A potential best way to answer bonus question during test

Introduction

Recently, when I review my peers’ posts, I found an interesting post about the B07 term test which I also took this test. Since I cannot find my previous test paper, so I will reuse Minqi’s diagram in his post here. The details of the bonus question are listed below.

Analysis

From the above diagram, we can see option 3 is a very special one and not many people will choose this option(since you will receive a negative bonus). Also, we can assume no one will leave this question blank(since you won’t get any bonus in any case). Therefore, we suppose the majority of people will choose between option 1 and 2. We will analyze option 3 independently later and right now we suppose we only have 2 strategies: over 1/3 people choose option 2(strategy B) and less than 1/3 people choose option 2(strategy A). By following these assumptions, we can make a payoff table below where I as player 1, my strategies are choosing option 1(A) or option 2(B) and the rest people in the test as player 2, their strategies are strategy A and strategy B(listed above).

From the table we can see there is a Nash Equilibrium at (B, A) where I choose strategy B and the rest of the class choose strategy A. In this case, both players can maximize their payoff. But in reality, this is a typical Prisoner’s Dilemma and no one can contact to each other, so everybody will be player 1 in this case and choose the best strategy B. Therefore, there will be more than 1/3 people in the class choosing option 2 and no one can get a bonus mark.

But this doesn’t mean choosing option 2 is always a bad decision. Consider that there will be 2 ways on deciding our final grade.

1. The first one is the professor will make some adjustments to your final grade because the head of the program will ask the students’ grades in this course should follow the normal distribution(I know some courses at SG campus have this requirement). In this case, the professor will add or minus some marks to everyone’s final grade, therefore, if you can get a little higher score than your classmates, you will benefit from this way. So choosing strategy B will make you always get the same or higher grade on this question. But choosing strategy A won’t.
2. The second way is professor won’t do any adjustments to your final grade, you will get what you should get. In this case, choosing strategy A is a better option because you have a higher probability to increase your grade.

Finally, let’s analyze option 3. Normal distribution is an interesting phenomenon and it exists among people’s IQs. According to Diana Mindrila’s research, the graph below shows the normal distribution among people’s IQs.

Here, I make 3 layers of students to represent their decision.

1.  The first layer of students are the normal students, they are classified as 68% of people in the graph above. They will consider “if there is another student choosing this option, I will lose my mark so I won’t choose this option”
2.  The second layer of students are clever students, they are classified as the right 13.6% of people in the graph above. They will consider “since students won’t choose this option, I might be the only one who will choose this option and I won’t lose a mark if there are more than 1/3 students choosing option 2, so I choose option 3”
3.  The third layer of students are more clever students, they are classified as the right 2.1% of people in the graph above. They will consider “since students in the second layer will choose option 3, so I will lose a mark if I also did this, therefore, I won’t”

There might exist a fourth or fifth layer of students and the loop will continue. Considering that CSCB07 is a mandatory course for all cs students, there will be 100-200 students taking this course at the same time, so there will be more than 1 student in the second layer, hence, choosing option 3 will highly make you lose mark in this case.

Conclusion

By using game theory, we can make a payoff table of choosing different options and expect students’ decisions based on their layers.

Depending on the professor’s different grading methods, we have different strategies. If the professor will adjust your final grade, choosing option 2 seems to be a better strategy. Otherwise, choosing option 1 is a more stable way to increase your grade. But in any case, option 3 is not a good one so don’t choose this.

Resource

https://www.westga.edu/academics/research/vrc/assets/docs/the_normal_distribution_notes.pdf

Prisoner’s Dilemma and Bonus Marks on Midterm Test