I am sorry if the title is not clear enough. I came across this question yesterday but I don't know how to give a title to it. It would be nice if you can (i) give me some ideas of solving the question and (ii) direct me to the related field in mathematics, as I would like to know more about this topic.
Suppose there are $30$ students in a class. Each student receives a piece of paper and they are asked to write down a rational number between $0$ and $100$ inclusively without talking to others. The teacher then collects the papers. She would find the average of the $30$ numbers and multiply the average by $0.7$. Denote $x$ as the final product. Five students with their numbers closest to $x$ will receive a prize. If you are one of the students aiming for the prize, what number should you write down?
I believe that it is related to game theory, but I am not sure. Please kindly shed some lights for me. Thanks in advance.
This is a pretty well known "paradox", you can find pretty good description on Wikipedia. In your case the average is multiplied by $0.7$ instead of $\frac{2}{3}$, but it doesn't change the conclusion.
Basically, if all students were acting perfectly rationally they would all write $0$. The argument goes a little bit like this: choosing any number over $0.7$ (in your case) is irrational, as for sure the "target" will not be that high (even if everyone wrote 100). But if no-one will choose a number higher than $0.7$ then the target will not exceed $0.49$ (even if everyone wrote $0.7$). By that logic you can show that it is not rational to write any number over $0$. Of course in real world we don't observe such extreme reasoning, with most people
If you wish to investigate the matter further, here is a pretty good paper.