of information. In order to remember something, the brain needs to “assemble” the memory, whether that is how to solve an equation or the contents of a textbook. During sleep, one’s brain can practice putting together these memories and even organize the individual “pieces” so the brain can put together memories more efficiently and with greater accuracy.
This issue can be modeled with a simple graphed example. Say one has 12 hours to prepare for an exam before having to leave and take it. The options are to sleep for 12 hours (Sleep), study for 12 hours (Study), sleep for 8 hours and study for 4 hours (Sleep > Study), or study for 8 hours and sleep for 4 hours (Study > Sleep). The exam is scaled against the rest of the class’s score, so the student benefits from the rest of the class doing worse, but the rest of the class is not significantly affected by the student’s score. The payoffs are scores from 0-10 (student is vertical and class is horizontal choices), and the payoffs in the matrix are of the format (Student score, Class score).
Sleep Study > Sleep Sleep > Study Study
Sleep (7, 7) (5, 8) (5, 10) (7, 7)
Study > Sleep (8, 7) (8, 8) (7, 10) (8, 7)
Sleep > Study (10, 7) (10, 8) (10, 10) (10, 7)
Study (7, 7) (6, 8) (6, 10) (7, 7)
In the above payoff matrix, a grade of 7 or above is passing. The student will fail if they just sleep or just study and the rest of the class does not (curve does not help them). The whole class will just barely pass due to the curve if everyone just studies or just sleeps. While studying for the majority of the night then sleeping for a bit is better than the two extremes, it is not quite as good as studying then sleeping for a solid 8 hours. This is for similar reasons as to why it is also not effective to pull an all-nighter. The student will get a perfect score if they study for a bit then go to sleep (Sleep > Study).
This situation can be analyzed for Nash equilibria in a similar fashion to the situations covered in class and the textbook. In this game, the student has a strictly dominant strategy of (Sleep > Study), with every other strategy being strictly dominated. This is because if the student just sleeps, they miss out on studying. If the student just studies or does not get enough sleep, the article suggests that they might not actually remember all that much and not do as well on the exam. The rest of the class also has a strictly dominant strategy of (Sleep > Study) for the same reasons, leading to a single Nash equilibrium of (Sleep > Study, Sleep > Study) for a payoff of (10, 10).