Game theory in curved classes
In the textbook, the book briefly described a scenario where you are given the choice to prepare for a presentation with a partner, or to study for an exam. In short, this is the payoff matrix below:
|
Presentation |
Exam |
|
|
Presentation |
90, 90 |
86, 92 |
|
Exam |
92, 86 |
88, 88 |
Source: https://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch06.pdf (page 4)
I was interested in how game theory applies in the classroom, and I decided to use CS 4780 as a case study.
In 4780, the grading scheme was composed of 60% exams and 40% projects. Homework was assigned and can be done up to groups of 5, and are only graded on a complete/incomplete basis. Turning in incomplete homework results in a 5% penalty to your total grade. Due to the stress exams place on students, the professors recently announced a new grading scheme: Homework will now count for 10% of the class, so the split will be 10% homework, 50% exams and 40% projects. They can only be done in pairs. At the end of the semester, two different grades will be computed: one grade will be computed under the new grading scheme (50% exams, 40% projects, 10% homework), and the other will be computed under the old grading scheme (60% exams, 40% projects). The higher of the two options will become your final grade.
The interesting part is that CS4780 is curved, so students are competing with each other to get a good grade. Under this new grading scheme, we can treat doing homework as a game. For the sake of simplicity, we can consider the case where only 2 students are in the class. The two students have two strategies: to take advantage of the new grading scheme (which we denote A) and the old grading scheme (which we denote B). We will assume the following:
- Both players only care about getting the best grade possible
- Both players get full marks in the homework
- Both players get 95% on their homework if they decide to use the new grading scheme.
- If a player decides to do the homework, they get 80% on their exams. Otherwise, they get 70% on their exams.
So the payoff matrix is as follows:
|
A (new grading) |
B (old grading) |
|
|
A (new grading) |
0.895, 0.895 |
0.895, 0.82 |
|
B (old grading) |
0.82, 0.895 |
0.82, 0.82 |
Suppose that the class is curved to a B+, then the median would be the average of the two scores. Furthermore, we get an A- if we score 3 points above the median and a B if we score 3 points below the median, so that would translate to:
|
A (new grading) |
B (old grading) |
|
|
A (new grading) |
B+, B+ |
A-, B |
|
B (old grading) |
B, A- |
B+, B+ |
We see that the only Nash equilibrium is when both players decide to use the new grading system, or when they do the homework for correctness. Hence, students are motivated to do the homework in order to better prepare for the exams and mitigate the effects of a relatively poor performance on the exams. Although this model has made many strong assumptions, all of them are reasonable and justifiable. (There is actually a third grading scheme which involves doing paper reading assignments, so the grading scheme would be 10% paper reading, 10% homework, 45% homework and 35% projects. Due to simplicity, I decided not to include the assignments, but the same argument should follow).
What does this mean for students? For students who want to get a good grade in the class, they must now do the homework for correctness in order to better prepare for the exams. Since exam scores are now higher, making careless mistakes cost much more, so students have to be more careful and likely more stressed. Overall, getting an A in the class requires more effort from the students. This analysis has only been done from the perspective of students who only care about getting an A in the class, however. From a professor’s point of view, this may be good news for them, since students are making more time to understand and master the material, which is the ultimate goal of a class anyway.
From the analysis above, we can see that curving classes does seem to motivate students to better learn the material. However, this also puts more stress on students. To determine whether curving is truly beneficial to a student’s learning, more analysis still needs to be done. For example, doing group work together or peer review grading schemes have not been examined in this post.
Source: https://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch06.pdf, https://www.cs.cornell.edu/courses/cs4780/2021fa/